# Nonempty Observation List Descriptive Statistics Foundations This batch has passed Schema, deduplication, source-snapshot, and independent-answer recomputation checks; it is still not marked as human-reviewed. This question bank has corresponding bilingual step-by-step derivations. Source candidate and English ai_candidate states are retained. AI review, finite computation and Lean formal proof are labelled separately. batch-20261005-advance-20261006-r07-descriptive-statistics-foundations · 2026-10-05 · 100 questions This English edition is an AI candidate and does not yet have an OCME Atlas (GPT-6) Twin closure receipt. ## 1. Arithmetic Mean Retaining Multiplicities The nonempty observation list is D=[0]. Retain every repetition. Find the arithmetic mean. Give a reduced fraction with positive denominator, retaining /1 for integers. View answer and explanation0/1 Sum all 1 observations with their multiplicities and divide by that positive count, obtaining 0/1. Do not average a deduplicated set or infer a population probability law. [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r07-descriptive-statistics-0001/) Descriptive summaries of nonempty observation lists · qf-statistics-observation-mean · candidate ## 2. Arithmetic Mean Retaining Multiplicities The nonempty observation list is D=[-3, -3, -3]. Retain every repetition. Find the arithmetic mean. Give a reduced fraction with positive denominator, retaining /1 for integers. View answer and explanation-3/1 Sum all 3 observations with their multiplicities and divide by that positive count, obtaining -3/1. Do not average a deduplicated set or infer a population probability law. [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r07-descriptive-statistics-0002/) Descriptive summaries of nonempty observation lists · qf-statistics-observation-mean · candidate ## 3. Arithmetic Mean Retaining Multiplicities The nonempty observation list is D=[-4, 1, 8]. Retain every repetition. Find the arithmetic mean. Give a reduced fraction with positive denominator, retaining /1 for integers. View answer and explanation5/3 Sum all 3 observations with their multiplicities and divide by that positive count, obtaining 5/3. Do not average a deduplicated set or infer a population probability law. [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r07-descriptive-statistics-0003/) Descriptive summaries of nonempty observation lists · qf-statistics-observation-mean · candidate ## 4. Arithmetic Mean Retaining Multiplicities The nonempty observation list is D=[-2, -2, 5, 5]. Retain every repetition. Find the arithmetic mean. Give a reduced fraction with positive denominator, retaining /1 for integers. View answer and explanation3/2 Sum all 4 observations with their multiplicities and divide by that positive count, obtaining 3/2. Do not average a deduplicated set or infer a population probability law. [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r07-descriptive-statistics-0004/) Descriptive summaries of nonempty observation lists · qf-statistics-observation-mean · candidate ## 5. Arithmetic Mean Retaining Multiplicities The nonempty observation list is D=[-7, -7, 2, 8, 8]. Retain every repetition. Find the arithmetic mean. Give a reduced fraction with positive denominator, retaining /1 for integers. View answer and explanation4/5 Sum all 5 observations with their multiplicities and divide by that positive count, obtaining 4/5. Do not average a deduplicated set or infer a population probability law. [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r07-descriptive-statistics-0005/) Descriptive summaries of nonempty observation lists · qf-statistics-observation-mean · candidate ## 6. Arithmetic Mean Retaining Multiplicities The nonempty observation list is D=[8, 8, 1, -7, -7, -7]. Retain every repetition. Find the arithmetic mean. Give a reduced fraction with positive denominator, retaining /1 for integers. View answer and explanation-2/3 Sum all 6 observations with their multiplicities and divide by that positive count, obtaining -2/3. Do not average a deduplicated set or infer a population probability law. [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r07-descriptive-statistics-0006/) Descriptive summaries of nonempty observation lists · qf-statistics-observation-mean · candidate ## 7. Arithmetic Mean Retaining Multiplicities The nonempty observation list is D=[9, 9, 9, -1, -1, -8]. Retain every repetition. Find the arithmetic mean. Give a reduced fraction with positive denominator, retaining /1 for integers. View answer and explanation17/6 Sum all 6 observations with their multiplicities and divide by that positive count, obtaining 17/6. Do not average a deduplicated set or infer a population probability law. [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r07-descriptive-statistics-0007/) Descriptive summaries of nonempty observation lists · qf-statistics-observation-mean · candidate ## 8. Arithmetic Mean Retaining Multiplicities The nonempty observation list is D=[10, 0, 0, 0, -9, -9]. Retain every repetition. Find the arithmetic mean. Give a reduced fraction with positive denominator, retaining /1 for integers. View answer and explanation-4/3 Sum all 6 observations with their multiplicities and divide by that positive count, obtaining -4/3. Do not average a deduplicated set or infer a population probability law. [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r07-descriptive-statistics-0008/) Descriptive summaries of nonempty observation lists · qf-statistics-observation-mean · candidate ## 9. Arithmetic Mean Retaining Multiplicities The nonempty observation list is D=[11, 11, 1, -10, -10, -10]. Retain every repetition. Find the arithmetic mean. Give a reduced fraction with positive denominator, retaining /1 for integers. View answer and explanation-7/6 Sum all 6 observations with their multiplicities and divide by that positive count, obtaining -7/6. Do not average a deduplicated set or infer a population probability law. [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r07-descriptive-statistics-0009/) Descriptive summaries of nonempty observation lists · qf-statistics-observation-mean · candidate ## 10. Arithmetic Mean Retaining Multiplicities The nonempty observation list is D=[12, 12, 12, -1, -1, -11]. Retain every repetition. Find the arithmetic mean. Give a reduced fraction with positive denominator, retaining /1 for integers. View answer and explanation23/6 Sum all 6 observations with their multiplicities and divide by that positive count, obtaining 23/6. Do not average a deduplicated set or infer a population probability law. [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r07-descriptive-statistics-0010/) Descriptive summaries of nonempty observation lists · qf-statistics-observation-mean · candidate ## 11. Arithmetic Mean Retaining Multiplicities The nonempty observation list is D=[13, 0, 0, 0, -12, -12]. Retain every repetition. Find the arithmetic mean. Give a reduced fraction with positive denominator, retaining /1 for integers. View answer and explanation-11/6 Sum all 6 observations with their multiplicities and divide by that positive count, obtaining -11/6. Do not average a deduplicated set or infer a population probability law. [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r07-descriptive-statistics-0011/) Descriptive summaries of nonempty observation lists · qf-statistics-observation-mean · candidate ## 12. Arithmetic Mean Retaining Multiplicities The nonempty observation list is D=[14, 14, 1, -13, -13, -13]. Retain every repetition. Find the arithmetic mean. Give a reduced fraction with positive denominator, retaining /1 for integers. View answer and explanation-5/3 Sum all 6 observations with their multiplicities and divide by that positive count, obtaining -5/3. Do not average a deduplicated set or infer a population probability law. [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r07-descriptive-statistics-0012/) Descriptive summaries of nonempty observation lists · qf-statistics-observation-mean · candidate ## 13. Arithmetic Mean Retaining Multiplicities The nonempty observation list is D=[15, 15, 15, -1, -1, -14]. Retain every repetition. Find the arithmetic mean. Give a reduced fraction with positive denominator, retaining /1 for integers. View answer and explanation29/6 Sum all 6 observations with their multiplicities and divide by that positive count, obtaining 29/6. Do not average a deduplicated set or infer a population probability law. [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r07-descriptive-statistics-0013/) Descriptive summaries of nonempty observation lists · qf-statistics-observation-mean · candidate ## 14. Arithmetic Mean Retaining Multiplicities The nonempty observation list is D=[16, 0, 0, 0, -15, -15]. Retain every repetition. Find the arithmetic mean. Give a reduced fraction with positive denominator, retaining /1 for integers. View answer and explanation-7/3 Sum all 6 observations with their multiplicities and divide by that positive count, obtaining -7/3. Do not average a deduplicated set or infer a population probability law. [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r07-descriptive-statistics-0014/) Descriptive summaries of nonempty observation lists · qf-statistics-observation-mean · candidate ## 15. Arithmetic Mean Retaining Multiplicities The nonempty observation list is D=[17, 17, 1, -16, -16, -16]. Retain every repetition. Find the arithmetic mean. Give a reduced fraction with positive denominator, retaining /1 for integers. View answer and explanation-13/6 Sum all 6 observations with their multiplicities and divide by that positive count, obtaining -13/6. Do not average a deduplicated set or infer a population probability law. [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r07-descriptive-statistics-0015/) Descriptive summaries of nonempty observation lists · qf-statistics-observation-mean · candidate ## 16. Arithmetic Mean Retaining Multiplicities The nonempty observation list is D=[18, 18, 18, -1, -1, -17]. Retain every repetition. Find the arithmetic mean. Give a reduced fraction with positive denominator, retaining /1 for integers. View answer and explanation35/6 Sum all 6 observations with their multiplicities and divide by that positive count, obtaining 35/6. Do not average a deduplicated set or infer a population probability law. [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r07-descriptive-statistics-0016/) Descriptive summaries of nonempty observation lists · qf-statistics-observation-mean · candidate ## 17. Arithmetic Mean Retaining Multiplicities The nonempty observation list is D=[19, 0, 0, 0, -18, -18]. Retain every repetition. Find the arithmetic mean. Give a reduced fraction with positive denominator, retaining /1 for integers. View answer and explanation-17/6 Sum all 6 observations with their multiplicities and divide by that positive count, obtaining -17/6. Do not average a deduplicated set or infer a population probability law. [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r07-descriptive-statistics-0017/) Descriptive summaries of nonempty observation lists · qf-statistics-observation-mean · candidate ## 18. Arithmetic Mean Retaining Multiplicities The nonempty observation list is D=[20, 20, 1, -19, -19, -19]. Retain every repetition. Find the arithmetic mean. Give a reduced fraction with positive denominator, retaining /1 for integers. View answer and explanation-8/3 Sum all 6 observations with their multiplicities and divide by that positive count, obtaining -8/3. Do not average a deduplicated set or infer a population probability law. [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r07-descriptive-statistics-0018/) Descriptive summaries of nonempty observation lists · qf-statistics-observation-mean · candidate ## 19. Arithmetic Mean Retaining Multiplicities The nonempty observation list is D=[21, 21, 21, -1, -1, -20]. Retain every repetition. Find the arithmetic mean. Give a reduced fraction with positive denominator, retaining /1 for integers. View answer and explanation41/6 Sum all 6 observations with their multiplicities and divide by that positive count, obtaining 41/6. Do not average a deduplicated set or infer a population probability law. [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r07-descriptive-statistics-0019/) Descriptive summaries of nonempty observation lists · qf-statistics-observation-mean · candidate ## 20. Arithmetic Mean Retaining Multiplicities The nonempty observation list is D=[22, 0, 0, 0, -21, -21]. Retain every repetition. Find the arithmetic mean. Give a reduced fraction with positive denominator, retaining /1 for integers. View answer and explanation-10/3 Sum all 6 observations with their multiplicities and divide by that positive count, obtaining -10/3. Do not average a deduplicated set or infer a population probability law. [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r07-descriptive-statistics-0020/) Descriptive summaries of nonempty observation lists · qf-statistics-observation-mean · candidate ## 21. Arithmetic Mean Retaining Multiplicities The nonempty observation list is D=[23, 23, 1, -22, -22, -22]. Retain every repetition. Find the arithmetic mean. Give a reduced fraction with positive denominator, retaining /1 for integers. View answer and explanation-19/6 Sum all 6 observations with their multiplicities and divide by that positive count, obtaining -19/6. Do not average a deduplicated set or infer a population probability law. [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r07-descriptive-statistics-0021/) Descriptive summaries of nonempty observation lists · qf-statistics-observation-mean · candidate ## 22. Arithmetic Mean Retaining Multiplicities The nonempty observation list is D=[24, 24, 24, -1, -1, -23]. Retain every repetition. Find the arithmetic mean. Give a reduced fraction with positive denominator, retaining /1 for integers. View answer and explanation47/6 Sum all 6 observations with their multiplicities and divide by that positive count, obtaining 47/6. Do not average a deduplicated set or infer a population probability law. [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r07-descriptive-statistics-0022/) Descriptive summaries of nonempty observation lists · qf-statistics-observation-mean · candidate ## 23. Arithmetic Mean Retaining Multiplicities The nonempty observation list is D=[25, 0, 0, 0, -24, -24]. Retain every repetition. Find the arithmetic mean. Give a reduced fraction with positive denominator, retaining /1 for integers. View answer and explanation-23/6 Sum all 6 observations with their multiplicities and divide by that positive count, obtaining -23/6. Do not average a deduplicated set or infer a population probability law. [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r07-descriptive-statistics-0023/) Descriptive summaries of nonempty observation lists · qf-statistics-observation-mean · candidate ## 24. Arithmetic Mean Retaining Multiplicities The nonempty observation list is D=[26, 26, 1, -25, -25, -25]. Retain every repetition. Find the arithmetic mean. Give a reduced fraction with positive denominator, retaining /1 for integers. View answer and explanation-11/3 Sum all 6 observations with their multiplicities and divide by that positive count, obtaining -11/3. Do not average a deduplicated set or infer a population probability law. [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r07-descriptive-statistics-0024/) Descriptive summaries of nonempty observation lists · qf-statistics-observation-mean · candidate ## 25. Arithmetic Mean Retaining Multiplicities The nonempty observation list is D=[27, 27, 27, -1, -1, -26]. Retain every repetition. Find the arithmetic mean. Give a reduced fraction with positive denominator, retaining /1 for integers. View answer and explanation53/6 Sum all 6 observations with their multiplicities and divide by that positive count, obtaining 53/6. Do not average a deduplicated set or infer a population probability law. [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r07-descriptive-statistics-0025/) Descriptive summaries of nonempty observation lists · qf-statistics-observation-mean · candidate ## 26. Exact Median of an Odd or Even Observation List The nonempty observation list is D=[0]. Retain every repetition. Find the median after sorting: one middle value for odd length, or the average of the two middle values for even length. Give a reduced fraction with positive denominator, retaining /1 for integers. View answer and explanation0/1 The sorted list retaining repetitions is [0]. Apply the stated odd/even convention to obtain 0/1; the even-length median need not be an observed value. [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r07-descriptive-statistics-0026/) Descriptive summaries of nonempty observation lists · qf-statistics-observation-median · candidate ## 27. Exact Median of an Odd or Even Observation List The nonempty observation list is D=[-3, -3, -3]. Retain every repetition. Find the median after sorting: one middle value for odd length, or the average of the two middle values for even length. Give a reduced fraction with positive denominator, retaining /1 for integers. View answer and explanation-3/1 The sorted list retaining repetitions is [-3, -3, -3]. Apply the stated odd/even convention to obtain -3/1; the even-length median need not be an observed value. [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r07-descriptive-statistics-0027/) Descriptive summaries of nonempty observation lists · qf-statistics-observation-median · candidate ## 28. Exact Median of an Odd or Even Observation List The nonempty observation list is D=[-4, 1, 8]. Retain every repetition. Find the median after sorting: one middle value for odd length, or the average of the two middle values for even length. Give a reduced fraction with positive denominator, retaining /1 for integers. View answer and explanation1/1 The sorted list retaining repetitions is [-4, 1, 8]. Apply the stated odd/even convention to obtain 1/1; the even-length median need not be an observed value. [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r07-descriptive-statistics-0028/) Descriptive summaries of nonempty observation lists · qf-statistics-observation-median · candidate ## 29. Exact Median of an Odd or Even Observation List The nonempty observation list is D=[-2, -2, 5, 5]. Retain every repetition. Find the median after sorting: one middle value for odd length, or the average of the two middle values for even length. Give a reduced fraction with positive denominator, retaining /1 for integers. View answer and explanation3/2 The sorted list retaining repetitions is [-2, -2, 5, 5]. Apply the stated odd/even convention to obtain 3/2; the even-length median need not be an observed value. [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r07-descriptive-statistics-0029/) Descriptive summaries of nonempty observation lists · qf-statistics-observation-median · candidate ## 30. Exact Median of an Odd or Even Observation List The nonempty observation list is D=[-7, -7, 2, 8, 8]. Retain every repetition. Find the median after sorting: one middle value for odd length, or the average of the two middle values for even length. Give a reduced fraction with positive denominator, retaining /1 for integers. View answer and explanation2/1 The sorted list retaining repetitions is [-7, -7, 2, 8, 8]. Apply the stated odd/even convention to obtain 2/1; the even-length median need not be an observed value. [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r07-descriptive-statistics-0030/) Descriptive summaries of nonempty observation lists · qf-statistics-observation-median · candidate ## 31. Exact Median of an Odd or Even Observation List The nonempty observation list is D=[8, 8, 1, -7, -7, -7]. Retain every repetition. Find the median after sorting: one middle value for odd length, or the average of the two middle values for even length. Give a reduced fraction with positive denominator, retaining /1 for integers. View answer and explanation-3/1 The sorted list retaining repetitions is [-7, -7, -7, 1, 8, 8]. Apply the stated odd/even convention to obtain -3/1; the even-length median need not be an observed value. [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r07-descriptive-statistics-0031/) Descriptive summaries of nonempty observation lists · qf-statistics-observation-median · candidate ## 32. Exact Median of an Odd or Even Observation List The nonempty observation list is D=[9, 9, 9, -1, -1, -8]. Retain every repetition. Find the median after sorting: one middle value for odd length, or the average of the two middle values for even length. Give a reduced fraction with positive denominator, retaining /1 for integers. View answer and explanation4/1 The sorted list retaining repetitions is [-8, -1, -1, 9, 9, 9]. Apply the stated odd/even convention to obtain 4/1; the even-length median need not be an observed value. [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r07-descriptive-statistics-0032/) Descriptive summaries of nonempty observation lists · qf-statistics-observation-median · candidate ## 33. Exact Median of an Odd or Even Observation List The nonempty observation list is D=[10, 0, 0, 0, -9, -9]. Retain every repetition. Find the median after sorting: one middle value for odd length, or the average of the two middle values for even length. Give a reduced fraction with positive denominator, retaining /1 for integers. View answer and explanation0/1 The sorted list retaining repetitions is [-9, -9, 0, 0, 0, 10]. Apply the stated odd/even convention to obtain 0/1; the even-length median need not be an observed value. [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r07-descriptive-statistics-0033/) Descriptive summaries of nonempty observation lists · qf-statistics-observation-median · candidate ## 34. Exact Median of an Odd or Even Observation List The nonempty observation list is D=[11, 11, 1, -10, -10, -10]. Retain every repetition. Find the median after sorting: one middle value for odd length, or the average of the two middle values for even length. Give a reduced fraction with positive denominator, retaining /1 for integers. View answer and explanation-9/2 The sorted list retaining repetitions is [-10, -10, -10, 1, 11, 11]. Apply the stated odd/even convention to obtain -9/2; the even-length median need not be an observed value. [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r07-descriptive-statistics-0034/) Descriptive summaries of nonempty observation lists · qf-statistics-observation-median · candidate ## 35. Exact Median of an Odd or Even Observation List The nonempty observation list is D=[12, 12, 12, -1, -1, -11]. Retain every repetition. Find the median after sorting: one middle value for odd length, or the average of the two middle values for even length. Give a reduced fraction with positive denominator, retaining /1 for integers. View answer and explanation11/2 The sorted list retaining repetitions is [-11, -1, -1, 12, 12, 12]. Apply the stated odd/even convention to obtain 11/2; the even-length median need not be an observed value. [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r07-descriptive-statistics-0035/) Descriptive summaries of nonempty observation lists · qf-statistics-observation-median · candidate ## 36. Exact Median of an Odd or Even Observation List The nonempty observation list is D=[13, 0, 0, 0, -12, -12]. Retain every repetition. Find the median after sorting: one middle value for odd length, or the average of the two middle values for even length. Give a reduced fraction with positive denominator, retaining /1 for integers. View answer and explanation0/1 The sorted list retaining repetitions is [-12, -12, 0, 0, 0, 13]. Apply the stated odd/even convention to obtain 0/1; the even-length median need not be an observed value. [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r07-descriptive-statistics-0036/) Descriptive summaries of nonempty observation lists · qf-statistics-observation-median · candidate ## 37. Exact Median of an Odd or Even Observation List The nonempty observation list is D=[14, 14, 1, -13, -13, -13]. Retain every repetition. Find the median after sorting: one middle value for odd length, or the average of the two middle values for even length. Give a reduced fraction with positive denominator, retaining /1 for integers. View answer and explanation-6/1 The sorted list retaining repetitions is [-13, -13, -13, 1, 14, 14]. Apply the stated odd/even convention to obtain -6/1; the even-length median need not be an observed value. [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r07-descriptive-statistics-0037/) Descriptive summaries of nonempty observation lists · qf-statistics-observation-median · candidate ## 38. Exact Median of an Odd or Even Observation List The nonempty observation list is D=[15, 15, 15, -1, -1, -14]. Retain every repetition. Find the median after sorting: one middle value for odd length, or the average of the two middle values for even length. Give a reduced fraction with positive denominator, retaining /1 for integers. View answer and explanation7/1 The sorted list retaining repetitions is [-14, -1, -1, 15, 15, 15]. Apply the stated odd/even convention to obtain 7/1; the even-length median need not be an observed value. [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r07-descriptive-statistics-0038/) Descriptive summaries of nonempty observation lists · qf-statistics-observation-median · candidate ## 39. Exact Median of an Odd or Even Observation List The nonempty observation list is D=[16, 0, 0, 0, -15, -15]. Retain every repetition. Find the median after sorting: one middle value for odd length, or the average of the two middle values for even length. Give a reduced fraction with positive denominator, retaining /1 for integers. View answer and explanation0/1 The sorted list retaining repetitions is [-15, -15, 0, 0, 0, 16]. Apply the stated odd/even convention to obtain 0/1; the even-length median need not be an observed value. [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r07-descriptive-statistics-0039/) Descriptive summaries of nonempty observation lists · qf-statistics-observation-median · candidate ## 40. Exact Median of an Odd or Even Observation List The nonempty observation list is D=[17, 17, 1, -16, -16, -16]. Retain every repetition. Find the median after sorting: one middle value for odd length, or the average of the two middle values for even length. Give a reduced fraction with positive denominator, retaining /1 for integers. View answer and explanation-15/2 The sorted list retaining repetitions is [-16, -16, -16, 1, 17, 17]. Apply the stated odd/even convention to obtain -15/2; the even-length median need not be an observed value. [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r07-descriptive-statistics-0040/) Descriptive summaries of nonempty observation lists · qf-statistics-observation-median · candidate ## 41. Exact Median of an Odd or Even Observation List The nonempty observation list is D=[18, 18, 18, -1, -1, -17]. Retain every repetition. Find the median after sorting: one middle value for odd length, or the average of the two middle values for even length. Give a reduced fraction with positive denominator, retaining /1 for integers. View answer and explanation17/2 The sorted list retaining repetitions is [-17, -1, -1, 18, 18, 18]. Apply the stated odd/even convention to obtain 17/2; the even-length median need not be an observed value. [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r07-descriptive-statistics-0041/) Descriptive summaries of nonempty observation lists · qf-statistics-observation-median · candidate ## 42. Exact Median of an Odd or Even Observation List The nonempty observation list is D=[19, 0, 0, 0, -18, -18]. Retain every repetition. Find the median after sorting: one middle value for odd length, or the average of the two middle values for even length. Give a reduced fraction with positive denominator, retaining /1 for integers. View answer and explanation0/1 The sorted list retaining repetitions is [-18, -18, 0, 0, 0, 19]. Apply the stated odd/even convention to obtain 0/1; the even-length median need not be an observed value. [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r07-descriptive-statistics-0042/) Descriptive summaries of nonempty observation lists · qf-statistics-observation-median · candidate ## 43. Exact Median of an Odd or Even Observation List The nonempty observation list is D=[20, 20, 1, -19, -19, -19]. Retain every repetition. Find the median after sorting: one middle value for odd length, or the average of the two middle values for even length. Give a reduced fraction with positive denominator, retaining /1 for integers. View answer and explanation-9/1 The sorted list retaining repetitions is [-19, -19, -19, 1, 20, 20]. Apply the stated odd/even convention to obtain -9/1; the even-length median need not be an observed value. [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r07-descriptive-statistics-0043/) Descriptive summaries of nonempty observation lists · qf-statistics-observation-median · candidate ## 44. Exact Median of an Odd or Even Observation List The nonempty observation list is D=[21, 21, 21, -1, -1, -20]. Retain every repetition. Find the median after sorting: one middle value for odd length, or the average of the two middle values for even length. Give a reduced fraction with positive denominator, retaining /1 for integers. View answer and explanation10/1 The sorted list retaining repetitions is [-20, -1, -1, 21, 21, 21]. Apply the stated odd/even convention to obtain 10/1; the even-length median need not be an observed value. [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r07-descriptive-statistics-0044/) Descriptive summaries of nonempty observation lists · qf-statistics-observation-median · candidate ## 45. Exact Median of an Odd or Even Observation List The nonempty observation list is D=[22, 0, 0, 0, -21, -21]. Retain every repetition. Find the median after sorting: one middle value for odd length, or the average of the two middle values for even length. Give a reduced fraction with positive denominator, retaining /1 for integers. View answer and explanation0/1 The sorted list retaining repetitions is [-21, -21, 0, 0, 0, 22]. Apply the stated odd/even convention to obtain 0/1; the even-length median need not be an observed value. [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r07-descriptive-statistics-0045/) Descriptive summaries of nonempty observation lists · qf-statistics-observation-median · candidate ## 46. Exact Median of an Odd or Even Observation List The nonempty observation list is D=[23, 23, 1, -22, -22, -22]. Retain every repetition. Find the median after sorting: one middle value for odd length, or the average of the two middle values for even length. Give a reduced fraction with positive denominator, retaining /1 for integers. View answer and explanation-21/2 The sorted list retaining repetitions is [-22, -22, -22, 1, 23, 23]. Apply the stated odd/even convention to obtain -21/2; the even-length median need not be an observed value. [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r07-descriptive-statistics-0046/) Descriptive summaries of nonempty observation lists · qf-statistics-observation-median · candidate ## 47. Exact Median of an Odd or Even Observation List The nonempty observation list is D=[24, 24, 24, -1, -1, -23]. Retain every repetition. Find the median after sorting: one middle value for odd length, or the average of the two middle values for even length. Give a reduced fraction with positive denominator, retaining /1 for integers. View answer and explanation23/2 The sorted list retaining repetitions is [-23, -1, -1, 24, 24, 24]. Apply the stated odd/even convention to obtain 23/2; the even-length median need not be an observed value. [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r07-descriptive-statistics-0047/) Descriptive summaries of nonempty observation lists · qf-statistics-observation-median · candidate ## 48. Exact Median of an Odd or Even Observation List The nonempty observation list is D=[25, 0, 0, 0, -24, -24]. Retain every repetition. Find the median after sorting: one middle value for odd length, or the average of the two middle values for even length. Give a reduced fraction with positive denominator, retaining /1 for integers. View answer and explanation0/1 The sorted list retaining repetitions is [-24, -24, 0, 0, 0, 25]. Apply the stated odd/even convention to obtain 0/1; the even-length median need not be an observed value. [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r07-descriptive-statistics-0048/) Descriptive summaries of nonempty observation lists · qf-statistics-observation-median · candidate ## 49. Exact Median of an Odd or Even Observation List The nonempty observation list is D=[26, 26, 1, -25, -25, -25]. Retain every repetition. Find the median after sorting: one middle value for odd length, or the average of the two middle values for even length. Give a reduced fraction with positive denominator, retaining /1 for integers. View answer and explanation-12/1 The sorted list retaining repetitions is [-25, -25, -25, 1, 26, 26]. Apply the stated odd/even convention to obtain -12/1; the even-length median need not be an observed value. [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r07-descriptive-statistics-0049/) Descriptive summaries of nonempty observation lists · qf-statistics-observation-median · candidate ## 50. Exact Median of an Odd or Even Observation List The nonempty observation list is D=[27, 27, 27, -1, -1, -26]. Retain every repetition. Find the median after sorting: one middle value for odd length, or the average of the two middle values for even length. Give a reduced fraction with positive denominator, retaining /1 for integers. View answer and explanation13/1 The sorted list retaining repetitions is [-26, -1, -1, 27, 27, 27]. Apply the stated odd/even convention to obtain 13/1; the even-length median need not be an observed value. [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r07-descriptive-statistics-0050/) Descriptive summaries of nonempty observation lists · qf-statistics-observation-median · candidate ## 51. Range of an Observation List The nonempty observation list is D=[0]. Retain every repetition. Find the range, defined as maximum minus minimum. View answer and explanation0 Subtract the least observation from the greatest to obtain 0. A singleton or constant list has range zero; this is neither variance nor a count of distinct values. [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r07-descriptive-statistics-0051/) Descriptive summaries of nonempty observation lists · qf-statistics-observation-range · candidate ## 52. Range of an Observation List The nonempty observation list is D=[-3, -3, -3]. Retain every repetition. Find the range, defined as maximum minus minimum. View answer and explanation0 Subtract the least observation from the greatest to obtain 0. A singleton or constant list has range zero; this is neither variance nor a count of distinct values. [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r07-descriptive-statistics-0052/) Descriptive summaries of nonempty observation lists · qf-statistics-observation-range · candidate ## 53. Range of an Observation List The nonempty observation list is D=[-4, 1, 8]. Retain every repetition. Find the range, defined as maximum minus minimum. View answer and explanation12 Subtract the least observation from the greatest to obtain 12. A singleton or constant list has range zero; this is neither variance nor a count of distinct values. [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r07-descriptive-statistics-0053/) Descriptive summaries of nonempty observation lists · qf-statistics-observation-range · candidate ## 54. Range of an Observation List The nonempty observation list is D=[-2, -2, 5, 5]. Retain every repetition. Find the range, defined as maximum minus minimum. View answer and explanation7 Subtract the least observation from the greatest to obtain 7. A singleton or constant list has range zero; this is neither variance nor a count of distinct values. [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r07-descriptive-statistics-0054/) Descriptive summaries of nonempty observation lists · qf-statistics-observation-range · candidate ## 55. Range of an Observation List The nonempty observation list is D=[-7, -7, 2, 8, 8]. Retain every repetition. Find the range, defined as maximum minus minimum. View answer and explanation15 Subtract the least observation from the greatest to obtain 15. A singleton or constant list has range zero; this is neither variance nor a count of distinct values. [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r07-descriptive-statistics-0055/) Descriptive summaries of nonempty observation lists · qf-statistics-observation-range · candidate ## 56. Range of an Observation List The nonempty observation list is D=[9, 9, 9, -1, -1, -8]. Retain every repetition. Find the range, defined as maximum minus minimum. View answer and explanation17 Subtract the least observation from the greatest to obtain 17. A singleton or constant list has range zero; this is neither variance nor a count of distinct values. [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r07-descriptive-statistics-0056/) Descriptive summaries of nonempty observation lists · qf-statistics-observation-range · candidate ## 57. Range of an Observation List The nonempty observation list is D=[10, 0, 0, 0, -9, -9]. Retain every repetition. Find the range, defined as maximum minus minimum. View answer and explanation19 Subtract the least observation from the greatest to obtain 19. A singleton or constant list has range zero; this is neither variance nor a count of distinct values. [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r07-descriptive-statistics-0057/) Descriptive summaries of nonempty observation lists · qf-statistics-observation-range · candidate ## 58. Range of an Observation List The nonempty observation list is D=[11, 11, 1, -10, -10, -10]. Retain every repetition. Find the range, defined as maximum minus minimum. View answer and explanation21 Subtract the least observation from the greatest to obtain 21. A singleton or constant list has range zero; this is neither variance nor a count of distinct values. [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r07-descriptive-statistics-0058/) Descriptive summaries of nonempty observation lists · qf-statistics-observation-range · candidate ## 59. Range of an Observation List The nonempty observation list is D=[12, 12, 12, -1, -1, -11]. Retain every repetition. Find the range, defined as maximum minus minimum. View answer and explanation23 Subtract the least observation from the greatest to obtain 23. A singleton or constant list has range zero; this is neither variance nor a count of distinct values. [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r07-descriptive-statistics-0059/) Descriptive summaries of nonempty observation lists · qf-statistics-observation-range · candidate ## 60. Range of an Observation List The nonempty observation list is D=[13, 0, 0, 0, -12, -12]. Retain every repetition. Find the range, defined as maximum minus minimum. View answer and explanation25 Subtract the least observation from the greatest to obtain 25. A singleton or constant list has range zero; this is neither variance nor a count of distinct values. [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r07-descriptive-statistics-0060/) Descriptive summaries of nonempty observation lists · qf-statistics-observation-range · candidate ## 61. Range of an Observation List The nonempty observation list is D=[14, 14, 1, -13, -13, -13]. Retain every repetition. Find the range, defined as maximum minus minimum. View answer and explanation27 Subtract the least observation from the greatest to obtain 27. A singleton or constant list has range zero; this is neither variance nor a count of distinct values. [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r07-descriptive-statistics-0061/) Descriptive summaries of nonempty observation lists · qf-statistics-observation-range · candidate ## 62. Range of an Observation List The nonempty observation list is D=[15, 15, 15, -1, -1, -14]. Retain every repetition. Find the range, defined as maximum minus minimum. View answer and explanation29 Subtract the least observation from the greatest to obtain 29. A singleton or constant list has range zero; this is neither variance nor a count of distinct values. [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r07-descriptive-statistics-0062/) Descriptive summaries of nonempty observation lists · qf-statistics-observation-range · candidate ## 63. Range of an Observation List The nonempty observation list is D=[16, 0, 0, 0, -15, -15]. Retain every repetition. Find the range, defined as maximum minus minimum. View answer and explanation31 Subtract the least observation from the greatest to obtain 31. A singleton or constant list has range zero; this is neither variance nor a count of distinct values. [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r07-descriptive-statistics-0063/) Descriptive summaries of nonempty observation lists · qf-statistics-observation-range · candidate ## 64. Range of an Observation List The nonempty observation list is D=[17, 17, 1, -16, -16, -16]. Retain every repetition. Find the range, defined as maximum minus minimum. View answer and explanation33 Subtract the least observation from the greatest to obtain 33. A singleton or constant list has range zero; this is neither variance nor a count of distinct values. [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r07-descriptive-statistics-0064/) Descriptive summaries of nonempty observation lists · qf-statistics-observation-range · candidate ## 65. Range of an Observation List The nonempty observation list is D=[18, 18, 18, -1, -1, -17]. Retain every repetition. Find the range, defined as maximum minus minimum. View answer and explanation35 Subtract the least observation from the greatest to obtain 35. A singleton or constant list has range zero; this is neither variance nor a count of distinct values. [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r07-descriptive-statistics-0065/) Descriptive summaries of nonempty observation lists · qf-statistics-observation-range · candidate ## 66. Range of an Observation List The nonempty observation list is D=[19, 0, 0, 0, -18, -18]. Retain every repetition. Find the range, defined as maximum minus minimum. View answer and explanation37 Subtract the least observation from the greatest to obtain 37. A singleton or constant list has range zero; this is neither variance nor a count of distinct values. [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r07-descriptive-statistics-0066/) Descriptive summaries of nonempty observation lists · qf-statistics-observation-range · candidate ## 67. Range of an Observation List The nonempty observation list is D=[20, 20, 1, -19, -19, -19]. Retain every repetition. Find the range, defined as maximum minus minimum. View answer and explanation39 Subtract the least observation from the greatest to obtain 39. A singleton or constant list has range zero; this is neither variance nor a count of distinct values. [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r07-descriptive-statistics-0067/) Descriptive summaries of nonempty observation lists · qf-statistics-observation-range · candidate ## 68. Range of an Observation List The nonempty observation list is D=[21, 21, 21, -1, -1, -20]. Retain every repetition. Find the range, defined as maximum minus minimum. View answer and explanation41 Subtract the least observation from the greatest to obtain 41. A singleton or constant list has range zero; this is neither variance nor a count of distinct values. [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r07-descriptive-statistics-0068/) Descriptive summaries of nonempty observation lists · qf-statistics-observation-range · candidate ## 69. Range of an Observation List The nonempty observation list is D=[22, 0, 0, 0, -21, -21]. Retain every repetition. Find the range, defined as maximum minus minimum. View answer and explanation43 Subtract the least observation from the greatest to obtain 43. A singleton or constant list has range zero; this is neither variance nor a count of distinct values. [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r07-descriptive-statistics-0069/) Descriptive summaries of nonempty observation lists · qf-statistics-observation-range · candidate ## 70. Range of an Observation List The nonempty observation list is D=[23, 23, 1, -22, -22, -22]. Retain every repetition. Find the range, defined as maximum minus minimum. View answer and explanation45 Subtract the least observation from the greatest to obtain 45. A singleton or constant list has range zero; this is neither variance nor a count of distinct values. [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r07-descriptive-statistics-0070/) Descriptive summaries of nonempty observation lists · qf-statistics-observation-range · candidate ## 71. Range of an Observation List The nonempty observation list is D=[24, 24, 24, -1, -1, -23]. Retain every repetition. Find the range, defined as maximum minus minimum. View answer and explanation47 Subtract the least observation from the greatest to obtain 47. A singleton or constant list has range zero; this is neither variance nor a count of distinct values. [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r07-descriptive-statistics-0071/) Descriptive summaries of nonempty observation lists · qf-statistics-observation-range · candidate ## 72. Range of an Observation List The nonempty observation list is D=[25, 0, 0, 0, -24, -24]. Retain every repetition. Find the range, defined as maximum minus minimum. View answer and explanation49 Subtract the least observation from the greatest to obtain 49. A singleton or constant list has range zero; this is neither variance nor a count of distinct values. [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r07-descriptive-statistics-0072/) Descriptive summaries of nonempty observation lists · qf-statistics-observation-range · candidate ## 73. Range of an Observation List The nonempty observation list is D=[26, 26, 1, -25, -25, -25]. Retain every repetition. Find the range, defined as maximum minus minimum. View answer and explanation51 Subtract the least observation from the greatest to obtain 51. A singleton or constant list has range zero; this is neither variance nor a count of distinct values. [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r07-descriptive-statistics-0073/) Descriptive summaries of nonempty observation lists · qf-statistics-observation-range · candidate ## 74. Range of an Observation List The nonempty observation list is D=[27, 27, 27, -1, -1, -26]. Retain every repetition. Find the range, defined as maximum minus minimum. View answer and explanation53 Subtract the least observation from the greatest to obtain 53. A singleton or constant list has range zero; this is neither variance nor a count of distinct values. [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r07-descriptive-statistics-0074/) Descriptive summaries of nonempty observation lists · qf-statistics-observation-range · candidate ## 75. Range of an Observation List The nonempty observation list is D=[28, 0, 0, 0, -27, -27]. Retain every repetition. Find the range, defined as maximum minus minimum. View answer and explanation55 Subtract the least observation from the greatest to obtain 55. A singleton or constant list has range zero; this is neither variance nor a count of distinct values. [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r07-descriptive-statistics-0075/) Descriptive summaries of nonempty observation lists · qf-statistics-observation-range · candidate ## 76. Set of All Maximum-Frequency Observed Values The nonempty observation list is D=[0]. Retain every repetition. Under the explicit all-frequency-maximizers convention, give every value with maximum frequency as an increasing set, including all values if every frequency is one. View answer and explanation{0} Count frequencies with all input repetitions retained, then list each maximizing value once. The result is {0}; ties are not broken by choosing just one value, and this stated convention is distinct from textbooks calling an all-distinct list no mode. [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r07-descriptive-statistics-0076/) Descriptive summaries of nonempty observation lists · qf-statistics-maximal-frequency-set · candidate ## 77. Set of All Maximum-Frequency Observed Values The nonempty observation list is D=[-3, -3, -3]. Retain every repetition. Under the explicit all-frequency-maximizers convention, give every value with maximum frequency as an increasing set, including all values if every frequency is one. View answer and explanation{-3} Count frequencies with all input repetitions retained, then list each maximizing value once. The result is {-3}; ties are not broken by choosing just one value, and this stated convention is distinct from textbooks calling an all-distinct list no mode. [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r07-descriptive-statistics-0077/) Descriptive summaries of nonempty observation lists · qf-statistics-maximal-frequency-set · candidate ## 78. Set of All Maximum-Frequency Observed Values The nonempty observation list is D=[-4, 1, 8]. Retain every repetition. Under the explicit all-frequency-maximizers convention, give every value with maximum frequency as an increasing set, including all values if every frequency is one. View answer and explanation{-4, 1, 8} Count frequencies with all input repetitions retained, then list each maximizing value once. The result is {-4, 1, 8}; ties are not broken by choosing just one value, and this stated convention is distinct from textbooks calling an all-distinct list no mode. [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r07-descriptive-statistics-0078/) Descriptive summaries of nonempty observation lists · qf-statistics-maximal-frequency-set · candidate ## 79. Set of All Maximum-Frequency Observed Values The nonempty observation list is D=[-2, -2, 5, 5]. Retain every repetition. Under the explicit all-frequency-maximizers convention, give every value with maximum frequency as an increasing set, including all values if every frequency is one. View answer and explanation{-2, 5} Count frequencies with all input repetitions retained, then list each maximizing value once. The result is {-2, 5}; ties are not broken by choosing just one value, and this stated convention is distinct from textbooks calling an all-distinct list no mode. [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r07-descriptive-statistics-0079/) Descriptive summaries of nonempty observation lists · qf-statistics-maximal-frequency-set · candidate ## 80. Set of All Maximum-Frequency Observed Values The nonempty observation list is D=[-7, -7, 2, 8, 8]. Retain every repetition. Under the explicit all-frequency-maximizers convention, give every value with maximum frequency as an increasing set, including all values if every frequency is one. View answer and explanation{-7, 8} Count frequencies with all input repetitions retained, then list each maximizing value once. The result is {-7, 8}; ties are not broken by choosing just one value, and this stated convention is distinct from textbooks calling an all-distinct list no mode. [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r07-descriptive-statistics-0080/) Descriptive summaries of nonempty observation lists · qf-statistics-maximal-frequency-set · candidate ## 81. Set of All Maximum-Frequency Observed Values The nonempty observation list is D=[8, 8, 1, -7, -7, -7]. Retain every repetition. Under the explicit all-frequency-maximizers convention, give every value with maximum frequency as an increasing set, including all values if every frequency is one. View answer and explanation{-7} Count frequencies with all input repetitions retained, then list each maximizing value once. The result is {-7}; ties are not broken by choosing just one value, and this stated convention is distinct from textbooks calling an all-distinct list no mode. [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r07-descriptive-statistics-0081/) Descriptive summaries of nonempty observation lists · qf-statistics-maximal-frequency-set · candidate ## 82. Set of All Maximum-Frequency Observed Values The nonempty observation list is D=[9, 9, 9, -1, -1, -8]. Retain every repetition. Under the explicit all-frequency-maximizers convention, give every value with maximum frequency as an increasing set, including all values if every frequency is one. View answer and explanation{9} Count frequencies with all input repetitions retained, then list each maximizing value once. The result is {9}; ties are not broken by choosing just one value, and this stated convention is distinct from textbooks calling an all-distinct list no mode. [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r07-descriptive-statistics-0082/) Descriptive summaries of nonempty observation lists · qf-statistics-maximal-frequency-set · candidate ## 83. Set of All Maximum-Frequency Observed Values The nonempty observation list is D=[10, 0, 0, 0, -9, -9]. Retain every repetition. Under the explicit all-frequency-maximizers convention, give every value with maximum frequency as an increasing set, including all values if every frequency is one. View answer and explanation{0} Count frequencies with all input repetitions retained, then list each maximizing value once. The result is {0}; ties are not broken by choosing just one value, and this stated convention is distinct from textbooks calling an all-distinct list no mode. [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r07-descriptive-statistics-0083/) Descriptive summaries of nonempty observation lists · qf-statistics-maximal-frequency-set · candidate ## 84. Set of All Maximum-Frequency Observed Values The nonempty observation list is D=[11, 11, 1, -10, -10, -10]. Retain every repetition. Under the explicit all-frequency-maximizers convention, give every value with maximum frequency as an increasing set, including all values if every frequency is one. View answer and explanation{-10} Count frequencies with all input repetitions retained, then list each maximizing value once. The result is {-10}; ties are not broken by choosing just one value, and this stated convention is distinct from textbooks calling an all-distinct list no mode. [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r07-descriptive-statistics-0084/) Descriptive summaries of nonempty observation lists · qf-statistics-maximal-frequency-set · candidate ## 85. Set of All Maximum-Frequency Observed Values The nonempty observation list is D=[12, 12, 12, -1, -1, -11]. Retain every repetition. Under the explicit all-frequency-maximizers convention, give every value with maximum frequency as an increasing set, including all values if every frequency is one. View answer and explanation{12} Count frequencies with all input repetitions retained, then list each maximizing value once. The result is {12}; ties are not broken by choosing just one value, and this stated convention is distinct from textbooks calling an all-distinct list no mode. [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r07-descriptive-statistics-0085/) Descriptive summaries of nonempty observation lists · qf-statistics-maximal-frequency-set · candidate ## 86. Set of All Maximum-Frequency Observed Values The nonempty observation list is D=[13, 0, 0, 0, -12, -12]. Retain every repetition. Under the explicit all-frequency-maximizers convention, give every value with maximum frequency as an increasing set, including all values if every frequency is one. View answer and explanation{0} Count frequencies with all input repetitions retained, then list each maximizing value once. The result is {0}; ties are not broken by choosing just one value, and this stated convention is distinct from textbooks calling an all-distinct list no mode. [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r07-descriptive-statistics-0086/) Descriptive summaries of nonempty observation lists · qf-statistics-maximal-frequency-set · candidate ## 87. Set of All Maximum-Frequency Observed Values The nonempty observation list is D=[14, 14, 1, -13, -13, -13]. Retain every repetition. Under the explicit all-frequency-maximizers convention, give every value with maximum frequency as an increasing set, including all values if every frequency is one. View answer and explanation{-13} Count frequencies with all input repetitions retained, then list each maximizing value once. The result is {-13}; ties are not broken by choosing just one value, and this stated convention is distinct from textbooks calling an all-distinct list no mode. [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r07-descriptive-statistics-0087/) Descriptive summaries of nonempty observation lists · qf-statistics-maximal-frequency-set · candidate ## 88. Set of All Maximum-Frequency Observed Values The nonempty observation list is D=[15, 15, 15, -1, -1, -14]. Retain every repetition. Under the explicit all-frequency-maximizers convention, give every value with maximum frequency as an increasing set, including all values if every frequency is one. View answer and explanation{15} Count frequencies with all input repetitions retained, then list each maximizing value once. The result is {15}; ties are not broken by choosing just one value, and this stated convention is distinct from textbooks calling an all-distinct list no mode. [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r07-descriptive-statistics-0088/) Descriptive summaries of nonempty observation lists · qf-statistics-maximal-frequency-set · candidate ## 89. Set of All Maximum-Frequency Observed Values The nonempty observation list is D=[16, 0, 0, 0, -15, -15]. Retain every repetition. Under the explicit all-frequency-maximizers convention, give every value with maximum frequency as an increasing set, including all values if every frequency is one. View answer and explanation{0} Count frequencies with all input repetitions retained, then list each maximizing value once. The result is {0}; ties are not broken by choosing just one value, and this stated convention is distinct from textbooks calling an all-distinct list no mode. [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r07-descriptive-statistics-0089/) Descriptive summaries of nonempty observation lists · qf-statistics-maximal-frequency-set · candidate ## 90. Set of All Maximum-Frequency Observed Values The nonempty observation list is D=[17, 17, 1, -16, -16, -16]. Retain every repetition. Under the explicit all-frequency-maximizers convention, give every value with maximum frequency as an increasing set, including all values if every frequency is one. View answer and explanation{-16} Count frequencies with all input repetitions retained, then list each maximizing value once. The result is {-16}; ties are not broken by choosing just one value, and this stated convention is distinct from textbooks calling an all-distinct list no mode. [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r07-descriptive-statistics-0090/) Descriptive summaries of nonempty observation lists · qf-statistics-maximal-frequency-set · candidate ## 91. Set of All Maximum-Frequency Observed Values The nonempty observation list is D=[18, 18, 18, -1, -1, -17]. Retain every repetition. Under the explicit all-frequency-maximizers convention, give every value with maximum frequency as an increasing set, including all values if every frequency is one. View answer and explanation{18} Count frequencies with all input repetitions retained, then list each maximizing value once. The result is {18}; ties are not broken by choosing just one value, and this stated convention is distinct from textbooks calling an all-distinct list no mode. [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r07-descriptive-statistics-0091/) Descriptive summaries of nonempty observation lists · qf-statistics-maximal-frequency-set · candidate ## 92. Set of All Maximum-Frequency Observed Values The nonempty observation list is D=[19, 0, 0, 0, -18, -18]. Retain every repetition. Under the explicit all-frequency-maximizers convention, give every value with maximum frequency as an increasing set, including all values if every frequency is one. View answer and explanation{0} Count frequencies with all input repetitions retained, then list each maximizing value once. The result is {0}; ties are not broken by choosing just one value, and this stated convention is distinct from textbooks calling an all-distinct list no mode. [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r07-descriptive-statistics-0092/) Descriptive summaries of nonempty observation lists · qf-statistics-maximal-frequency-set · candidate ## 93. Set of All Maximum-Frequency Observed Values The nonempty observation list is D=[20, 20, 1, -19, -19, -19]. Retain every repetition. Under the explicit all-frequency-maximizers convention, give every value with maximum frequency as an increasing set, including all values if every frequency is one. View answer and explanation{-19} Count frequencies with all input repetitions retained, then list each maximizing value once. The result is {-19}; ties are not broken by choosing just one value, and this stated convention is distinct from textbooks calling an all-distinct list no mode. [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r07-descriptive-statistics-0093/) Descriptive summaries of nonempty observation lists · qf-statistics-maximal-frequency-set · candidate ## 94. Set of All Maximum-Frequency Observed Values The nonempty observation list is D=[21, 21, 21, -1, -1, -20]. Retain every repetition. Under the explicit all-frequency-maximizers convention, give every value with maximum frequency as an increasing set, including all values if every frequency is one. View answer and explanation{21} Count frequencies with all input repetitions retained, then list each maximizing value once. The result is {21}; ties are not broken by choosing just one value, and this stated convention is distinct from textbooks calling an all-distinct list no mode. [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r07-descriptive-statistics-0094/) Descriptive summaries of nonempty observation lists · qf-statistics-maximal-frequency-set · candidate ## 95. Set of All Maximum-Frequency Observed Values The nonempty observation list is D=[22, 0, 0, 0, -21, -21]. Retain every repetition. Under the explicit all-frequency-maximizers convention, give every value with maximum frequency as an increasing set, including all values if every frequency is one. View answer and explanation{0} Count frequencies with all input repetitions retained, then list each maximizing value once. The result is {0}; ties are not broken by choosing just one value, and this stated convention is distinct from textbooks calling an all-distinct list no mode. [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r07-descriptive-statistics-0095/) Descriptive summaries of nonempty observation lists · qf-statistics-maximal-frequency-set · candidate ## 96. Set of All Maximum-Frequency Observed Values The nonempty observation list is D=[23, 23, 1, -22, -22, -22]. Retain every repetition. Under the explicit all-frequency-maximizers convention, give every value with maximum frequency as an increasing set, including all values if every frequency is one. View answer and explanation{-22} Count frequencies with all input repetitions retained, then list each maximizing value once. The result is {-22}; ties are not broken by choosing just one value, and this stated convention is distinct from textbooks calling an all-distinct list no mode. [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r07-descriptive-statistics-0096/) Descriptive summaries of nonempty observation lists · qf-statistics-maximal-frequency-set · candidate ## 97. Set of All Maximum-Frequency Observed Values The nonempty observation list is D=[24, 24, 24, -1, -1, -23]. Retain every repetition. Under the explicit all-frequency-maximizers convention, give every value with maximum frequency as an increasing set, including all values if every frequency is one. View answer and explanation{24} Count frequencies with all input repetitions retained, then list each maximizing value once. The result is {24}; ties are not broken by choosing just one value, and this stated convention is distinct from textbooks calling an all-distinct list no mode. [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r07-descriptive-statistics-0097/) Descriptive summaries of nonempty observation lists · qf-statistics-maximal-frequency-set · candidate ## 98. Set of All Maximum-Frequency Observed Values The nonempty observation list is D=[25, 0, 0, 0, -24, -24]. Retain every repetition. Under the explicit all-frequency-maximizers convention, give every value with maximum frequency as an increasing set, including all values if every frequency is one. View answer and explanation{0} Count frequencies with all input repetitions retained, then list each maximizing value once. The result is {0}; ties are not broken by choosing just one value, and this stated convention is distinct from textbooks calling an all-distinct list no mode. [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r07-descriptive-statistics-0098/) Descriptive summaries of nonempty observation lists · qf-statistics-maximal-frequency-set · candidate ## 99. Set of All Maximum-Frequency Observed Values The nonempty observation list is D=[26, 26, 1, -25, -25, -25]. Retain every repetition. Under the explicit all-frequency-maximizers convention, give every value with maximum frequency as an increasing set, including all values if every frequency is one. View answer and explanation{-25} Count frequencies with all input repetitions retained, then list each maximizing value once. The result is {-25}; ties are not broken by choosing just one value, and this stated convention is distinct from textbooks calling an all-distinct list no mode. [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r07-descriptive-statistics-0099/) Descriptive summaries of nonempty observation lists · qf-statistics-maximal-frequency-set · candidate ## 100. Set of All Maximum-Frequency Observed Values The nonempty observation list is D=[27, 27, 27, -1, -1, -26]. Retain every repetition. Under the explicit all-frequency-maximizers convention, give every value with maximum frequency as an increasing set, including all values if every frequency is one. View answer and explanation{27} Count frequencies with all input repetitions retained, then list each maximizing value once. The result is {27}; ties are not broken by choosing just one value, and this stated convention is distinct from textbooks calling an all-distinct list no mode. [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r07-descriptive-statistics-0100/) Descriptive summaries of nonempty observation lists · qf-statistics-maximal-frequency-set · candidate [Canonical question JSONL](https://ocme.evemisslab.com/data/questions/batches/batch-20261005-advance-20261006-r07-descriptive-statistics-foundations/questions.jsonl) · [English JSONL](https://ocme.evemisslab.com/data/i18n/questions/batch-20261005-advance-20261006-r07-descriptive-statistics-foundations.en.jsonl) · [Validation report](https://ocme.evemisslab.com/data/questions/batches/batch-20261005-advance-20261006-r07-descriptive-statistics-foundations/validation-report.json)