Question

compare the three data sets on the right. (a) which data set has the greatest sample standard deviation? a. data set (ii), because it has more entries that are farther away from the mean. b. data set (i), because it has two entries that are far away from the mean. c. data set (iii), because it has more entries that are close to the mean. which data set has the least sample standard deviation? a. data set (ii), because it has more entries that are farther away from the mean. b. data set (i), because it has less entries that are farther away from the mean. c. data set (iii), because it has more entries that are close to the mean. (b) how are the data sets the same? how do they differ? a. the three data sets have the same mode but have different standard deviations and means. b. the three data sets have the same mean, median and mode but have different standard deviations. c. the three data sets have the same mean and mode but have different medians and standard deviations. d. the three data sets have the same standard deviations but have different means.

Explanation:

Step1: Understand standard deviation

Standard deviation measures the spread of data. More spread means larger standard deviation. Data set (i) has two data points far from the mean.

Step2: Analyze each option for part (a)

• Option A: Data set (ii) does not have more entries far from the mean.
• Option B: Data set (i) has two entries far from the mean. This causes a larger spread compared to (iii) which has data close to the mean.
• Option C: Data set (iii) has data close to the mean, so standard deviation is small.

Step3: Analyze each option for part (a) (least standard deviation)

• Option A: Incorrect as more entries far from mean would give larger standard deviation.
• Option B: Incorrect as data set (i) has some entries far from mean.
• Option C: Data set (iii) has more entries close to the mean. So standard deviation is least.

Step4: Analyze part (b)

• Calculate mean, median and mode.
• For symmetric data (as these seem to be), mean = median.
• Mode is the most frequent value.
• All three data - sets have the same center (mean, median, mode) but different spreads (standard deviations)

Answer:

(a) B. Data set (i), because it has two entries that are far away from the mean. C. Data set (iii), because it has more entries that are close to the mean.
(b) B. The three data sets have the same mean, median and mode but have different standard deviations.