Let's analyze the problem in your image. We have three data sets (i, ii, iii) displayed as bar graphs showing their frequency distributions, and the following questions:

Part (a)

(1) Which data set has the greatest sample standard deviation?

• Answer: Data set (i), because it has more entries that are farther away from the mean.
  • The reason for this choice is that data set (i) has entries spread more widely across the data values (from 3 to 11) compared to the others, indicating greater variability. Standard deviation measures how much the data is spread out, and the farther the values are from the mean, the greater the standard deviation.

(2) Which data set has the least sample standard deviation?

• Answer: Data set (ii), because it has more entries that are close to the mean.
  • Data set (ii) is more tightly clustered around the center (mean), with most of its data points centered near 7. This implies that the variability is low, and thus it has the smallest standard deviation.

Part (b)

How are the data sets the same? How do they differ?

• Answer: The three data sets have the same range and mean but have different standard deviations.
  • This matches Option B. All data sets appear to cover the same range (values from 3 to 11) and have the same mean. However, their spread or variability is different, as reflected in the differing shapes of the frequency distributions. Data set (i) is more spread out, (ii) is tightly clustered, and (iii) has moderate variability, which implies different standard deviations.

Clarification:

• Standard deviation: A measure of how spread out numbers are from the mean. Data sets with more values far from the mean will have a higher standard deviation.

Would you like more details on standard deviation, or do you have any other questions about this problem?

Here are 5 follow-up questions to deepen understanding:

1. How can we calculate the sample standard deviation of a data set manually?
2. Why does data set (ii) have a smaller standard deviation even though all three sets have the same range?
3. How is range different from standard deviation in describing data distribution?
4. How would the shape of a frequency distribution change if we reduced the standard deviation of a data set?
5. What other statistics, besides the mean and standard deviation, could we use to compare these data sets?

Tip: The standard deviation is sensitive to outliers—extreme values can significantly increase it even if most data points are close to the mean.