Questions: Measures of Spread

Question 1

A dataset of exam scores has a mean of 75 but includes one outlier score of 15. Which measure of spread best represents the variability experienced by most students?

• A. Range
• B. Variance
• C. Interquartile range (IQR)
• D. None — all measures are equally affected

Answer: (C)

IQR measures the spread of the middle 50% of data (Q3 - Q1), making it resistant to outliers. Range = max - min, so the single score of 15 would dominate it entirely. Variance and standard deviation are also pulled by extreme values because they use squared deviations from the mean.

Question 2

The standard deviation of a dataset is typically smaller than its variance.

• T. True
• F. False

Answer: False

Standard deviation = √variance. When variance is less than 1, its square root is larger — for example, a variance of 0.25 gives a standard deviation of 0.5. Whether std dev is larger or smaller than variance depends entirely on the scale of the data, not a fixed rule.

Question 3

Why is standard deviation usually reported instead of variance when describing how spread out data is?

Model answer: Standard deviation is in the same units as the original data, making it directly interpretable, while variance is in squared units that have no intuitive meaning.

If scores are measured in points, variance is in points-squared, which cannot be compared to the data directly. Standard deviation returns to the original unit, so you can say 'scores varied by about 8 points on average' — a statement that is immediately meaningful to anyone who knows the scale.