3. Describing Data Numerically

Given four histograms representing different data sets, each with the same mean $\mu$ but varying spreads, which histogram would correspond to the largest standard deviation $\sigma$?

Which of the following values can the variance of a dataset never be?

Suppose you have two histograms, A and B, each representing the distribution of exam scores for two different classes. Histogram $A$ shows scores tightly clustered around the mean, while Histogram $B$ shows scores spread out over a wider range. Which histogram depicts a higher $\mathrm{standard} \mathrm{deviation}$?

Given the population data set $2,0,1,9$, what is the variance of this population?

Given a population with mean $\mu$ = $65$ and standard deviation $\sigma$ = $4$, what is the z-score of a value $x$ = $73$?

Given the sample data $8$, $6$, $2$, $8$, what is the variance of the sample?

Variance of Roller Coaster Speeds The standard deviation of the sample values in Exercise 1 is 43.1 km/h. What is the variance (including units)?

Comparing Values. In Exercises 13–16, use z scores to compare the given values.

Tallest and Shortest Men The tallest adult male was Robert Wadlow, and his height was 272 cm. The shortest adult male was Chandra Bahadur Dangi, and his height was 54.6 cm. Heights of men have a mean of 174.12 cm and a standard deviation of 7.10 cm. Which of these two men has the height that is more extreme?

Birth Weights Babies born after a gestation period of 32–35 weeks have a mean weight of 2600 grams and a standard deviation of 660 grams. Babies born after a gestation period of 40 weeks have a mean weight of 3500 grams and a standard deviation of 470 grams. Suppose a 34-week gestation period baby weighs 3000 grams and a 40-week gestation period baby weighs 3900 grams. What is the z-score for the 34-week gestation period baby? What is the z-score for the 40-week gestation period baby? Which baby weighs less relative to the gestation period?

Which of the following correctly represents the formula for the population standard deviation?

Why is the $standard deviation$ generally considered a better measure of variation than the $range$?

If a set of data has a standard deviation of $0$, which of the following must be true about the data values?

Which of the following is true for the quantity $\sqrt{x-\mu }$?

Given four normal curves with the same mean ($\mu$) but different spreads, which one of the following curves has the largest standard deviation ($\sigma$)?

If a set of data has a standard deviation of $0$, which of the following must be true about the data?