Question 1

Which of the following statements about the standard deviation is TRUE?

Accepted Answer

Standard deviation is always greater than or equal to zero and shares the same units as the original data.

Question 2

A dataset of home prices has a mean of $$250,000 and a standard deviation of 50$$,000. If an unusually high home price of $1,000,000 is removed from the dataset, what happens to the standard deviation?

Accepted Answer

The standard deviation decreases because the spread of the data narrows.

Question 3

Which formula correctly calculates the sample mean ($$\bar{x}) $$for a dataset?

Accepted Answer

$$\bar{x} = \frac{\sum x}{n}$$

Question 4

Suppose you add 10 to every value in a dataset. What happens to the mean and standard deviation?

Accepted Answer

The mean increases by 10, but the standard deviation remains unchanged.

Question 5

A distribution of exam scores is strongly skewed to the right. Which measure of center is most appropriate to represent a 'typical' score?

Accepted Answer

The median, because it is resistant to outliers and skewness.

Question 6

Given the following sorted heights of 8 children: 48, 48, 53, 53.5, 54, 60, 62, 71 inches, what is the interquartile range (IQR)?

Accepted Answer

$$IQR = 61.0 - 50.5 = 10.5$$

Question 7

Which formula is used to calculate a z-score for a value $$x in a $$distribution with mean $$\mu$$ and standard deviation $$\sigma$$?

Accepted Answer

$$z = \frac{x - \mu}{\sigma}$$

Question 8

If a dataset has all values equal to 7, what is the standard deviation?

Accepted Answer

The standard deviation is 0.

Question 9

Which measure of spread is most robust to outliers in a strongly skewed dataset?

Accepted Answer

Interquartile range (IQR)

Question 10

A city has a mean daily smog level of $$\mu = 8$$ micrograms/m$$^3$$ and a standard deviation of $$\sigma = 2$$ micrograms/m$$^3. $$According to the Empirical Rule, what percentage of days are expected to have smog levels between 6 and 10 micrograms/m$$^3$$?

Accepted Answer

About 68%

Question 11

A newborn baby weighs 2500 grams. The mean birth weight is 3200 grams with a standard deviation of 400 grams. What is the baby's z-score?

Accepted Answer

$$z = \frac{2500 - 3200}{400} = -1.75$$

Question 12

Which of the following is NOT a property of the standard deviation?

Accepted Answer

It changes when a constant is added to every data point.

Question 13

A dataset has a mean of 100 and a standard deviation of 15. What value corresponds to a z-score of 2?

Accepted Answer

$$x = 100 + (2 \cdot 15) = 130$$

Question 14

Which measure is most affected by a single extreme value in a dataset?

Accepted Answer

Mean

Question 15

In a bell-shaped distribution, what proportion of data falls within two standard deviations of the mean according to the Empirical Rule?

Accepted Answer

95%

Question 16

A student scores 85 on a test where the mean is 75 and the standard deviation is 5. Another student scores 90 on a test where the mean is 80 and the standard deviation is 10. Who performed better relative to their group?

Accepted Answer

The first student, because their z-score is $$z = \frac{85-75}{5} = 2$$

Question 17

Which formula is used to calculate the sample variance ($$s^2) $$for a dataset?

Accepted Answer

$$s^2$$ = \frac{\sum (x - \bar{x}$$)^2$$}{n-1}$$

Question 18

Which measure of spread is calculated as the difference between the maximum and minimum values?

Accepted Answer

Range

Question 19

A dataset has a mean of 60 and a standard deviation of 8. What is the z-score for a value of 76?

Accepted Answer

$$z = \frac{76 - 60}{8} = 2$$

Question 20

Which of the following best describes the purpose of the interquartile range (IQR)?

Accepted Answer

To measure the spread of the middle 50% of the data, making it resistant to outliers.

Center and Spread: Numerical Summaries of Data Distributions

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