AP Statistics z-Scores and the Normal Distribution Worksheet (PDF)

AP Statistics math worksheet: z-Scores and the Normal Distribution – Maths PDF

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This ap statistics normal distribution worksheet for AP Statistics covers z-scores and the normal distribution (Common Core AP STAT 1.10, AP STAT 1.11, HSS-ID.A.4), with a free PDF to print.

These 20 problems practice AP Statistics topics 1.10 and 1.11. You will compute z-scores and percentiles, apply the empirical rule, find normal probabilities, and work backward from a percentile to a value.

AP Statistics · Common Core State Standards: AP STAT 1.10, AP STAT 1.11, HSS-ID.A.4

Ap statistics normal distribution worksheet: exercises 1 to 10

Exercise 1 – Compute a z-score

Heights of adult women are approximately normal with mean 64 inches and standard deviation 2.5 inches. Find the z-score of a woman who is 69 inches tall.

Exercise 2 – Value from a z-score

Scores on a driving test have mean 78 and standard deviation 6. Find the score that has z = -1.5.

Exercise 3 – Compare two scores

Jamal scored 84 on a math test (mean 75, sd 6). Sofia scored 640 on a science test (mean 600, sd 50). Who did relatively better?

Exercise 4 – Empirical rule

Weights of a bag of rice in grams.

The weights of rice bags are approximately normal with mean 500 g and standard deviation 40 g. What percent of bags weigh between 420 g and 580 g? Between 460 g and 540 g?

Exercise 5 – Empirical rule counts

Battery life is approximately normal with mean 12 hours and standard deviation 1.5 hours. Of 400 batteries, about how many last more than 15 hours?

Exercise 6 – Area to the left

Area left of 280.

A variable is normal with mean 250 and standard deviation 20. Find the proportion of values below 280.

Exercise 7 – Area to the right

For the same distribution (mean 250, sd 20), find the proportion of values above 215.

Exercise 8 – Area between

Area between 230 and 270.

Find the proportion of values between 230 and 270 for the same distribution.

Exercise 9 – Value from a percentile

The 90th percentile.

IQ scores are normal with mean 100 and standard deviation 15. Find the IQ at the 90th percentile.

Exercise 10 – Percentile of a value

A runner finishes a 5K in 22 minutes. Times are normal with mean 27 minutes and standard deviation 4 minutes. Find her percentile and interpret it.

Exercises 11 and up

Exercise 11 – Quartiles of a normal

Find Q1 and Q3 for the IQ distribution (mean 100, sd 15) and the IQR.

Exercise 12 – Change of units

Heights have mean 64 inches and sd 2.5 inches. A woman is 69 inches tall. If the heights are converted to centimeters (1 inch = 2.54 cm), what is her z-score? Why?

Exercise 13 – Is it approximately normal?

A data set of 200 values has mean 50 and sd 8. There are 138 values between 42 and 58 and 192 values between 34 and 66. Does the empirical rule suggest approximate normality?

Exercise 14 – Bottle filling

Bottle volume in ounces.

A machine fills bottles with mean 12.0 oz and standard deviation 0.15 oz (normal). A bottle with less than 11.8 oz is rejected. What proportion is rejected? How many of 5,000 bottles?

Exercise 15 – Tall women

Using mean 64 inches and sd 2.5 inches, what proportion of women are taller than 70 inches? Is a height of 70 inches unusual?

Exercise 16 – Commute times

Commute times in a city are normal with mean 60 minutes and sd 4 minutes. Find the percent of commutes between 55 and 65 minutes.

Exercise 17 – Top 5 percent

Top 5 percent of IQ scores.

A gifted program admits students scoring in the top 5% (IQ mean 100, sd 15). Find the minimum IQ for admission.

Exercise 18 – Find the standard deviation

The weights of apples are normal with mean 3.2 oz. About 2.5% of apples weigh more than 4.0 oz. Use the empirical rule to estimate the standard deviation.

Exercise 19 – Two-step

The rainfall in a region in one month is normal with mean 3.5 in. and sd 0.8 in. Find the proportion of months with rainfall within 0.5 in. of the mean, and the proportion more than 1 in. above the mean.

Exercise 20 – Scores above a cutoff

Scores on a placement exam are normal with mean 72 and sd 9. A student needs at least 90 to place out of the course. What proportion of students do so?

Key facts behind this ap statistics normal distribution worksheet

Key facts

z = (x – mean) / sd measures position in standard deviations.
Empirical rule: 68% within 1 sd, 95% within 2 sd, 99.7% within 3 sd.
Area under the curve over an interval is the proportion of values in it; total area is 1.
Use normalcdf for areas and invNorm for percentiles; always sketch and shade first.
A percentile is the percent of values at or below a given value.

How to use this ap statistics normal distribution worksheet

Work through the exercises in order: they go from simple to challenging. Write every step on paper, then compare with the solutions page and redo any exercise you missed the next day.

Download the ap statistics normal distribution worksheet as a PDF with the button below and keep it with your notes.

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