Worksheets · Statistics

Normal distributions worksheet for Statistics

In Statistics the normal model turns a measurement into a percent: standardize with a z-score, then use the 68-95-99.7 rule or a table of areas. Answers are percents or proportions, often rounded to a tenth of a percent, and each one is an area under the curve.

Also taught in Algebra 2, at its own level and with its own examples.

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Free math worksheets at sigmaprep.io/worksheets© 2026 Sigma Prep

Sigma Prep · sigmaprep.io/worksheets

Name

Normal Distributions

Date Period

Find each value.

  1. Convert between values and z-scores

    A set is normal with mean 30 and standard deviation 10. Find the value with a z-score of −2.
  2. A set is normal with mean 20 and standard deviation 5. Find the z-score of x=30.
  3. A set is normal with mean 15 and standard deviation 10. Find the value with a z-score of 2.
  4. A set is normal with mean 55 and standard deviation 2. Find the value with a z-score of 3.
  5. A set is normal with mean 20 and standard deviation 10. Find the z-score of x=−10.
  6. A set is normal with mean 50 and standard deviation 2. Find the value with a z-score of −2.
  7. A set is normal with mean 45 and standard deviation 2. Find the z-score of x=47.

  1. Use the empirical rule

    The lengths of a species of fish are approximately normal with mean 47 and standard deviation 5 centimeters. About what percent are between 42 and 57?
  2. The heights of adult women are approximately normal with mean 62 and standard deviation 2 inches. About what percent are between 64 and 68?
  3. Resting heart rates are approximately normal with mean 69 and standard deviation 8 beats per minute. About what percent are between 69 and 77?
  4. Scores on a statistics exam are approximately normal with mean 69 and standard deviation 7 points. About what percent are between 69 and 83?
  5. The weights of bags of flour are approximately normal with mean 32 and standard deviation 1 ounce. About what percent are between 30 and 33?
  6. Commute times in a city are approximately normal with mean 22 and standard deviation 8 minutes. About what percent are between 22 and 38?
  7. The weights of bags of flour are approximately normal with mean 32 and standard deviation 2 ounces. About what percent are between 32 and 34?

  1. Find percents from z-scores

    Weights of apples are normal with mean 187 g and standard deviation 8 g. What percent are between 177.4 and 193.4 g? Round to the nearest tenth of a percent.
  2. Test scores are normal with mean 85 points and standard deviation 8 points. What percent are between 75.4 and 83.4 points? Round to the nearest tenth of a percent.
  3. Battery lives are normal with mean 56 hours and standard deviation 5 hours. What percent are more than 62 hours? Round to the nearest tenth of a percent.
  4. Test scores are normal with mean 81 points and standard deviation 8 points. What percent are more than 85 points? Round to the nearest tenth of a percent.
  5. Heights of adults are normal with mean 64 in and standard deviation 4 in. What percent are more than 65.6 in? Round to the nearest tenth of a percent.
  6. Heights of adults are normal with mean 63 in and standard deviation 2 in. What percent are between 61 and 62 in? Round to the nearest tenth of a percent.

Free math worksheets at sigmaprep.io/worksheets© 2026 Sigma Prep

Sigma Prep · sigmaprep.io/worksheets

Name

Normal DistributionsAnswer keyVersion 2

Date Period

Find each value.

  1. Convert between values and z-scores

    A set is normal with mean 30 and standard deviation 10. Find the value with a z-score of −2.10
  2. A set is normal with mean 20 and standard deviation 5. Find the z-score of x=30.2
  3. A set is normal with mean 15 and standard deviation 10. Find the value with a z-score of 2.35
  4. A set is normal with mean 55 and standard deviation 2. Find the value with a z-score of 3.61
  5. A set is normal with mean 20 and standard deviation 10. Find the z-score of x=−10.−3
  6. A set is normal with mean 50 and standard deviation 2. Find the value with a z-score of −2.46
  7. A set is normal with mean 45 and standard deviation 2. Find the z-score of x=47.1

  1. Use the empirical rule

    The lengths of a species of fish are approximately normal with mean 47 and standard deviation 5 centimeters. About what percent are between 42 and 57?81.5%
  2. The heights of adult women are approximately normal with mean 62 and standard deviation 2 inches. About what percent are between 64 and 68?15.85%
  3. Resting heart rates are approximately normal with mean 69 and standard deviation 8 beats per minute. About what percent are between 69 and 77?34%
  4. Scores on a statistics exam are approximately normal with mean 69 and standard deviation 7 points. About what percent are between 69 and 83?47.5%
  5. The weights of bags of flour are approximately normal with mean 32 and standard deviation 1 ounce. About what percent are between 30 and 33?81.5%
  6. Commute times in a city are approximately normal with mean 22 and standard deviation 8 minutes. About what percent are between 22 and 38?47.5%
  7. The weights of bags of flour are approximately normal with mean 32 and standard deviation 2 ounces. About what percent are between 32 and 34?34%

  1. Find percents from z-scores

    Weights of apples are normal with mean 187 g and standard deviation 8 g. What percent are between 177.4 and 193.4 g? Round to the nearest tenth of a percent.z=−1.2 and 0.8; about 67.3%
  2. Test scores are normal with mean 85 points and standard deviation 8 points. What percent are between 75.4 and 83.4 points? Round to the nearest tenth of a percent.z=−1.2 and −0.2; about 30.6%
  3. Battery lives are normal with mean 56 hours and standard deviation 5 hours. What percent are more than 62 hours? Round to the nearest tenth of a percent.z=1.2; about 11.5%
  4. Test scores are normal with mean 81 points and standard deviation 8 points. What percent are more than 85 points? Round to the nearest tenth of a percent.z=0.5; about 30.9%
  5. Heights of adults are normal with mean 64 in and standard deviation 4 in. What percent are more than 65.6 in? Round to the nearest tenth of a percent.z=0.4; about 34.5%
  6. Heights of adults are normal with mean 63 in and standard deviation 2 in. What percent are between 61 and 62 in? Round to the nearest tenth of a percent.z=−1 and −0.5; about 15.0%

Free math worksheets at sigmaprep.io/worksheets© 2026 Sigma Prep

How to do these

Scores are normal with mean 50 and standard deviation 6. About what percent are above 62?

  1. 62 is (62 - 50) ÷ 6 = 2 standard deviations above the mean.
  2. 95% lie within 2 standard deviations, leaving 5% in the two tails.
  3. Half of that is in the upper tail.

The answer is 2.5%.

Where students go wrong

Forgetting to halve the tails. Within 2 standard deviations is 95%, so 5% is outside, but that 5% is split between both ends; only 2.5% is above two standard deviations.

Also called z-score, standard deviations from the mean, empirical rule or bell curve.

What you can put on this worksheet

Z-scores, both ways
A set is normal with mean 40 and standard deviation 5. Find the z-score of x=50. → 2
The empirical rule (68-95-99.7)
Commute times in a city are approximately normal with mean 34 and standard deviation 8 minutes. About what percent are between 26 and 42? → 68%
Percents from z
Battery lives are normal with mean 52 hours and standard deviation 4 hours. What percent are more than 44 hours? Round to the nearest tenth of a percent. → z=−2; about 97.7%

Questions about these worksheets

Yes. Every normal distributions sheet is free to print and free to download as a PDF. There is no account to make, no email to hand over and no limit on how many you take.

Yes. The answer key prints on a second page, and you can choose whether it shows each question beside its answer or just the answers in a list.

Yes. Print straight from the page, or download the PDF and print that. The sheet is laid out for paper rather than squeezed off a screen, so there is room to work under each question, and the answer key comes out on its own page.

Statistics is usually taken around 11th or 12th grade, though schools vary and plenty of students meet it earlier or later. Pick the difficulty rather than the grade: the easy level suits a first lesson on it, the hard level suits review before a test.

Yes, as many as you like. The questions are built fresh each time rather than picked from a fixed set of files, so pressing Generate gives a new sheet. That is what you want for a second class, a retake, or two students sitting next to each other.

That is the closest comparison, and Kuta Software is good software. It is also paid software you install on a computer. These worksheets run in a browser tab for free, with no account and nothing to install. Like Kuta, the normal distributions questions are generated when you ask for them rather than pulled from a fixed set of files, so two classes never get the same sheet, and the answer key comes with it.

Yes. There are three levels, and you can tick more than one for a sheet that mixes them, in which case the questions come out easiest first. The harder levels are not just larger numbers: they ask for something the easy ones do not.

You pick from 3: z-scores, both ways, the empirical rule (68-95-99.7) and percents from z. Tick as many as you want and set how many of each, or let it spread them evenly.

Dividing by the mean instead of the standard deviation. The mean gets subtracted; the standard deviation is what you divide by.