A z-score says how many standard deviations a value sits from the mean. Subtract the mean, divide by the standard deviation, and the sign tells you which side you are on. Going the other way, multiply and add back.
Also taught in Statistics, at its own level and with its own examples.
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A set is normal with mean 15 and standard deviation 5. Find the value with a z-scoreof 1.
A set is normal with mean 20 and standard deviation 2. Find the z-scoreof x=18.
A set is normal with mean 25 and standard deviation 5. Find the value with a z-scoreof −3.
A set is normal with mean 25 and standard deviation 10. Find the z-scoreof x=−5.
A set is normal with mean 20 and standard deviation 2. Find the value with a z-scoreof −3.
A set is normal with mean 35 and standard deviation 10. Find the z-scoreof x=55.
A set is normal with mean 35 and standard deviation 10. Find the value with a z-scoreof 2.
Use the empirical rule
The lengths of a species of fish are approximately normal with mean 35 and standard deviation 6 centimeters. About what percent are between 35 and 47?
The weights of bags of flour are approximately normal with mean 33 and standard deviation 1 ounce. About what percent are between 33 and 34?
Commute times in a city are approximately normal with mean 42 and standard deviation 5 minutes. About what percent are between 47 and 57?
The heights of adult women are approximately normal with mean 64 and standard deviation 3 inches. About what percent are between 61 and 70?
The weights of bags of flour are approximately normal with mean 32 and standard deviation 2 ounces. About what percent are between 32 and 36?
Resting heart rates are approximately normal with mean 63 and standard deviation 8 beats per minute. About what percent are between 63 and 71?
Scores on a statistics exam are approximately normal with mean 81 and standard deviation 7 points. About what percent are between 67 and 88?
Find percents from z-scores
Test scores are normal with mean 71 points and standard deviation 10 points. What percent are between 76 and 86 points? Round to the nearest tenth of a percent.
Heights of adults are normal with mean 67 in and standard deviation 2.5 in. What percent are between 71 and 72.25 in? Round to the nearest tenth of a percent.
Battery lives are normal with mean 39 hours and standard deviation 4 hours. What percent are more than 44.6 hours? Round to the nearest tenth of a percent.
Weights of apples are normal with mean 200 g and standard deviation 8 g. What percent are more than 220 g? Round to the nearest tenth of a percent.
Battery lives are normal with mean 57 hours and standard deviation 4 hours. What percent are between 60.2 and 64.2 hours? Round to the nearest tenth of a percent.
Heights of adults are normal with mean 65 in and standard deviation 2.5 in. What percent are more than 68.75 in? Round to the nearest tenth of a percent.
A set is normal with mean 15 and standard deviation 5. Find the value with a z-scoreof 1.20
A set is normal with mean 20 and standard deviation 2. Find the z-scoreof x=18.−1
A set is normal with mean 25 and standard deviation 5. Find the value with a z-scoreof −3.10
A set is normal with mean 25 and standard deviation 10. Find the z-scoreof x=−5.−3
A set is normal with mean 20 and standard deviation 2. Find the value with a z-scoreof −3.14
A set is normal with mean 35 and standard deviation 10. Find the z-scoreof x=55.2
A set is normal with mean 35 and standard deviation 10. Find the value with a z-scoreof 2.55
Use the empirical rule
The lengths of a species of fish are approximately normal with mean 35 and standard deviation 6 centimeters. About what percent are between 35 and 47?47.5%
The weights of bags of flour are approximately normal with mean 33 and standard deviation 1 ounce. About what percent are between 33 and 34?34%
Commute times in a city are approximately normal with mean 42 and standard deviation 5 minutes. About what percent are between 47 and 57?15.85%
The heights of adult women are approximately normal with mean 64 and standard deviation 3 inches. About what percent are between 61 and 70?81.5%
The weights of bags of flour are approximately normal with mean 32 and standard deviation 2 ounces. About what percent are between 32 and 36?47.5%
Resting heart rates are approximately normal with mean 63 and standard deviation 8 beats per minute. About what percent are between 63 and 71?34%
Scores on a statistics exam are approximately normal with mean 81 and standard deviation 7 points. About what percent are between 67 and 88?81.5%
Find percents from z-scores
Test scores are normal with mean 71 points and standard deviation 10 points. What percent are between 76 and 86 points? Round to the nearest tenth of a percent.z=0.5and 1.5;about 24.2%
Heights of adults are normal with mean 67 in and standard deviation 2.5 in. What percent are between 71 and 72.25 in? Round to the nearest tenth of a percent.z=1.6and 2.1;about 3.7%
Battery lives are normal with mean 39 hours and standard deviation 4 hours. What percent are more than 44.6 hours? Round to the nearest tenth of a percent.z=1.4;about 8.1%
Weights of apples are normal with mean 200 g and standard deviation 8 g. What percent are more than 220 g? Round to the nearest tenth of a percent.z=2.5;about 0.6%
Battery lives are normal with mean 57 hours and standard deviation 4 hours. What percent are between 60.2 and 64.2 hours? Round to the nearest tenth of a percent.z=0.8and 1.8;about 17.6%
Heights of adults are normal with mean 65 in and standard deviation 2.5 in. What percent are more than 68.75 in? Round to the nearest tenth of a percent.z=1.5;about 6.7%
A set is normal with mean 70 and standard deviation 5. Find the z-scoreof 80
80 minus the mean 70 is 10.
10 divided by the standard deviation 5.
The answer is 2.
Where students go wrong
Dividing by the mean instead of the standard deviation. The mean gets subtracted; the standard deviation is what you divide by.
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-scoreof 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.
Algebra 2 is usually taken around 10th or 11th 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.