AP Stats Normal Distribution Calculator

What Is the Normal Distribution?

The normal distribution is the bell-shaped curve that describes many natural measurements, such as heights or test scores. It is symmetric around its mean, and about 68% of values fall within one standard deviation of the mean, about 95% within two, and about 99.7% within three. Any normal distribution can be converted to the standard normal distribution, with mean 0 and standard deviation 1, by computing z-scores.

How to Use This Calculator

1. Enter the mean of your normal distribution — the center of the curve.
2. Enter the standard deviation. It must be greater than 0, because a standard deviation of 0 would mean every value is exactly the mean and there is no spread to shade.
3. Enter the lower bound and upper bound for the interval you care about. The lower bound must be less than or equal to the upper bound.
4. Click Calculate Probability, or just change any input — the probability, z-scores, and shaded curve update live.
5. Read the result as a decimal and as a percentage, and use the z-scores to check your work against a standard normal table or your calculator's normalcdf function.

Example to try first: mean 50, standard deviation 10, lower 40, upper 60. You should get about 0.6827, or 68.27%.

How to Calculate

For a value x from a normal distribution with mean μ and standard deviation σ, first convert it to a z-score:

Then the probability between a lower bound a and an upper bound b is the area under the standard normal curve between their z-scores:

Here Φ(z) is the standard normal cumulative distribution function (CDF) — the area to the left of z. This calculator computes it with an error-function (erf) approximation, the same idea behind normalcdf on a graphing calculator.

Worked Example

Suppose test scores are approximately normal with mean 50 and standard deviation 10. What proportion of scores fall between 40 and 60?

• z for 40: (40 − 50) / 10 = −1
• z for 60: (60 − 50) / 10 = 1
• P(40 < X < 60) = Φ(1) − Φ(−1) ≈ 0.8413 − 0.1587 = 0.6827

Answer: about 0.6827, or 68.27%. This matches the empirical rule: about 68% of values in a normal distribution fall within one standard deviation of the mean.

Tips and Common Mistakes

• Do not confuse standard deviation with variance. If a problem gives variance, take its square root first to get the standard deviation before using this calculator or normalcdf.
• Check that the distribution is (approximately) normal before using normal probabilities. A skewed distribution, a small sample with outliers, or a categorical variable is not a normal-distribution problem.
• Keep lower and upper in the right order. If you swap them, the probability is still the same area, but your z-scores will be backwards and your work will not match the expected setup.
• Use σ, not the standard error, for an individual value. The standard error σ / √n is for a sample mean, not for one observation. Using it here will make your interval look far too narrow.
• Watch for "less than" and "greater than" wording. For P(X < b), use a very low lower bound; for P(X > a), use a very high upper bound — or subtract from 1. Do not just enter one bound and leave the other at its default.
• Round only at the end. Carry full z-scores through the calculation and round the final probability to four decimal places, as AP Statistics usually expects.
• Sanity-check with the empirical rule. Within 1 SD should be near 68%, within 2 SD near 95%, and within 3 SD near 99.7%. If your answer is far from those landmarks for a 1-, 2-, or 3-SD interval, recheck your inputs.

Frequently Asked Questions

What is the difference between normalcdf and invNorm?

normalcdf starts with bounds and gives you an area (a probability) — exactly what this calculator does. invNorm works in reverse: you give it an area (a percentile) and it gives you the boundary value or z-score that leaves that area to its left. Use normalcdf when you know the values and want a probability; use invNorm when you know the probability and want the value.

What is the empirical rule (68-95-99.7 rule)?

For an approximately normal distribution, about 68% of values fall within 1 standard deviation of the mean, about 95% within 2 standard deviations, and about 99.7% within 3 standard deviations. It is a quick check, not a replacement for normalcdf when a problem asks for an exact probability for bounds that are not whole numbers of standard deviations away.

What does a z-score tell me?

A z-score tells you how many standard deviations a value is above or below the mean. A z-score of 1 means one standard deviation above the mean; −1.5 means one and a half standard deviations below it. Z-scores let you compare values from different normal distributions on the same scale, and they are the input to the standard normal table.

What is the difference between standard deviation and standard error?

Standard deviation describes the spread of individual values in a distribution. Standard error describes the spread of a statistic — most often a sample mean — across repeated samples, and it equals σ / √n. For a probability about one individual, use the standard deviation. For a probability about a sample mean, use the standard error and the sampling distribution of the mean.

Can I use this calculator for a "less than" or "greater than" problem?

Yes, with a workaround. For P(X < 60) with mean 50 and SD 10, set the lower bound far below the mean (for example, 50 − 10 × 10 = −50) and the upper bound to 60. For P(X > 40), set the lower bound to 40 and the upper bound far above the mean. The tiny area beyond 10 standard deviations is effectively zero, so the result matches normalcdf with −1E99 or 1E99.

Why does my graphing calculator give a slightly different answer?

Small differences in the fourth decimal place usually come from rounding z-scores before looking them up in a table, or from different internal approximations. If the difference is larger, check whether you entered the standard deviation or the variance, whether lower and upper are in the right order, and whether the problem was about a sample mean (which needs the standard error).

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