Normal Distribution Calculator

Probability between, below, and above any bounds on a normal curve.

What it does

The normal curve, or bell curve, describes measurements that cluster around a mean: heights, test scores, measurement errors, and countless process outputs. Its shape is fixed once you know two numbers, the mean and the standard deviation, and most questions about it reduce to finding an area under the curve.

This tool answers the three common area questions at once. Give it a range and it reports the probability between the bounds, the probability below the lower bound, and the probability above the upper bound. The three pieces add up to 100 percent, which makes a useful check on any bar chart or claim.

The method behind it is standardization. Each bound converts to a z score by subtracting the mean and dividing by the standard deviation. A z score of zero sits at the mean, a z of 1 is one standard deviation above, and the standard normal table turns those z values into tail areas. The tool does that conversion and lookup for both bounds at once.

The default example uses a mean of 100 and a standard deviation of 15, the classic IQ style scale, with bounds of 85 and 115. Those sit exactly one standard deviation either side of the mean, so the middle holds about 68.27 percent, and each tail holds about 15.87 percent. The empirical 68, 95, 99.7 rule is just this math at one, two, and three standard deviations.

Normal models fail when the data are skewed or heavy tailed, and the tool flags bounds beyond 20 standard deviations where the math stops being meaningful. Within its range, though, it handles everything from quality control limits to standardized test percentiles.

How to use

1. Enter the mean of the distribution.
2. Enter the standard deviation, which must be greater than zero.
3. Enter the lower bound of the range you care about.
4. Enter the upper bound.
5. Read the middle probability first, then the two tails.
6. Check that the three probabilities add to 100 percent.

Formula

mean

Center of the curve, where the peak sits.

standard deviation

Spread of the curve; larger values make it wider and flatter.

z

Standardized distance from the mean, in standard deviations.

Phi

Standard normal cumulative distribution, the area to the left of a z score.

The standard normal table is the engine behind all of it: z scores convert any normal curve to the one with mean 0 and standard deviation 1, then Phi(z) reads the area to the left. Working in percentages, the three outputs always sum to 100.

Worked example

Reference table

Limits

• Assumes a perfectly normal shape; real data rarely match it exactly.
• Skewed or heavy tailed data give misleading tail probabilities.
• Uses the mean and standard deviation as given, so bad inputs give confident but wrong answers.
• Flags bounds beyond 20 standard deviations, where the model is not useful.

Common mistakes

• Entering the variance instead of the standard deviation.
• Swapping the bounds; the tool catches it but the first run is wasted.
• Reading the middle percentage as a count instead of a share of the whole curve.
• Forgetting that the two tails are separate; 68.27 percent in the middle leaves 31.73 percent split across both ends.
• Applying the curve to clearly skewed data such as incomes or waiting times.

Try it yourself

Quick checks. Answers sit tucked under each question.

Questions

Sources

• NIST/SEMATECH e-Handbook: normal distribution
• OpenStax Introductory Statistics: the standard normal distribution

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