The normal distribution (bell curve) explained
The normal distribution, often called the bell curve, is the most important pattern in statistics. Many natural measurements, from heights to test scores to measurement errors, follow its symmetric, bell-shaped curve. This guide explains what it is, the 68-95-99.7 rule, and how z-scores let you read it.
What the normal distribution is
A normal distribution is a symmetric, bell-shaped curve where most values cluster near the center and become rarer as you move away in either direction. It is defined entirely by two numbers: the mean, which sets the center, and the standard deviation, which sets the width. Because it is symmetric, the mean, median, and mode all sit at the same central point. Heights, blood pressure, and many measurement errors approximate this shape closely.
The 68-95-99.7 rule
The most useful fact about the normal curve is how predictably values spread from the mean, summarized by the empirical rule.
| Within | Percentage of values |
|---|---|
| 1 standard deviation | about 68% |
| 2 standard deviations | about 95% |
| 3 standard deviations | about 99.7% |
Putting the rule to work
Take IQ scores, designed with a mean of 100 and a standard deviation of 15. The empirical rule says about 68 percent of people score between 85 and 115 (within one standard deviation), about 95 percent between 70 and 130 (within two), and almost everyone, 99.7 percent, between 55 and 145 (within three). The same logic turns any normal data into quick probability estimates.
Z-scores: a common ruler
A z-score expresses how many standard deviations a value sits from the mean, computed as z = (x − mean) ÷ standard deviation. A z-score of 0 is exactly average, +1 is one standard deviation above, and −2 is two below. Converting to z-scores puts any normal distribution on the same standard scale, so you can look up the probability of a value in a single table regardless of the original units.
If exam scores average 70 with a standard deviation of 8, a score of 86 has z = (86 − 70) ÷ 8 = 2.0, two standard deviations above the mean, better than about 97.5 percent of scores.
Why it matters
The normal distribution appears everywhere partly because of the central limit theorem, which says that averages of many independent random quantities tend toward a normal shape even when the underlying data is not normal. This is why it underpins so much of statistics, from confidence intervals to hypothesis testing. Recognizing normal data lets you make precise statements about how likely or unusual any particular value is.
- The normal distribution is a symmetric bell curve set by its mean and standard deviation.
- Mean, median, and mode all coincide at the center.
- The 68-95-99.7 rule: about 68, 95, and 99.7 percent of values fall within 1, 2, and 3 standard deviations.
- A z-score, z = (x − mean) ÷ standard deviation, measures distance from the mean in standard deviations.
- The central limit theorem explains why the normal curve appears so widely.
Related statistics guides
Read the Standard Deviation Guide, or use the statistics calculators to compute z-scores and spread.
FAQ
What is the 68-95-99.7 rule?
For a normal distribution, about 68 percent of values lie within one standard deviation of the mean, 95 percent within two, and 99.7 percent within three.
What is a z-score?
The number of standard deviations a value is from the mean, found with z = (x − mean) ÷ standard deviation. It standardizes any normal distribution.
Why is the normal distribution so common?
The central limit theorem: averages of many independent random effects tend toward a normal shape, so it arises naturally in many measurements.