Statistics › 🔔 Random Variables & Distributions

The Normal Distribution

The bell curve, the two numbers that draw it, and the 68–95–99.7 rule that lets you read it at a glance.

Intro StatisticsAP Statistics levelCollege Stats 101
💡
The big idea: The normal distribution is a single, symmetric bell-shaped curve whose entire shape is fixed by just two numbers: the mean μ, which slides it left or right, and the standard deviation σ, which sets how wide it spreads. Once you know those two, a simple rule of thumb — 68% of the data within one standard deviation, 95% within two, 99.7% within three — tells you roughly where almost everything lands. And because averaging tends to produce this shape, the bell turns up almost everywhere you look.
🎯 By the end, you'll be able to
  • Describe the shape of a normal distribution and the separate jobs of the mean μ and standard deviation σ
  • Read the center and spread of a bell curve directly from its μ and σ
  • Apply the 68–95–99.7 empirical rule to estimate what proportion of values fall in a given range
  • Explain, through the Central Limit Theorem, why the normal curve shows up in so many places
📎 You should already know
  • Mean and standard deviation
  • Histograms and the shape of data

A shape you have already met

Plot the heights of thousands of adults, the errors a scale makes when you weigh the same object again and again, or the total on a stack of dice rolls, and something striking happens: the histogram keeps settling into the same lop-free, single-humped shape. Most values crowd near a central peak; fewer and fewer appear as you move out to either side; and the left half mirrors the right.

That shape is the normal distribution — the famous bell curve. It is not just one curve but a whole family, and every member of the family is drawn by choosing only two numbers.

🔑 What makes a distribution 'normal'
A distribution is normal when it is a smooth, symmetric bell: a single peak in the middle, tails that fall away at the same rate on both sides, and mean = median = mode all sitting together at the center. Its exact width and location are controlled entirely by the mean μ (mu) and the standard deviation σ (sigma).

Two numbers, and only two, draw the curve

Every normal curve is set by exactly two dials:

  • The mean μ is the balance point. Change μ and the whole bell slides left or right without changing shape — the peak sits right above μ.
  • The standard deviation σ is the spread. A small σ gives a tall, narrow, sharp bell; a large σ gives a short, wide, flat one. Change σ and the bell stretches or squeezes around the mean.

That is the whole recipe. Tell me μ and σ, and I can draw your curve — no other information needed.

\[ f(x) = \frac{1}{\sigma\sqrt{2\pi}}\, e^{-\frac{1}{2}\left(\frac{x-\mu}{\sigma}\right)^2} \]
The normal density. You will rarely plug numbers into this by hand — but notice that μ and σ are the only quantities in it that describe the data. Everything else is a fixed constant.

The 68–95–99.7 rule

Because every normal curve has the same shape, the fraction of data caught inside a band measured in standard deviations is always the same, no matter what μ and σ happen to be. That gives a rule of thumb worth memorizing — the empirical rule:

  • About 68% of values lie within ±1 SD of the mean.
  • About 95% lie within ±2 SD.
  • About 99.7% lie within ±3 SD.

In the tool below, the horizontal axis is measured in standard deviations from the mean. Pick a band — the central ±1, ±2, or ±3 SD — and watch the area under the curve get shaded and reported. Notice how quickly the band swallows almost the entire curve.

🎮 Interactive: the 68–95–99.7 rule LIVE
A standard bell curve marked off in standard deviations. Choose the ±1, ±2, or ±3 SD band and the enclosed area is shaded and labeled with the 68% / 95% / 99.7% proportion it captures. The thin leftover slivers in the tails are all that remain outside ±3 SD.
✨ Why the bell is everywhere
The normal curve is not a lucky coincidence. Whenever a measurement is the sum or average of many small, independent influences — dozens of genes plus diet nudging a height, countless tiny vibrations nudging a scale reading — those influences tend to cancel and pile up into a bell. That is the promise of the Central Limit Theorem: averages become approximately normal even when the raw ingredients are not. It is the deep reason this one shape keeps reappearing across nature, measurement, and data.
\[ z = \frac{x - \mu}{\sigma} \]
The z-score restates any value as 'how many standard deviations from the mean.' It is what lets a single bell — the axis in the tool above — stand in for every normal distribution at once.

Reading proportions off the curve

Because the curve is symmetric, the empirical rule does more than count the middle. Whatever percentage sits inside a band, the rest lives in the two tails — and those tails hold equal shares. For example, 95% inside ±2 SD leaves 5% outside, split into 2.5% in each tail. That single idea turns the rule into a quick way to estimate almost any proportion.

📝 Worked example: Adult heights in a population are approximately normal with mean μ = 175 cm and standard deviation σ = 7 cm. (a) Between which two heights do about 95% of people fall? (b) Roughly what percent of people are taller than 182 cm?
  1. Part (a): 95% corresponds to the ±2 SD band. Two standard deviations is 2 × 7 = 14 cm.
  2. So the middle 95% runs from 175 − 14 = 161 cm up to 175 + 14 = 189 cm.
  3. Part (b): 182 cm is exactly 175 + 7, that is mean + 1 SD. About 68% of people lie within ±1 SD, leaving 32% split between the two tails.
  4. By symmetry that 32% divides evenly, so the upper tail beyond +1 SD holds about 32% ÷ 2 = 16%.
✓ About 95% of people fall between 161 cm and 189 cm, and roughly 16% are taller than 182 cm.
⚠️ Not everything is a bell
The normal distribution is a model, and plenty of real data does not fit it — incomes are strongly right-skewed, wait times pile up near zero, and counts of rare events are not symmetric at all. Before leaning on the 68–95–99.7 rule, sketch a histogram and check that the data actually looks roughly bell-shaped and symmetric. Using the rule on lopsided data will mislead you.

Check your understanding

1. According to the empirical rule, about what percent of values in a normal distribution fall within one standard deviation of the mean?
The 68–95–99.7 rule starts with ±1 SD capturing about 68% of the data.
2. Exam-style scores are approximately normal with mean 500 and standard deviation 100. About 95% of scores fall between which two values?
95% corresponds to ±2 SD. Two SDs is 2 × 100 = 200, so the band is 500 − 200 = 300 to 500 + 200 = 700.
3. In a normal distribution, roughly what percent of values lie MORE than 2 standard deviations ABOVE the mean?
About 95% lie within ±2 SD, leaving 5% in the two tails combined. By symmetry that splits evenly, so about 2.5% sits in the upper tail beyond +2 SD.
4. Two normal curves describe the same measurement. Curve A has a larger standard deviation than Curve B, and both share the same mean. Compared with B, curve A is…
The mean sets location; the standard deviation sets spread. A larger σ spreads the same total area over a wider range, making the bell shorter and wider (same mean, so no left–right shift).
✅ Key takeaways
  • A normal distribution is a symmetric, single-peaked bell curve; its mean, median, and mode all sit together at the center.
  • Just two numbers fix the whole curve: the mean μ sets where the peak is, and the standard deviation σ sets how wide the bell spreads.
  • The empirical rule: about 68% of values fall within ±1 SD of the mean, 95% within ±2 SD, and 99.7% within ±3 SD.
  • Because the curve is symmetric, the leftover area in the tails splits evenly between the two sides.
  • The bell appears so often because averaging many small, independent effects tends to produce it — that is the Central Limit Theorem at work.