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The normal curve

Quetelet's 5,738 chest sizes have mean 39.83 inches and standard deviation 2.05; a normal curve with those two numbers predicts 1,102 men at 40 inches, and the table has 1,079.

Many measured quantities, from heights to test scores to errors in repeated measurements, pile up in the same bell shape: most values near the middle, fewer and fewer toward either side, and the two sides mirror each other. The smooth curve that describes this shape is called the normal curve, and a variable that follows it has a normal distribution. It is the most used model in statistics.

Two numbers fix the curve

Every normal curve is pinned down by two numbers. The mean μ sets where the peak stands, and the standard deviation σ sets how wide the bell is. OpenStax writes X ~ N(μ, σ). Change μ and the whole curve slides along the axis without changing shape. Change σ and the curve stretches or squeezes: a larger σ gives a lower, wider bell, because the total area under every normal curve is exactly 1. The curve is symmetric about μ, so the mean, the median and the mode are all the same number. It bends from curving down to curving up at μ − σ and μ + σ, which is a handy way to spot σ on a drawing.

Drag to turn. Scroll or pinch to zoom.

The normal curve fitted to Quetelet's chests, N(39.83, 2.05). Move the μ slider and the curve slides; move σ and it widens or narrows. Set Shade to "Left of x" and drag on the graph: the shaded area is the proportion of soldiers below x.

Area is proportion

For a histogram drawn with relative frequencies, the share of values in a range is the area of the bars over that range. A normal curve works the same way. The area under the curve between two values is the proportion of the population between them, which is also the probability that one randomly chosen value lands there. The area to the left of μ is exactly 0.5, because of the symmetry. Areas for other ranges come from a table or a calculator; the lab computes them for you and prints them in the panel.

Quetelet's 5,738 chest measurements (bars) and the normal curve with the same mean and standard deviation (red).

BackgroundThe curve was first found as a shortcut. In the 1700s Abraham de Moivre, a mathematician in London who was often asked by gamblers to work out odds, faced sums such as the chance of 60 or more heads in 100 coin flips, which means adding dozens of binomial probabilities by hand. He noticed that as the number of flips grows, the bars of the binomial distribution trace an ever smoother curve, and in 1733 he found a formula for that curve. A century later the Belgian astronomer Adolphe Quetelet took the curve from games and the sky to people. In 1846 he published the chest sizes of 5,738 Scottish militiamen, copied from an 1817 source, and showed that they followed the bell closely. He built his idea of "the average man" on such curves. Later, Stephen Stigler compared Quetelet's table with the original and found copying errors; his corrected table has 168 men at 44 inches, not 92, which closes most of the gap with the curve at that point.

A model, not a law

No real data set is exactly normal. The chests come in whole inches while the curve is smooth, the curve has tails that never quite reach zero while chests have a smallest and largest size, and the counts wobble around the curve. Statisticians call a distribution "approximately normal" when a curve with the same mean and standard deviation fits well enough for the question at hand. Skewed data, such as the restaurant staff counts, are not approximately normal, and normal-curve answers for them can be far off.

Normal distribution	Symmetric bell-shaped distribution N(μ, σ)
μ	Mean: where the peak stands
σ	Standard deviation: how wide the bell is
Total area	Exactly 1
Area between a and b	Proportion (probability) of values between a and b
de Moivre, 1733	Found the curve as an approximation to coin-flip counts

Which number moves the normal curve left or right without changing its shape?
- μ
- σ
- n
- The area
σ doubles. What happens to the peak height?
- Doubles
- Halves
- Stays the same
- Becomes 1
Area under the whole normal curve?
Area to the left of the mean?
Quetelet chests: what fraction measured 39, 40 or 41 inches (1,073 + 1,079 + 934 of 5,738)? Percent, one decimal.
How many soldiers were 37 inches or smaller? (3 + 18 + 81 + 185 + 420)
In which year did de Moivre find the formula for the curve?
- 1654
- 1733
- 1846
- 1954
In a normal distribution the mean, median and mode are...
- All equal
- Mean largest
- Median largest
- Unrelated
Where does the curve change from bending down to bending up?
- At μ
- At μ ± σ
- At μ ± 3σ
- Nowhere
Restaurant staff counts are skewed right. Should you use normal-curve areas for them?
- Yes, always
- No, they are not approximately normal
- Only for the mean
- Only with 3D

Key points

A normal curve is a symmetric bell fixed by μ (centre) and σ (width); total area 1.
Area under the curve between two values = proportion of values between them.
Real data are at best approximately normal; check the shape first.

Sources for this course (7)

OpenStax, Introductory Statistics 2e by Barbara Illowsky and Susan Dean (, per the page notice on 2026-10-17): 2.2 frequency tables, relative and cumulative frequency, frequency polygons (calculus final test scores 5, 10, 30, 40, 15; ages of US presidents at inauguration in six classes)
the 100 soccer player heights, the 50 students' books, the 25 classmates' weekend video game hours and the 22 New York restaurant staff counts (openstax.org/books/introductory-statistics-2e/pages/2-2-histograms-frequency-polygons-and-time-series-graphs)
6.1 the standard normal distribution, z-scores, the empirical rule, X ~ N(5, 6)
heights of Chilean males aged 15 to 18 N(170, 6.28) for 2009 to 2010 and N(172.36, 6.34) for 1984 to 1985, SAT verbal 2012 N(496, 114) with 1,664,479 test takers, Jerome N(16, 4) (openstax.org/books/introductory-statistics-2e/pages/6-1-the-standard-normal-distribution)
De Moivre and the normal approximation to coin flips: D. Lane, Online Statistics Education, 7.2 History of the Normal Distribution (stats.libretexts.org, Introductory Statistics (Lane) 7.2)
EBSCO Research Starters, "De Moivre Describes the Bell-Shaped Curve" (1733)
Quetelet's chest measurements of 5,738 Scottish militiamen (Quetelet 1846, from an 1817 source; Stigler 1986 corrections): R package HistData, ChestSizes and ChestStigler (vincentarelbundock.github.io/Rdatasets/doc/HistData/ChestSizes.html).

Checked: 2026-10-17