The Normal Distribution

Bars that settle into a shape

Toss a fair coin 6 times and count the heads. Of the $2^6=64$2⁶ = 64 equally likely sequences of heads and tails, 1 has no heads, 6 have one, 15 have two, 20 have three, 15 have four, 6 have five and 1 has six. These are the binomial probabilities of B(6, ½) in 64ths.

The bars rise to a single peak in the middle and fall away in the same way on both sides. Toss the coin more times and the bars become narrower and more numerous, and their tops trace out a smooth curve shaped like a bell. A binomial with p far from ½ starts out lopsided, but it settles into the same bell once n is large enough.

The number of heads in 6 tosses of a fair coin, counted out of 64 equally likely sequences: one peak at 3, and the same fall on both sides.

The bars are B(n, 0.15), and the curve is the normal curve with the same mean and standard deviation. At n = 8 the mean is np = 1.2 and the bars are pressed against 0. Drag n up: once np passes 5, at n = 34, the bars have spread out to follow the bell.

The normal curve

That bell is the normal curve. It describes many measured quantities: the masses of apples from one orchard, the heights of adults, the errors in repeated measurements. A normal distribution has two numbers, its mean $\mu$μ and its standard deviation $\sigma$σ, and is written X ~ $N\left(\mu ,{\sigma }^2\right)$N(μ, σ²). The second number is the variance, so $N\left(150,10^2\right)$N(150, 10²) has standard deviation 10.

The curve is symmetric about the mean, and its single peak is at the mean, so the mean, the median and the mode are the same value. Moving away from the mean on either side, the curve falls, first steeply and then more and more slowly, and comes closer and closer to the axis without ever reaching it.

The standard deviation can be seen on the curve. Near the peak the curve bends downward, and further out it bends upward as it flattens toward the axis. It changes from one to the other exactly one standard deviation from the mean, at $\mu -\sigma$μ − σ and $\mu +\sigma$μ + σ.

The mean places it, sigma sets its width

Changing $\mu$μ slides the whole curve along the axis without changing its shape. Changing $\sigma$σ makes it wider or narrower. The total area under the curve is always 1, so a wider curve must also be lower: doubling $\sigma$σ halves the height of the peak.

Two normal curves with mean 0. The gold one has $\sigma =1$σ = 1 and a peak 0.40 high. The plain one has $\sigma =2$σ = 2: it is twice as wide and half as high, with a peak 0.20 high, so both enclose an area of 1.

Within one sigma: about 68%

Every normal curve has the same shape, so the same share of its area always lies within a given number of standard deviations of the mean. Within one standard deviation either side, from $\mu -\sigma$μ − σ to $\mu +\sigma$μ + σ, lies about 68% of the area. More exactly, it is 68.27%.

Take apples whose masses follow $N\left(150,10^2\right)$N(150, 10²), in grams. One standard deviation either side of the mean is 140 g to 160 g, so about 68% of the apples weigh between 140 g and 160 g. By symmetry, half of that, about 34%, weigh between 150 g and 160 g, and the 32% left over is split between the two tails: about 16% heavier than 160 g and 16% lighter than 140 g.

The masses of the apples, $N\left(150,10^2\right)$N(150, 10²). The shaded band, from 140 g to 160 g, is one standard deviation either side of the mean, and holds about 68% of the area.

Within two sigmas: about 95%

Widen the band to two standard deviations either side, from $\mu -2\sigma$μ − 2σ to $\mu +2\sigma$μ + 2σ, and it holds about 95% of the area, more exactly 95.45%. For the apples that is 130 g to 170 g.

About 5% lies outside, split between the tails, so about 2.5% of the apples are heavier than 170 g and about 2.5% are lighter than 130 g.

The band from 130 g to 170 g, two standard deviations either side of the mean, holds about 95% of the area. The unshaded tails hold about 2.5% each.

Within three sigmas: about 99.7%

Three standard deviations either side, from $\mu -3\sigma$μ − 3σ to $\mu +3\sigma$μ + 3σ, hold about 99.7% of the area, more exactly 99.73%. For the apples that is 120 g to 180 g. Only about 0.3% lies outside, 0.15% in each tail: about 1 apple in 740 is lighter than 120 g.

This is why a value more than three standard deviations from the mean is rare enough to be worth checking: it may be a mistake in the measurement, or a sign that the model does not fit.

Reading the rule in parts

The three percentages, with the symmetry, give many other areas. Between 150 g and 170 g, the mean to two standard deviations above it, lies half of 95%, which is 47.5%. Between 160 g and 170 g, one to two standard deviations above, lies (95% − 68%) ÷ 2 = 13.5%.

Two slips are common. The 68% is the band on both sides of the mean together, not on each side; on one side it is 34%. And 50% is not one of the bands at all: it is the half of the curve on one side of the mean.

The rule gives only these few areas, and only roughly. Any other area, such as the share of apples heavier than 155 g, is found by standardizing the mass and reading a table.