The normal distribution.

Step 1 of 5

A histogram

When there are many readings, it is hard to look at each one. A histogram splits the values into bins of equal width, and over each bin stands a bar whose height is the number of values that fell in it. At a glance you see where most of the data are, how far they spread, and whether they have a tail to one side.

A Galton board builds a histogram by itself: each ball bounces $10$⁠ times, and here only $25\%$⁠ of the time to the right. A bin’s number is how many times the ball went right. The amber bars are the expected count in each bin, and the blue curve is the bell curve that approximates them.

The histogram leans left, with a longer tail to the right. Find its mode.

Step 2 of 5

The width of the bell

In the probability Unit you saw that the board’s mean is $n \cdot p$⁠: the number of rows times the chance of bouncing right. The standard deviation has a formula too:

$\sigma = \sqrt{n \cdot p \cdot (1 - p)}$$\sigma =$$\sqrt{n \cdot p \cdot (1 - p)}$⁠

On a fair board, $p = 0.5$⁠, that is $\sqrt{\frac{n}{4}}$⁠. The Greek letter $\sigma$⁠ (sigma) is the standard deviation of a whole distribution, as $\mu$⁠ (mu) is its mean.

There are $8$⁠ rows now. Change the number of rows.

Step 3 of 5

One standard deviation

With many rows, the histogram is almost the same as the smooth bell curve. A curve like this is called a normal distribution, and it turns up wherever many small, independent effects add up: people’s heights, measuring errors, the weight of fruit. Every normal distribution has the same shape, and only $\mu$⁠ and $\sigma$⁠ change: where the centre is and how wide it is.

So one question answers for all of them. Here $\mu = 8$⁠ and $\sigma = 2$⁠.

Step 4 of 5

Two standard deviations

About $68\%$⁠ within one standard deviation: about two thirds. And if the range widens to two standard deviations each side?

Step 5 of 5

Real data

These are the three numbers of the $68$⁠–$95$⁠–$99.7$⁠ rule: in a normal distribution, about $68\%$⁠ of the data lie within $\mu \pm \sigma$⁠, $95\%$⁠ within $\mu \pm 2\sigma$⁠, and $99.7\%$⁠ within $\mu \pm 3\sigma$⁠. A value farther out than three standard deviations is very rare.

The rule works for every normal distribution, so knowing the mean and the standard deviation tells you a lot. Try it on people’s heights.