The Normal Distribution, Explained Simply

Full transcript

Word for word what is said in the video — the same text as the subtitles. Times are minutes:seconds into the video.

0:00Intro ▶

Welcome to Clearstistics. This is the normal distribution, the most famous shape in all of statistics. Let's learn to read it. We start with an experiment.

0:16250 people, one chart ▶

Suppose we stop 250 random people on the street and measure how tall they are. Every dot on this chart is one person, placed above their height in centimetres. Watch what happens as the dots pile up. Most people land somewhere in the middle, around 170 centimetres. That's where the colours run darkest. Only a few are very short or very tall. The pile grows into a hill, high in the middle, sloping down on both sides. And this shape isn't special to heights. You'll find it in birth weights, in measurement errors, in test scores. It shows up everywhere in nature. If we draw a smooth line over our

1:06The bell curve and the mean μ ▶

hill of dots, we get this elegant curve. It's called the bell curve. This is the picture of the normal distribution. Its highest point sits exactly at the average. In statistics, the average, the mean, is written with the Greek letter μ, pronounced as μ. Here, the mean μ is 170 centimeters. Now, the centre can actually mean three different things.

1:39Mean = median = mode ▶

The mean, the average we just met. The median, line everyone up by height and take the person exactly in the middle. And the mode, the height you see most often. In most datasets, those three are different numbers. But in a perfect normal distribution, something elegant happens. Mean, median and mode all land on the exact same spot, right at the centre of the bell. The second number we need tells us

2:20The standard deviation σ ▶

how spread out the heights are. Are most people huddled close to the mean or scattered far away from it? That number is the standard deviation, written with the Greek letter sigma. Think of sigma as the typical distance from the average. In our group, sigma is 8 centimetres.

2:48The 68–95–99.7 rule ▶

And now the magic. For every normal distribution, heights, weights, test scores, anything, the same rule holds. Take the region within one sigma of the mean, from 162 to 178 centimeters. That dark blue slice contains 68 percent of all people. Widen it to two sigma on each side and you capture 95 percent. Go out to three sigma and you've covered 99.7 percent, almost everyone. This is the 68–95–99.7 rule. Know just two numbers, mu and sigma. And you know where nearly everyone will be.

3:45What μ and σ actually do ▶

So what do those two numbers actually do? Watch the mean first, the orange line. When mu changes, the whole curve slides left and right, but its shape stays exactly the same. Mu sets the position. Now watch sigma. The green arrow. A small sigma pulls the curve in, narrow and tall. Everyone is close to average. A large sigma stretches it out, wide and flat. Much more variety. Sigma sets the spread. One last question.

4:27Does more data change σ? ▶

What happens if we measure more people, not 250, but thousands? Here is a common misconception. The bell does not get narrower. Sigma is a fact about people and it stays around 8 centimetres, no matter how many we measure. What does shrink is our uncertainty about the average? Look at the zoomed-in box. That uncertainty narrows like sigma divided by the square root of n, the sample size. Four times the data makes your estimate of mu twice as precise. So, in one picture, the bell curve is symmetric

5:15Everything in one picture ▶

around its centre, where mean, median and mode all coincide at mu. Sigma measures the spread. 68, 95 and 99.7% of everything falls within one, two and three sigma. And more data sharpens your estimate of the average, but never changes the spread itself. That's the normal distribution.

5:51What's next ▶

In the next video, we'll use it to answer real questions like how special is someone who is two metres tall? Until then, happy learning!