Why do statisticians
divide by n−1?
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The full narration of the video.
Same five students' heights, two calculator keys, two different answers. One divides by five, the other by four. The textbook says: if you only measured some people, divide by four. Why one fewer?
Say a school of two thousand: how much do their heights vary? You only measure five. To measure spread, first find the middle: the average height of the five. Then see how far each person is from it, and average those distances.
Here's the catch: the true middle is the average of all two thousand. And you don't know it. All you have is the average of five, so that becomes your middle. Try an extreme case: measure just one person.
One person's average is himself, zero centimeters from the middle. By that math, the whole school is the same height. Clearly wrong. So where's the flaw? Picture an archer.
He shoots first. Then he walks over and paints the bullseye right in the middle of his arrows. Now every arrow looks close, like he's a sharpshooter. But the real bullseye is elsewhere, and the arrows are much farther from it.
The average of your five is that painted bullseye. It always lands dead center among them, so they look tightly packed. Put the center anywhere else, the real one included, and the spread only grows.
So how much too low? Take a guess. Roll a die just twice. Divide by two to estimate how spread out its numbers are. Do that over and over, and on average you get only half the true spread.
Divide by one instead, and on average it's spot on. Same with five people: divide by five and you average eighty percent. Divide by four and it's just right. Why exactly one fewer? Because the painted bullseye wobbles too.
These five average 168. Pick five others, maybe 172. That wobble, on average, eats exactly one person's worth of spread. Another way to see it: once the average is set, give me four heights. I can work out the fifth without measuring.
Of five people, only four are free to vary. The average stole one. So you divide by four. This trick is called Bessel's correction. Two times you skip it. Measure the whole school and you hold the real bullseye, so just divide by two thousand.
And with lots of people, one fewer barely matters. It isn't right every time; it just isn't off on average. Shoot first, paint later: it's not just a math thing.
Test yourself on problems you've already done, and your score runs high too. Where else have you seen someone shoot first and paint the target later?
Two keys on
the calculator
Same five students’ heights, two calculator keys, two different answers: one divides by five, the other by four.
The textbook says: if you only measured some of the people, use the one that divides by four.
Calculators often label them σx and sx; spreadsheets call them STDEV.P and STDEV.S.
What we want:
how much heights vary
A school has two thousand students. No time to measure them all, so you pick five.
First find the middle: the average height of the five. Then see how far each person is from it and average those distances. That’s the “spread.”
Strictly, each distance is squared before averaging, so people far away count more.
The catch: the true middle is the average of all two thousand. You don’t know it, so the average of five stands in.
Measure one person,
everyone’s the same?
Go extreme: measure just one person. He is his own average, zero centimeters from the middle.
Divide by n and the whole school is the same height. Clearly wrong.
With n−1, one person means dividing by zero: no answer. That’s honest. One person can’t tell you how much people differ.
Shoot first,
paint the target later
An archer shoots first, then walks over and paints a bullseye right in the middle of his arrows. Every arrow looks close, like he’s a sharpshooter. But the real bullseye is somewhere else.
The average of your five is that painted bullseye.
It always lands dead center among those five. Move the center anywhere else, the true middle included, and they only look more spread out. So dividing by n comes out too small.
Roll a die twice,
miss half
Same with five people: divide by five and you average 80% of the truth; divide by four and it’s just right. In general, dividing by n gives (n−1)/n on average.
The average
stole one person
So you divide by four.
That’s Bessel’s correction.
Named after the German astronomer Friedrich Bessel, though Gauss used the same divisor in 1823.
When you
don’t need it
Where else have you seen someone shoot first and paint the target later?
Quick check
Why does dividing by the number of people make the spread come out too small?
Like painting the bullseye after shooting: drawn in the middle of the arrows, every arrow looks close.
