You get off work at 5, and you hit the gym on the way home. It’s always packed. You wait for a bench, and every treadmill is taken.
Your gym’s website says the average crowd is just 35 people. To get that, the gym counts how many people are inside during every hour it’s open, from 5 a.m. to 11 p.m. That number is the hour’s crowd.
One day at the gym
Each bar is one hour, in order through the day, and its height is that hour’s crowd. Everyone’s workout fits inside one hour, so no one shows up in two bars.
The gym’s average counts each hour once. Over its 18 hours, the crowds add up to 630 people, and 630 ÷ 18 is 35.0. The dashed line marks that average. Most hours fall below it: 12 of the 18.
Now ask the people instead. You’re one of them. Ask each of the 630 people how big the crowd was when they worked out, and average their 630 answers. What do you get?
You chose about 35.
That’s too low. Each hour counts once in the gym’s 35, but the people’s answers don’t work that way. The busy hours have more people to ask, so they give more of the answers.
You chose more than 35.
That’s right. The busy hours have more people to ask, so they give more of the answers.
You said you’re not sure.
That’s okay. The 630 answers average more than 35. The busy hours have more people to ask, so they give more of the answers.
Two ways to count
Here’s the same day, with the dashed line for the average and a strip under the bars. The strip has a piece for each hour, and the wider the piece, the more that hour counts.
Watch the strip and the average line, and press Count each person.
The busy hours’ pieces of the strip got wider, and the average line moved, but the bars didn’t. Nothing about the day changed, only how much each hour counts. Counting each person gives an average crowd of 60.0, not 35.0.
At 6 p.m., 115 people each say 115. Their answers add up to 13,225. At 10 p.m., 5 people each say 5, for just 25. All 630 answers add up to 37,800. Divide by 630, and you get 60.0.
That’s a weighted mean. A weighted mean is a mean in which some numbers count more than others. In the people’s average, each hour’s crowd counts once for each person who was there.
What it adds up to
So why is the gym always packed when you go? You go after work, the busiest time of day. And most people come at busy times: two-thirds of them work out in the 6 busiest hours.
That’s why the two averages disagree, and both are right. The gym’s 35.0 describes its hours. The people’s 60.0, a weighted mean, describes the crowds they find.
Before you trust an average, ask what it counts: the hours or the people.