Study Numbers Before You Rely on Them
Many people think average is a number that matters. Mistake unless you know about the population of observations. Then does that population match your situation. If it might, the average might have meaning.
The intuitive idea
Average contains many bits of information and so the average must be useful. But no! It doesn’t have the depth people believe it does.
How no depth works
- Let’s take the municipality of Medina Washington. Zipcode 98039. There are a few more than 3,000 people who live there. If you live in a town of 3,000 people would you expect the average net worth or income of people in Medina would be like yours? It is possible, but not likely. Medina is an outlier. It is home to Bill Gates, Jeff Bezos ex-wife Mackenzie Scott, a founder of Costco, some retired professional athletes, and many corporate executives. Their average may not be relevant to what should be true on average in your town. When Jeff Bezos lived there, Bill Gates was the second wealthiest person in the zipcode.
- Okay, maybe too variable. How about the average height of all the people who work for the Los Angeles Lakers. I hear the aha. You got the idea that it would be very tall because it would include a crop of folks who are 6’10.” Good thinking but wrong. You would need to know how many people work for the Lakers. There are about 15 basketball players and they would all be far right outliers on the distribution. There are at least 160 employees in total. From the director of marketing, to the PA announcer, to the season ticket processor. The other 145 are what we think of as normal. If we guess that the mix of males and females is normal in the 145, we could guess their average height at 5’9″” if the 15 players averaged 6’7″ then there will be 150 extra inches arising from the players. so the overall average is likely about 5’10”. The number of people in the population matters too. The tall folks only influence it if there are not many in the total sample.
Okay that makes sense. The quantity of the population and the quality of the factor you are measuring matter.
- Let’s try this one. You are looking at buying a cellphone plan. You do some research and find one that on a scale of 1 to 10, the average satisfaction level is 7. Pretty good right? Maybe. When you look deeper you find there 100 people who have commented and of those 67 rated it 10 and the other 33 rated it 1. The problem now is the distribution is not a bell curve. It is u-shaped. Either you love it or you hate it. You don’t know which side you will fall on. Don’t assume a “normal distribution” – Bell curve.
- Another variation of the cellphone plan idea is this. What is the average number of left arms Canadians have. Not quite 2, is it. Most have one, a few have zero. There is not one person who has the average number of left arms. When there is no example of the average, you should step back and think. U-shaped distributions are a near guess.
- You cannot average averages. Suppose you have a moped that has trouble with hills. You want to drive two miles. One mile uphill, one mile downhill. You want to average 30 miles an hour over the whole journey. You can drive 15 miles per hour uphill. How fast must you drive downhill? I know you would like to say 45 mph, but you know it’s wrong. The answer is you cannot do it at all. Here’s why. Two miles averaging 30 mph takes four minutes. One mile uphill at 15 mph takes four minutes. You reach the top of the hill at the same time you must be at the bottom of the hill. No speed will let you be in two places at once. Don’t average averages.
A normal distribution has an average and a median – the middle value, that are equal. There will be as many observations on each side of the number. The third number is standard deviation. Basically how flat is the curve. If all the observations are close to the average, the standard deviation is small and the curve will be a pointy bell curve. A high standard deviation means the curve is flatter.
If you know all three and the size of the population, you can get an intuitive idea of what the average means.
Be careful with averages. Some people using them to mislead you.
Understand the shape of the distribution.
Understand the size of the population.
Not all small sections of the total population are like the whole thing. That’s what statistics is about. How many random samples would you need to make you believe the sample represents the whole? If you understand the nature of the population, the shape of the distribution, and you can pick randomly, the sample size will be smaller than you think.
When things turn out not as they seem, you can be very wrong.
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