Mean vs. Median Salary: Why was mean ever used?

In a previous post, I showed how the mean salary and median salary can be very different.

The median is much better than the arithmetic mean for giving a “typical" annual salary; median is the method that we favor in our PayScale salary survey. In fact, the median is better for characterizing “typical” in almost any data set. So then why and how did the mean become the standard for “average” or “typical?”

A Mean Mistake

A historical accident caused the mean to be used for typical or average. Before the first personal computers were introduced, it was much easier to calculate the mean than the median. Scientists in the 1800's, when statistics was being developed, like today, were lazy (I speak from experience). Hence they settled on the easier, but less accurate, way of computing typical values.

Why is the mean so much easier to calculate with pencil and paper? Consider the following example. Say, you find a job, are paid on Friday, and open a new checking account with $500. Over the next week, you write 15 checks. By the next Friday, you have $50 left.

Online Banking Checking Account

I know it is very old school to talk about a checking account, using a personal check and paying for things with money that you actually have already earned.

For the youngsters in the audience, replace "open a new online banking checking account with $500" with "get a new credit card with a $500 limit," "write 15 checks" with "use the credit card 15 times," and "have $50 left" with "have $50 worth of credit card limit left" :-)

At the end of the week, you would like to know how you are spending your money.

For example, what is the size of the “typical” personal check that you wrote? To calculate the mean check size, all you have to do is divide the $450 you spent ($500 to start - $50 left) by 15 checks. The mean check size is $30.

Finding the Median Salary or Median Check

Since you have been reading my posts, you know the mean can be misleading. Just like the median salary is more typical than than the mean salary, the median check size is more typical than the mean. To calculate the median, you have to:

1. Look through the 15 checks and sort them into an ordered list from smallest personal check to largest.
2. Count to the 8th from the bottom. Since 7 checks are below and 7 above the 8th, it is the median check. (The same method using salaries is used to determine a median salary.)

This is harder than calculating the mean, but not a lot. Let's go a step further in difficulty: what is the typical number of checks you wrote per day?

For the mean number of checks per day, the calculation is simple: 15 checks/7 days equals 2.14 checks per day. For the median number of checks per day, you need to:

1. Count the number of checks written on each day (e.g., Saturday you wrote 3 checks, Sunday 0 checks, Monday 4 checks, etc.).
2. Sort the counts of checks per day into an ordered list from fewest to most.
3. Count up 4 from the bottom to find the median number per day (4 is the middle of 7).

With seven days and 15 checks, this calculation likely will take 5 minutes or so to do. In the next post, I'll look at calculating this for a whole year of 365 days and ~750 checks. For 15 checks, scientists just look lazy using the mean instead of median.

Cheers,

Dr. Al Lee

p.s., By the way, this example can show why the mean can be misleading. Say you paid your rent on Monday with a personal check of $408, and the other 14 purchases were your twice-a-day Starbucks latte habit. There were 14 checks for $3 and one check for $408, so the median check was $3.

If your concern is that you are spending too much money, the problem is not because you are writing too many $30 checks, the conclusion from mean $/check. You only actually wrote checks that are 10X small or larger than $30. You either need to cut back on the latte habit, apply for another credit card, or find a cheaper apartment. :-)

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[sylvia scott]

I m trying to teach teens about salary negotiation, is there a standard percentage that your benefits represent as a part of your salary??

[Sonia]

Al, This is the best math lesson I've ever had. I love that using mean was basically a historical efficiency and is completely baseless. Here's a question though - what's the OED deliniation of the concepts of mean, average and median?