Choose an Honest Average: Mean vs Median

You can now add values into a trustworthy total and read each part’s share of it. This chapter tackles the quantity people reach for most and misuse most often: the average. The word feels safe, one number that stands in for a whole group, but there is more than one kind of average, and choosing the wrong one can quietly bend the truth. Your job here is to compute both of the everyday averages and, more importantly, to know which one is honest for the data in front of you.

What “Average” Really Means

When someone says “on average,” they mean a single value that represents a typical value in a group. But “typical” can be measured in more than one way, and the two you will meet constantly are the mean and the median.

Both are honest attempts to answer the same question, “what is normal here?”, and much of the time they land close together. A club where everyone paid between 20 and 40 euros will have a mean and a median only a euro or two apart. The trouble starts when the values are lopsided, and that is where you have to choose.

Learn to Spot Skew and Outliers

Two words explain almost every case where the mean and the median part ways.

Here is the key fact that drives the whole chapter. The mean adds every value, so a single extreme number pulls it toward that extreme; the median only cares about position, so one runaway value barely moves it. That difference in how they react is exactly what makes one of them honest and the other misleading when the data is lopsided.

Because using every value is a strength only when every value is representative. When one value is an outlier, letting it count fully drags the mean somewhere no ordinary member actually is. The median resists that pull, so it keeps pointing at where the bulk of the group truly sits. Using all the data faithfully reports the outlier too; sometimes that is the last thing you want a “typical” figure to do.

Compare the Two at a Glance

Keep this comparison in view whenever you have to pick an average.

Notice the median is not automatically the winner. If you are budgeting for a total cost, the one big spender is real money you must account for, and a middle value that ignores them will leave you short. Honesty depends on the question, not on a rule that one average always beats the other.

See It in Action: A Book Club’s Yearly Spending

A small book club of nine members tracks how much each person spent on books over the year. Here are the amounts, in euros, already sorted from smallest to largest.

18, 22, 25, 28, 30, 33, 35, 40, 210

Eight members spent between 18 and 40 euros. One member, hunting a rare collector’s edition, spent 210. Now compute both averages.

The mean: add the nine values, 18 + 22 + 25 + 28 + 30 + 33 + 35 + 40 + 210 = 441, then divide by 9, which gives 49.

The median: the values are already sorted, and with nine of them the middle one is the fifth. Counting in, the fifth value is 30, so the median is 30.

Now read the two results side by side. The mean says the typical member spent 49 euros, yet eight of the nine members spent 40 or less. Not one ordinary member is anywhere near 49; the figure exists only because the single 210 dragged it upward. The median, 30, sits right in the middle of what people actually spent. If you had to tell a newcomer what a normal year of book spending looks like in this club, 30 is the honest answer and 49 is the misleading one.

Watch Out for Averaging Averages

There is one more trap worth naming now, because it hides inside innocent-looking sums. It is tempting, when you already have the average of two groups, to average those two averages to get an overall figure. That shortcut is usually wrong when the groups are different sizes.

Imagine two reading circles. Circle A has 2 members who averaged 10 euros; Circle B has 8 members who averaged 40 euros. Averaging the averages gives (10 + 40) ÷ 2 = 25. But the true overall mean weighs each member equally: (2 × 10 + 8 × 40) ÷ 10 = 340 ÷ 10 = 34, not 25. The naive shortcut silently pretended the small circle counted as much as the large one.

Choose the Honest One: A Simple Rule

Put the pieces together into a habit you can reuse on any dataset.

• Compute both the mean and the median; it takes seconds once the data is sorted.

• If they are close, either one is fine, and the mean is the familiar choice.

• If they are far apart, look at why: a skew or an outlier is almost always the cause, and the median is usually the more honest “typical” figure.

• When you are unsure, report both and say which one better represents a typical value, and why.

Treat the last point as your safety net rather than a failure. Prefer-the-median-when-skewed is a reliable rule of thumb, not an unbreakable law, so inspecting the data and, when in doubt, showing both numbers is what keeps you honest.

Your Turn!

Seven flats in a small building report last month’s water bill, in euros. One flat had a hidden leak that month.

20, 24, 25, 27, 29, 31, 96

Do the following, writing your answers down before you check:

1. Compute the mean of the seven bills.

2. Compute the median.

3. Decide which of the two is the honest “typical” bill for a flat in this building, and explain in one sentence why the other one misleads.

Then check your answer against the model solution found at the end of this chapter.

Let’s Recap!

• An average is one number meant to represent a typical value, and the two everyday kinds are the mean and the median.

• The mean adds every value and so is pulled by extremes, while the median takes the middle value and mostly ignores them.

• When data is skewed or holds an outlier, the median is usually the honest choice and the mean can mislead even when it is calculated perfectly.

• Preferring the median under skew is a rule of thumb, not a law, so when in doubt report both and say which better represents a typical value.

• You cannot fairly average two averages when the groups behind them are different sizes.

You can now pick an average that tells the truth about your data. Next you will handle the quantity that trips people up more than any other, the percentage, and learn to compute change over time without falling for the wrong-base error.

Model Solution

1. The mean: 20 + 24 + 25 + 27 + 29 + 31 + 96 = 252, divided by 7, gives 36.

2. The median: the bills are already sorted, and with seven values the middle one is the fourth, which is 27. So the median is 27.

3. The honest typical bill is the median, 27 euros. Six of the seven flats paid between 20 and 31, so 27 sits right among them. The mean of 36 misleads because the single leaking flat’s 96-euro bill, an outlier, dragged it upward to a figure no ordinary flat actually paid. (If the building instead wanted the total to budget for, it would use the sum, 252, not an average, because the leak is real money that has to be covered.)