Sizu A small firm advertises: “Average salary $115,000.” You are tempted. Then you ask for the five actual salaries and learn that four of the five people earn under $50,000. Nobody lied. They used the mean, and one very large number dragged it far from what most people earn.
Here are the five salaries, in thousands of dollars (invented for this example): 40, 42, 45, 48 and 400.
Step 1: work out the mean
The mean is the “share it out equally” average. Add everything up, then divide by how many there are.
40 + 42 + 45 + 48 + 400 = 575
575 ÷ 5 = 115, so the mean is $115k.
If the firm paid its money out equally, everyone would get $115k. In real life, four people get far less and one gets a lot more.
Step 2: work out the median
The median is the middle value when you line the numbers up in order. Sort them (these already are), then count in from both ends. With five values, the middle one has two on each side:
The median is $45k.
Step 3: see what the big salary did
Take the $400k salary away and compare. The four remaining salaries are 40, 42, 45 and 48.
- Their mean is 175 ÷ 4 = 43.75, which is $43,750.
- Their median is halfway between the middle two (42 and 45): (42 + 45) ÷ 2 = 43.5.
Now put the $400k salary back.
The median moved from 43.5 to 45, a shift of 1.5. The mean jumped from 43.75 to 115, a shift of 71.25. One extreme value hardly touches the median, because the median only cares about the order, not how far out the biggest value sits. The mean cares about every size, so an outlier tugs it hard.
Which should you trust?
Ask what the question is.
- “What does a typical person earn here?” Use the median. Here that is $45k.
- “How much money goes out in total?” Use the mean, because total pay is the mean times the headcount: 115 × 5 = 575.
When a list has a few very big or very small values, the two averages split apart, and the headline tends to pick whichever flatters the writer. Pay, house prices, rents and waiting times are all like this. Take five houses on a street valued at 310, 320, 335, 350 and 1,400 (thousands of dollars). The mean is 2,715 ÷ 5 = 543, but the median is 335. A buyer who sees “average value $543k” will be disappointed by four of the five houses.
A good habit: whenever you see “the average”, ask which one, and ask for the median as well. If the two are close, the list is fairly even. If they are far apart, a few extreme values are doing the pulling.
And the mode?
There is a third average, the mode, which is the value that turns up most often. In this list every salary is different, so there is no mode, and it would not help. It is more useful for things like shoe sizes or the most popular order, where you want the commonest item rather than a middle one.
Questions to ask next time
When someone quotes an average pay, rent or price, a few questions cut through it:
- Is that the mean or the median?
- How many people (or houses) is it based on? With five numbers, one outlier decides everything. With five thousand it matters less.
- Is there one very high or very low value? A quick look at the sorted list answers that.
- What would the average be without the biggest value? Here it falls from $115k to $43.75k, a sign that one number was doing most of the work.
A rule I find handy: if the mean is well above the median, a few big values are pulling it up, and if it is well below, a few small values are dragging it down. Pay and prices usually fall in the first group, because there is a floor at zero but no ceiling on the top.
None of this means the mean is wrong. In the firm above, $115k really is the total pay divided by the staff. It just answers a different question from “what will I earn?”.
A quick way to spot the gap
You do not always have the whole list. If someone gives you only the mean, ask for the range as well: the lowest and highest values. A range of 40 to 400 next to a mean of 115 tells you at once that the mean sits far from most of the numbers, because the top value is ten times the bottom one (400 ÷ 40 = 10).
Try it with your own numbers
The free Mean, Median and Mode calculator takes a pasted list and gives all of them at once, so you can see the gap for yourself. The same idea runs through Statistics Didn’t Have to Be This Hard, which explains which average to trust and when.
Rule of thumb: mean = total shared out equally, and it chases extreme values. Median = the middle one, and it stays put. If a few big values could skew a list, look at the median.
This is arithmetic for reading numbers, not pay or financial advice. Salaries in this example are invented.