MISSION · Data and probability
Mean, Median, and Mode
Three different answers to "what is a typical value?"
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LEVEL 2 · LEARN
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The idea in one sentence
“Average” isn’t one thing — it’s three, and they disagree on purpose.
The three measures
Mean. Add everything, divide by how many. It’s the fair-share value: if you pooled all the numbers and split them evenly, everyone would get the mean.
\[\text{mean} = \frac{\text{sum of the values}}{\text{how many values}}\]Median. Line the values up in order and take the one in the middle. With an even count, take the mean of the two middle ones. It’s the value that splits the group in half — as many above as below.
Mode. The value that shows up most often. A set can have two modes, or none at all. It’s the only one of the three that works on things that aren’t numbers, like the most common shoe size sold or the most frequent answer on a survey.
Why the mean and median disagree
Take five houses on a street worth $200k, $220k, $240k, $250k, and $260k. The mean and the median are both around $234k. Now a mansion worth $5 million is built at the end. The median barely moves — it’s still a quarter-million house. The mean jumps past $1 million, a figure no house on that street costs.
That’s the whole difference. The mean counts every value’s size, so one extreme number drags it. The median counts every value’s position, so one extreme number moves it by one place at most.
Which one to use
- Mean when the values are bunched together and you want every value to count — test scores, heights, daily temperatures.
- Median when a few extreme values would mislead — incomes, house prices, response times. This is why news reports say “median income,” not “mean income.”
- Mode when you want the most common category rather than a middle — the most popular size, colour, or choice.
Common mix-ups
- Forgetting to sort before finding the median. The middle of an unsorted list is just a random number.
- Dividing by the wrong count for the mean. Divide by how many values there are, not by the largest one.
- Assuming the mean is one of the values. The mean number of children per family can be 2.3; no family has 2.3 children.
- Treating “no repeats” as mode zero. If nothing repeats, there is no mode. That’s not the same as a mode of 0.
Check yourself
Find the mean, median, and mode of 4, 7, 7, 9, 13.
Mean: \(\frac{40}{5} = 8\). Median: the middle of the sorted list is 7. Mode: 7, because it appears twice.
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