Statistics: mean, median, mode · Lập trình với Martin

Three ways to describe a pile of data with one number — and why the mean is easy to fool.

One number for a whole pile

Ten people in the class wrote a test. How do you describe the result of the whole class with ONE number? Statistics has three candidates for that — and each says something different.

Mean: add everything and divide by how many. You know it from school reports.

Median: order the values from smallest to largest and take the middle one. Half the people are under it, half above it.

Mode: the value that appears most often. No calculating — just looking for the most frequent.

Why three? Because the mean is easy to fool. In a class where nine people get a 1 and one a 5, the mean jumps, even though almost everyone can do it. An extreme value pulls the mean towards itself. The median does not notice extremes.

When you see “average pay” in the news, remember this lesson: a few huge salaries pull the mean up, while the median says what the middle person earns. Both numbers are true — but they tell a different story.

Jonáš reports at home: “The class mean from the test was 2.8 — and I have a 2, I am above average!” Dad asks: “And what was the median?” Jonáš does not know, so the next day he calculates: grades ordered — lots of 1s and 2s, then three 4s and two 5s from those who did not study at all. The middle grade: 2. “The median is 2. So half the class had a 1 or a 2 and a few 5s spoiled the mean.” Dad nods: “That is why you always ask for both numbers. The mean is sensitive.”

💡

The median NEEDS ordered values. An unordered middle is just a random number in the middle of the writing. Order first, only then look for the centre.

How to calculate all three

Data: dice rolls — 3, 1, 4, 1, 5, 1, 2.

Mean:

Add: 3 + 1 + 4 + 1 + 5 + 1 + 2 = 17.
Divide by the number of values: 17 : 7 ≈ 2.43.

Median:

Order: 1, 1, 1, 2, 3, 4, 5.
Find the middle. Seven values → the fourth: 2.

With an EVEN number of values there are two in the middle — the median is their mean. For 1, 2, 4, 6 it is (2 + 4) : 2 = 3.

Mode: the most frequent value. One came up three times → the mode is 1.

Which when? The mean when the data is “even” without extremes (grades, temperatures). The median when a few values fly off (salaries, flat prices, time on a video game). The mode for things you cannot average: the best-selling shoe size is a mode — mean size 41.3 fits nobody.

One extra rule: before you calculate, look through the data with your eyes. A typo (150 instead of 15) ruins the mean just as much as a real extreme.

Example 1: pocket money in a group

Weekly pocket money of six friends: 100, 150, 100, 200, 150, 100 Kč.

Mean: sum 800, divided by 6 → 133.33 Kč.

Median: I order — 100, 100, 100, 150, 150, 200. Even number, in the middle the third and fourth value: (100 + 150) : 2 = 125 Kč.

Mode: 100 Kč (three times).

Three different numbers for the same data — and all correct. The mean is pulled up by those two extra hundreds from the richest.

Example 2: an extreme fools the mean

Five people wait for a bus: 4 students and a lorry driver. Monthly incomes: 0, 0, 0, 0 and 50,000 Kč.

Mean: 50,000 : 5 = 10,000 Kč. “The average waiting person earns ten thousand!” True, but useless.

Median: 0, 0, 0, 0, 50,000 → the middle is 0 Kč. A typical waiting person earns nothing — and that is exactly what the median said.

Remember this mini example whenever you read a headline with a mean. One extreme, five people, and the mean is outside the reality of all five.

Example 3: temperatures over a week

Morning temperatures: 12, 14, 13, 15, 14, 16, 14 °C.

Mean: sum 98, divided by 7 → 14 °C.

Median: ordered 12, 13, 14, 14, 14, 15, 16 → the middle 14 °C.

Mode: 14 °C (three times).

All three numbers the same! When the data is even without extremes, the measures agree — and you can trust the mean. They start to disagree just when an extreme flies into the data.

⚠️

Unordered data has no median. And never take the mean as a “typical value” until you have looked whether there is an extreme in the data — one runaway number completely knocks it.

Practice: now you do it 🎲

Now try it for real. You will see all the questions at once, like on paper. Bắt đầu with ten. When it goes well, add more.

Calculate the mean of the given numbers. Write a decimal result with a point.

Practise the questions

On paper

Statistics is xong on real data. Collect your own.

Calculate the mean, median and mode: 2, 5, 2, 8, 2, 5, 4. Write the working: sum, ordering, counting frequencies.
Invent five numbers whose mean is higher than the median. (Hint: one very big.)
Ask five people from family or class how many minutes a day they spend on videos. Calculate the mean and the median and write which number better describes a “typical” person from your sample.
Record morning temperatures for a week (or find them in the forecast). Calculate all three measures.

Now you 💪

I can calculate the mean, median and mode and I know how they differ.
Before looking for the median I order the data; with an even count I average the two middle ones.
I know an extreme value pulls the mean, but barely touches the median.

✅

Done when: You have a triple of measures calculated on given and your own data, and a run of five correct in both practices.

What to take from this lesson

Mean = sum divided by the count. Sensitive to extremes.
Median = the middle value of ordered data. Extremes do not knock it.
Mode = the most frequent value. The only one that works on shoe sizes too.
Even number of values: the median is the mean of the two middle ones.
With headlines about an “average” always ask what the median is.