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The mean of marks

The arithmetic mean: add, divide by how many — and understand the result.

One number for a whole list

“What is your mean in maths?” You will hear that question many times — on a report, at entrance tests, in sport. The arithmetic mean is one number that stands for a whole pile of numbers.

The recipe is short: add all the numbers and divide by how many there are. Marks 1, 2, 2, 3: sum 8, count 4, mean 8 : 4 = 2.

What does that number mean? If all the values were the same and together they made the same sum, each would equal the mean. The mean is a fair share: four children have 8 sweets together → on average 2 each, even if someone has 3 and someone 1.

Two things we will watch today, because even adults slip on them:

The mean does not have to be a “real” value — 2.4 can come out, even though a mark of 2.4 does not exist.

Everything counts in the count, even zeros. Who saved nothing one day, that zero stays in the mean — it must not be hidden.

Jonáš has maths marks 1, 2, 2, 3 and at home he reports: “I have a mean of two!” Dad nods, but sister Sofie is waiting: “Wait, tomorrow you write a test. What if it is a 5?” Jonáš counts a new version: sum 1 + 2 + 2 + 3 + 5 = 13, count 5, mean 13 : 5 = 2.6. “One five and I dropped from a two almost to a three?!” Sofie nods: “The mean pulls every number, a big one more. And the other way — a one would pull you to 1.8.” Jonáš would rather go practise fractions.

💡

The mean must sit between the smallest and the biggest number in the list. If it comes out outside, there is a mistake in the sum, or in the count.

The recipe, a check and counting backwards

Method:

  1. Add all the values. Carefully — one forgotten value spoils everything.
  2. Count how many values there are. Even zeros, even repeated numbers!
  3. Divide the sum by the count. Happily with a decimal point (lesson five) — a mean of marks 2.6 is a normal result.

Checks:

Counting backwards — the most useful trick of the lesson: mean × count = sum.

You want after five tests a mean of 2.0? You need a sum of 5 × 2 = 10. So far you have 1 + 2 + 2 + 3 = 8 from four tests → the fifth may be at most 10 − 8 = 2.

What to watch with a mean in life: one extreme number can pull the mean. Five friends with pocket money 100 Kč and one with 700 Kč have “average pocket money” of 200 Kč — yet five out of six do not have even half of it. The mean is a useful servant, but it sometimes hides how spread out the numbers are.

The mean of marks 1, 2, 2, 3, 2

Sum: 1 + 2 + 2 + 3 + 2 = 10.

Count: 5 marks.

Mean: 10 : 5 = 2.0.

Check: all the marks are between 1 and 3, the mean 2 sits between them ✓. And the estimate sat — the marks wander around two.

Notice: both twos and the third one each count on their own. The mean does not take “kinds of marks”, but every write-up in the mark book.

The mean temperature of a week (even below zero)

Morning temperatures: −2, 0, 1, −1, 2, 3, 4 °C.

Sum (integers from lesson 25!): −2 + 0 + 1 − 1 + 2 + 3 + 4. Negatives together −3, positives 10 → sum 7.

Count: 7 days. Zero counts — it is a valid measurement!

Mean: 7 : 7 = 1 °C.

The average morning was just above zero, even though two mornings froze. The mean can mix frost with a thaw.

Backwards: what do I need for the report?

Ema has Czech marks 2, 1, 3, 2 and wants a mean of at most 2.0 with five marks.

Needed sum: 5 × 2 = 10.

Sum so far: 2 + 1 + 3 + 2 = 8.

Fifth mark: at most 10 − 8 = 2.

A one or a two will keep the mean (with a one it drops to 1.8), a three will lift it to 2.2. Mean × count = sum — this reverse solves all “what must I get so that…” questions.

⚠️

Do not forget zeros and repeated values. The mean of “0, 0, 6” is 2, not 6 or 3. In the count of values every write-up belongs — even a zero, even a fifth two in a row. Who leaves out “uninteresting” numbers counts a mean from a different list.

Practice: now you do it 🎲

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

Work out the mean: add the numbers and divide by how many there are. The answer is a number.

Practise the questions

On paper

Take real numbers — a mark book, game scores, temperatures from a phone.

  1. Work out the mean of your marks in one subject (or from a made-up list 1, 2, 2, 1, 3). Check that it sits between the best and the worst mark.

  2. Adam threw darts in five throws 20, 0, 15, 25 and 0 points. What is his mean per throw? Do not forget the zeros!

  3. Tereza has marks 2, 2, 3. What mark does she need from the fourth test to have a mean of exactly 2.25? Count backwards through the sum.

  4. Measure (or make up) the heights of family members and work out the mean height. Who is over the mean and who is under it?

Now you 💪

  1. I can work out the mean with the recipe add–divide by the count, including zeros in the list.
  2. I check the result: the mean sits between the smallest and the biggest value.
  3. I use the reverse mean × count = sum on a “what do I need to get” question.
✅

Done when: You work out the arithmetic mean of any list of numbers (even with negatives and zeros), you check it, and you can count backwards from it.

What to take from this lesson

© 2026 Ing. Martin Polak / AlgoRhino · 콘텐츠 이용 약관