What you'll learn
The mean and the mode are two different ways of describing a whole set of numbers with a single figure. The mean is the average: add all the values and divide by how many there are. The mode is the value that appears most often. This topic covers finding both, working backwards from a mean to a total or a missing value, and knowing which measure describes a set of data more fairly. Statistics is worth 6 of the 40 items on the SEA Mathematics paper, and mean and mode questions are among the most predictable marks on it — the method never changes, so the marks go to whoever applies it carefully.
Key terms and definitions
Data — a collection of values, such as a set of test scores or the masses of five pupils.
Mean — the average. Found by adding all the values and dividing by how many values there are.
Mode — the value that occurs most often in a set of data.
Total (sum) — all the values added together.
Frequency — how many times a particular value appears.
Range — the difference between the largest and the smallest value. Useful for describing how spread out the data is.
Outlier — a value much larger or much smaller than the rest, which can pull the mean away from the middle.
Core concepts
Finding the mean
The mean takes everything the set contains and shares it out evenly. There are always exactly two steps.
For the data 4, 8, 6 and 10:
- Add: 4 + 8 + 6 + 10 = 28.
- Divide by how many values there are: 28 ÷ 4 = 7.
Stopping after the addition is the single most common error in this topic. The total is 28; the mean is 7. They are different numbers answering different questions.
Note that the mean does not have to be one of the values in the set. Here, 7 does not appear in the data at all, and that is perfectly normal.
Finding the mode
The mode is simply the value that turns up most often. Count how many times each value appears and pick the winner.
For 12, 15, 12, 18 and 20: the value 12 appears twice and everything else appears once, so the mode is 12.
Three things about the mode surprise children:
- The mode can be the smallest number in the set. In 5, 5, 6, 8, 9 the mode is 5. Being most common has nothing to do with being large.
- A set can have two modes. In 4, 4, 6, 8, 8 both 4 and 8 appear twice, so there are two modes.
- A set can have no mode at all. In 2, 4, 6, 8 every value appears once, so no value is more common than any other.
Working backwards from the mean
If you know the mean and how many values there are, you can find the total by multiplying:
total = mean × number of values
Five pupils have a mean mass of 32 kg, so the total mass is 32 × 5 = 160 kg.
This is the same rule as finding the mean, used in reverse, and it unlocks the harder questions. If four numbers have a mean of 10, their total must be 40. If three of them are 8, 9 and 12, which add to 29, then the missing number is 40 − 29 = 11.
What happens when a new value is added
Adding a value that is equal to the mean leaves the mean unchanged. Adding a value above the mean pulls it up; adding one below pulls it down.
Kern's five test scores are 6, 7, 8, 9 and 10, which total 40, so his mean is 8. He then scores 14. The new total is 54 and there are now 6 scores, so the new mean is 54 ÷ 6 = 9. The high score pulled the mean up by one.
Mean or mode — which describes the data better?
Usually the mean, because it uses every value. But a single extreme value can drag it away from the middle.
Take 3, 5, 5, 7 and 20. The total is 40, so the mean is 8 — yet four of the five values are below 8. The mode is 5, which sits much closer to most of the data. The 20 is an outlier, and it is what pulls the mean upwards.
The mode is also the only sensible measure when the data is not numbers at all. You cannot take the mean of shoe colours or favourite fruits, but you can say which was most common.
Worked examples
Example 1: Mean and mode of the same set
Anisa scored 7, 8, 9, 8 and 8 in five spelling tests. Find the mean and the mode of her scores.
Mean: add first, 7 + 8 + 9 + 8 + 8 = 40. Then divide by 5: 40 ÷ 5 = 8.
Mode: the score 8 appears three times, more than any other. The mode is 8.
Here the mean and the mode happen to be the same number, which does not always happen and is worth noticing rather than assuming.
Example 2: Finding a missing value from the mean
Four numbers have a mean of 10. Three of them are 8, 9 and 12. What is the fourth number?
Work out what the four must total: 10 × 4 = 40.
Add the three you have: 8 + 9 + 12 = 29.
The missing number is 40 − 29 = 11.
Check it: 8 + 9 + 12 + 11 = 40, and 40 ÷ 4 = 10. The mean is right, so the answer is right. Checking backwards like this takes ten seconds and catches almost every arithmetic slip.
Example 3: When an outlier pulls the mean
For the data 3, 5, 5, 7 and 20, find the mean and the mode, and say which better describes the set.
Mean: 3 + 5 + 5 + 7 + 20 = 40, and 40 ÷ 5 = 8.
Mode: 5 appears twice and everything else once, so the mode is 5.
The mode describes the set better here. Four of the five values are 7 or below, so a mean of 8 sits above almost all the data. The single value of 20 is an outlier, and it has pulled the mean up by 3 above the mode.
Common mistakes and how to avoid them
Giving the total instead of the mean. Adding 4, 8, 6 and 10 to get 28 and stopping. Fix: the word mean always requires two steps — add, then divide.
Dividing by the wrong number. Dividing a total of 60 by 4 when there are 5 values. Fix: count the values before you divide, and write that count down.
Choosing the largest value as the mode. Picking 20 in 12, 15, 12, 18, 20. Fix: the mode is about how often, not how big. Count the appearances.
Assuming every set has exactly one mode. Fix: two values can tie, and a set where everything appears once has no mode at all.
Dividing when you should multiply. Finding the total from a mean of 32 across 5 pupils by working out 32 ÷ 5. Fix: total = mean × number of values. Check whether your answer is sensible — 6.4 kg cannot be the mass of five pupils.
Expecting the mean to be one of the values. Fix: it usually is not, and that is fine.
How parents can help at home
Mean and mode are easy to practise in conversation, because any small set of numbers will do.
Use scores and times. Test marks, cricket scores, minutes spent reading over a week. Ask "what was the average?" and then the harder question, "did anything unusual pull it up or down?"
Ask the two-step question out loud. When your child gives an answer, ask "did you add, and then divide?" Most lost marks in this topic are a total given where a mean was wanted, and that one question catches it every time.
Practise the backwards version. "If the average of four numbers is 10, what do they add up to?" This is the step that turns a routine question into a hard one, and it is worth rehearsing until multiplying feels obvious.
Talk about the odd number in a set. If one week your child reads for 10 minutes a day and then 90 minutes on Saturday, work out the mean together and ask whether it really describes the week. That conversation teaches outliers better than any definition.
Exam technique for mean and mode
Write down the number of values before you divide. Most mean errors are division by the wrong count, and having the number on the page prevents it.
Check a mean by looking at where it sits. The mean must always fall between the smallest and the largest value in the set. If it does not, something has gone wrong, and you will know before the marker does.
Check backwards on missing-value questions. Put your answer into the set, add, and divide. If the mean comes back to the figure stated in the question, your answer is right.
Read whether the question wants the mean, the mode or the total. In multiple-choice items all three usually appear among the options, because the question is testing which one you were asked for.
Quick revision summary
- Mean = total ÷ number of values. Always two steps: add, then divide.
- Mode = the value that appears most often.
- The mode can be the smallest value, there can be two modes, or there can be none.
- The mean does not have to be one of the values in the set.
- Total = mean × number of values — use this to work backwards.
- Adding a value equal to the mean leaves the mean unchanged.
- The mean must lie between the smallest and largest values — use this as a check.
- An outlier pulls the mean away from the middle; the mode can describe such a set better.
- Only the mode works for data that is not numbers, such as favourite colours.