Mean, median, mode. Three ways to summarise a set of numbers. Plus range for the spread.
The six facts you need
Every averages question uses one of these. You don't need to remember anything else.
1 · MeanSUM ÷ COUNT. Add all values, divide by how many.
2 · MedianThe MIDDLE value after sorting from smallest to biggest.
3 · ModeThe value that appears MOST OFTEN.
4 · RangeBiggest − smallest. Range is SPREAD, not an average.
5 · Even count medianIf there are two middles, take their MEAN. The median can be a value not in the list.
6 · Mode edge casesData can have no mode, one mode, or more than one mode.
Worked example 1 - mean of a small set
Find the mean of 4, 6, 8, 10, 12.
Read it. Find the mean of five values.
The fact. Mean = sum ÷ count.
Sum: 4 + 6 + 8 + 10 + 12 = 40.
Count: 5 values.
40 ÷ 5 = 8.
Answer: 8
Watch outAdd ALL values first. Then divide.
Worked example 2 - mean with repeats
Find the mean of 2, 5, 7, 6, 5.
Read it. Find the mean.
The fact. Add all values (including repeats), then divide by count.
Sum = 2 + 5 + 7 + 6 + 5 = 25.
Count = 5.
25 ÷ 5 = 5.
Answer: 5
Watch outInclude every value, even repeats.
Worked example 3 - median with odd count
Find the median of 3, 7, 4, 9, 5.
Read it. Find the median of five values.
The fact. Median = middle value after sorting.
Sort: 3, 4, 5, 7, 9.
5 values, middle position is 3rd: 5.
Answer: 5
Watch outAlways sort FIRST. The middle of the original list is NOT the median.
Worked example 4 - median with even count
Find the median of 2, 6, 4, 8.
Read it. Find the median of four values.
The fact. Even count: take the MEAN of the two middle values.
Sort: 2, 4, 6, 8.
Two middles are 4 and 6.
Mean: (4 + 6) ÷ 2 = 5.
Answer: 5
Watch outFor even counts, the median can be a value NOT in the list.
Worked example 5 - finding the mode
Find the mode of 2, 5, 3, 5, 8, 5, 6.
Read it. Find the mode.
The fact. Mode = the value that appears most often.
Counts: 2 → 1, 5 → 3, 3 → 1, 8 → 1, 6 → 1.
5 is the most common.
Answer: 5
Watch outThe mode is the value that repeats MOST. Not the highest or lowest.
Worked example 6 - no mode
What is the mode of 1, 2, 3, 4, 5?
Read it. Find the mode.
The fact. Mode = most common value. But every value appears only ONCE here.
Every value appears once.
No value is more common than another.
So there is NO MODE.
Answer: no mode
Watch outWhen no value repeats, there is no mode. It is NOT zero.
Worked example 7 - finding the range
Find the range of 12, 5, 18, 8, 15.
Read it. Find the range.
The fact. Range = biggest − smallest.
Biggest: 18.
Smallest: 5.
18 − 5 = 13.
Answer: 13
Watch outRange is a measure of SPREAD, not an average.
Worked example 8 - all four measures
For the set 3, 5, 4, 7, 5, 6, 4, 5, find the mean, median, mode and range.
Read it. Find all four measures.
The fact. Apply each rule in turn.
Sum = 39, count = 8. Mean = 39 ÷ 8 = 4.875.
Sorted: 3, 4, 4, 5, 5, 5, 6, 7. Two middles: 5, 5. Median = 5.
Mode: 5 appears 3 times — most common.
Range = 7 − 3 = 4.
Answer: mean 4.875 · median 5 · mode 5 · range 4
Watch outDifferent measures can give different numbers. That’s the point.
Practice · Section A
Mean
Sum ÷ count
Practice · Section B
Median & mode
Sort for median · Most common for mode
Practice · Section C
Range & mixed
Biggest − smallest · Different sized sets
Ready for more? 100 questions
When you have done these 12, you are ready for the bigger practice paper.
It has 100 questions. They start very easy and get harder near the end. Every question is still Grade 2 — nothing harder. You can stop at any time and come back.