Cambridge Primary Checkpoint · Mathematics 0096 · Stage 6 · Statistics and Probability

Mode, Median, Mean and Range, and Interpreting Data

Find and interpret the mode (including bimodal data), median, mean and range, choose the most suitable average, and compare data sets to draw conclusions and check predictions.

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In the curriculum

Understand that the mode, median, mean and range describe and summarise a set of data; find and interpret each one (including data with two modes), and decide which is most suitable for the situation (6Ss.03)
Interpret data and find patterns within one data set and between two data sets to answer statistical questions; discuss conclusions, think about why data varies, and check predictions (6Ss.04)

When you have collected a lot of data, you need a few numbers that sum it up. An average tells you what is typical. The three averages are the mode, the median and the mean. The range is not an average: it tells you how spread out the data is. We will use this data all the way through: the number of goals a school team scored in 7 matches.

Goals scored: 2, 5, 1, 3, 5, 0, 5

Key words

Word / phrase	Meaning / use
average	متوسط / قيمة نموذجية — one number that represents the whole data set
mode	المنوال — the value that appears most often
bimodal	ثنائي المنوال — data with two modes
median	الوسيط — the middle value when the data is in order
mean	المتوسط الحسابي — the total of all values ÷ the number of values
range	المدى — largest value − smallest value; shows how spread out the data is
data set	مجموعة بيانات
consistent	منتظم / ثابت — the values are close together (small range)
outlier (extreme value)	قيمة متطرفة — a value much bigger or smaller than the rest
variation	التباين — the way data values differ from each other

1. The mode

The mode is the value that occurs most often. In 2, 5, 1, 3, 5, 0, 5 the value 5 appears three times — more than any other — so the mode is 5.

Two modes (bimodal): shoe sizes 35, 36, 36, 37, 38, 38, 39 — both 36 and 38 appear twice, so the modes are 36 and 38. The data is bimodal.
No mode: if every value appears once (e.g. 4, 7, 9, 12), there is no mode.
The mode is the only average you can use for categorical data: “The modal colour is blue.”
From a frequency table, bar chart or dot plot, the mode is the value with the highest frequency (the tallest bar), not the frequency itself.

2. The median

The median is the middle value when the data is written in order from smallest to largest.

Write the values in order: 0, 1, 2, 3, 5, 5, 5. Count them to make sure none is missing (7 values).
Cross off one value from each end, again and again, until you reach the middle.
With an odd number of values there is one middle value. Here it is the 4th value: 3.

The 7 goals data in order. There are 3 values on each side of the median.

Even number of values: there are two middle values. The median is halfway between them: add them and divide by 2. Heights of 6 plants (cm): 12, 15, 9, 14, 11, 17 → in order 9, 11, 12, 14, 15, 17 → median = (12 + 14) ÷ 2 = 13 cm. The median does not have to be one of the values.

Where is the middle?

For n values in order, the median is the value in position (n + 1) ÷ 2. For 7 values: (7 + 1) ÷ 2 = 4th. For 6 values: (6 + 1) ÷ 2 = 3.5, which means halfway between the 3rd and 4th values. For 25 values: the 13th value.

3. The mean

The mean shares the total out equally. Mean = total of all the values ÷ number of values.

Goals: total = 2 + 5 + 1 + 3 + 5 + 0 + 5 = 21. Number of matches = 7. Mean = 21 ÷ 7 = 3 goals per match. Remember to count the 0 — it is a value!

The mean is a “levelling out”. If the cubes above the red line (yellow) were moved into the gaps below it, every tower would be 3 cubes tall.

Mean from a frequency table or dot plot. Multiply each value by its frequency, add these, then divide by the total frequency. Number of brothers and sisters of 25 students:

Number of brothers and sisters	Frequency	Value × frequency
0	3	0
1	8	8
2	7	14
3	4	12
4	2	8
5	1	5
Total	25	47

Mean = 47 ÷ 25 = 1.88. The mean can be a decimal even when the data is whole numbers — nobody has 1.88 brothers, but it is a useful summary. The mode here is 1 (frequency 8). The median is the 13th value: the first 3 are 0, the next 8 (4th to 11th) are 1, the next 7 (12th to 18th) are 2, so the median is 2. The range is 5 − 0 = 5.

Working backwards. If you know the mean, you can find the total: total = mean × number of values. Example: the mean of 4 numbers is 6, so their total is 24. If three of them are 4, 7 and 5 (total 16), the fourth is 24 − 16 = 8.

4. The range

Range = largest value − smallest value. Goals: 5 − 0 = 5. The range is always one number (not “0 to 5”). A small range means the values are close together — the data is consistent. A large range means the values are spread out.

5. Which average should you use?

Measure	Good for	Watch out
Mode	categorical data; the most popular or most common item (e.g. which shoe size a shop should stock most)	there may be no mode, or two modes; it can be far from the centre (goals: mode 5, but most matches had fewer)
Median	data with an extreme value (outlier), e.g. house prices or pocket money with one very large value	uses only the middle of the data
Mean	numerical data with values fairly close together; it uses every value	one extreme value can pull it a long way up or down
Range	describing how spread out or consistent the data is; use it with an average to compare data sets	it is not an average; one extreme value makes it large

For the goals data: mode 5, median 3, mean 3. The median and mean describe a “typical” match better than the mode, because 4 of the 7 matches had fewer than 5 goals.

بالعربي: المتوسطات الثلاثة والمدى

المنوال (mode) = الأكثر تكرارًا. ممكن يبقى في منوالين (bimodal) أو مافيش منوال.
الوسيط (median) = الرقم اللي في النص بعد الترتيب. لو العدد زوجي ناخد الرقمين اللي في النص ونجمعهم ونقسم على 2.
المتوسط الحسابي (mean) = المجموع ÷ العدد.
المدى (range) = الأكبر − الأصغر، وده مش متوسط — ده بيقيس قد إيه البيانات متفرقة.

6. Interpreting and comparing data

To compare two data sets, use one average (to say which is generally higher) and the range (to say which is more consistent). Then write a sentence in the context of the question.

Spelling scores out of 10 for 7 students in each class. Both classes have a total of 49, so both means are 49 ÷ 7 = 7. Class A: range 8 − 6 = 2. Class B: range 10 − 3 = 7.

Conclusion: “On average, both classes did equally well (mean 7). Class A’s scores were much more consistent (range 2) than Class B’s (range 7), where some students scored very high and some very low.”

Why does data vary? When you discuss a conclusion, think about the sources of variation:

Natural differences — people, plants and days are not all the same.
Measuring — different people may measure in slightly different ways or round differently; instruments are not perfectly accurate.
Chance — a test on a different day, or a different group of students, could give different results.
Sample size — with only a few values, one unusual value changes the results a lot; more data gives a more reliable conclusion.
Mistakes — a value recorded wrongly (e.g. 200 cm for a child’s height) should be checked.

Checking a prediction. Compare what you predicted with the averages and ranges you found. Say whether the data supports the prediction, quote the numbers, and say how sure you can be (e.g. “only 7 students in each class, so we should collect more data”).

Worked examples

Worked example 1 — All four measures [3 marks] The number of books 8 students read in a month: 4, 2, 7, 3, 4, 6, 4, 2. Find the mode, the median, the mean and the range.

In order: 2, 2, 3, 4, 4, 4, 6, 7 (8 values ✓).
Mode: 4 appears three times → mode = 4.
Median: 8 values, so the middle is between the 4th and 5th values: 4 and 4 → (4 + 4) ÷ 2 = 4.
Mean: total = 2 + 2 + 3 + 4 + 4 + 4 + 6 + 7 = 32; 32 ÷ 8 = 4.
Range: 7 − 2 = 5.

Answer: mode 4, median 4, mean 4, range 5. Common mistake: finding the median from the unordered list (the 4th and 5th values there are 3 and 4, giving 3.5 — wrong).

Worked example 2 — Median of an even number of values [1 mark] The times (in seconds) for 6 children to run 60 m: 11.4, 10.8, 12.0, 11.0, 13.2, 10.5. Find the median time.

In order: 10.5, 10.8, 11.0, 11.4, 12.0, 13.2.
Middle two: 11.0 and 11.4. (11.0 + 11.4) ÷ 2 = 22.4 ÷ 2 = 11.2.

Answer: 11.2 seconds.

Worked example 3 — Bimodal data and choosing the mode [2 marks] A shop sold these sizes of school shoes on Monday: 36, 38, 35, 38, 37, 36, 39, 38, 36. (a) Find the mode. (b) Why is the mode the most useful average for the shop?

Tally: 35 → 1, 36 → 3, 37 → 1, 38 → 3, 39 → 1.
36 and 38 both appear 3 times → the data is bimodal: modes 36 and 38.
The shop needs to know which sizes are most often bought so it can order more of them. A mean shoe size such as 37.0 is not useful for ordering stock and might not even be a real size.

Answer: (a) 36 and 38 (both needed). (b) It shows the most popular sizes, which the shop should stock most.

Worked example 4 — An extreme value [2 marks] The weekly pocket money of 6 friends (in pounds): 20, 25, 25, 30, 30, 200. (a) Work out the mean and the median. (b) Which one better describes the typical pocket money? Explain.

Mean: total = 20 + 25 + 25 + 30 + 30 + 200 = 330; 330 ÷ 6 = 55.
Median: middle values are 25 and 30 → (25 + 30) ÷ 2 = 27.5.
Five of the six friends get 30 pounds or less, but the mean is 55 because the one value of 200 pulls it up.

Answer: (a) mean 55 pounds, median 27.5 pounds. (b) The median, because the extreme value 200 makes the mean bigger than what almost everyone gets.

Worked example 5 — Finding a missing value from the mean [2 marks] The mean mass of 5 parcels is 3 kg. Four of the parcels have masses 2.5 kg, 4 kg, 1.5 kg and 3 kg. Find the mass of the fifth parcel.

Total of all 5 = mean × number = 3 × 5 = 15 kg.
Total of the four known = 2.5 + 4 + 1.5 + 3 = 11 kg.
Fifth parcel = 15 − 11 = 4 kg.

Check: (2.5 + 4 + 1.5 + 3 + 4) ÷ 5 = 15 ÷ 5 = 3 ✓ Common mistake: answering 3 kg (“the mean”), or adding the four values and dividing by 4.

Worked example 6 — Comparing and checking a prediction [2 marks] Omar predicted: “Class B is better at spelling than Class A.” Use the dot plots in section 6 to say whether the data supports his prediction.

Compare an average: both means are 7, so on average the classes are the same.
Compare the range: Class A 2, Class B 7. Class B has the two highest scores (10) but also the lowest (3).
Conclusion in context.

Answer: The data does not support Omar’s prediction: both classes have the same mean score (7). Class B’s scores are more spread out (range 7 compared with 2), so Class B is less consistent, not better. With only 7 students in each class, more data would make the conclusion more reliable.

Common mistakes

Finding the median without putting the data in order first.
Giving the frequency instead of the value for the mode (e.g. “3” instead of “shoe size 36”).
Forgetting values of 0 when finding the mean, or dividing by the wrong number of values.
Writing the range as “0 – 5” instead of one number, 5.
Saying there is “no mode” when there are two modes — two modes means bimodal.
Comparing two data sets with only one number, or without writing in context (“Class A is more consistent”, not just “A has a smaller range”).

How Checkpoint asks this

“Find the median / mode / mean / range of …” — show your working (the ordered list, or the total and the division).
“The mean of five numbers is 8. Four of the numbers are … Find the fifth number.”
“Write a set of four numbers with a mode of 5, a median of 6 and a range of 4” — build the list and check every condition.
“Which average is the best to use? Explain why.”
“Compare the two sets of data” — one comment on an average and one on the range, in context.
“Was the prediction correct? Use the data to explain your answer.”

Practice

Remember

Mode = most common (there can be two: bimodal). Median = middle value after ordering. Mean = total ÷ number of values.
Range = largest − smallest: it measures spread, not a typical value.
Use the mode for categorical data, the median when there is an extreme value, and the mean when the values are close together.
To compare two data sets: compare an average and the range, and write your conclusion in context.
Data varies because of natural differences, measuring, chance and sample size — more data gives more reliable conclusions.

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