📖 Theory notes

Every note is built the same way: I can…, a quick recall, 🔑 key ideas, ✨ tips, ⚠️ common mistakes, ✏️ worked examples and 🧠 Remember in 3 lines.

🎯 Learn, then practise from two books

Each LO starts with 📘 a learning sheet (worked examples one step at a time), then 📗 Collins practice (Book 1 exercises) and, where Book 2 covers the topic, 📙 Pearson practice (Book 2 exercises, including starred questions).

🧪 LO tests · 30 minutes each

When the practice feels easy, take the timed test for that LO. Tests run in Quiz mode, submit themselves at 00:00 and can be taken once. Your report shows each LO as a pie chart.

About this chapter

An average describes a whole set of data with one typical value, so two sets can be compared quickly. This chapter covers the three averages — mode, median and mean — and two measures of spread: the range and the interquartile range. You will also work with frequency tables, grouped data and cumulative frequency diagrams, which are the forms in which large data sets usually appear.

All practice comes from 📗 Collins Book 1, Chapter 32. Pearson Book 2 has no unit on averages, so this chapter has no 📙 Pearson sheets. LO tests use Collins "Examination questions on Statistics and probability" (past Edexcel papers).

LO	📗 Collins Book 1	📙 Pearson Book 2
LO 1	32.1 The mode · 32.2 The median (Exercises 32A, 32B)	—
LO 2	32.3 The mean · 32.4 The range (Exercises 32C, 32D)	—
LO 3	32.5 Which average to use (Exercise 32E)	—
LO 4	32.6 Using frequency tables (Exercise 32F)	—
LO 5	32.7 Grouped data (Exercise 32G)	—
LO 6	32.8 Measuring spread · 32.9 Cumulative frequency diagrams (Exercises 32H, 32I)	—

Learning objectives

LO 1 · 32.1–32.2 Mode and medianI can find the mode and the median of a list of data and from a frequency table, including lists with an even number of values.

LO 2 · 32.3–32.4 Mean and rangeI can calculate the mean and the range, use totals (mean × number) to solve problems, and compare data sets by average and spread.

LO 3 · 32.5 Which average to useI can choose the most suitable average for a set of data and explain my choice, including the effect of extreme values.

LO 4 · 32.6 Averages from frequency tablesI can find the mode, median and mean from a frequency table, using running totals and Σfx ÷ Σf.

LO 5 · 32.7 Grouped dataI can find the modal class and estimate the mean of grouped data using midpoints.

LO 6 · 32.8–32.9 Quartiles and cumulative frequencyI can find quartiles and the interquartile range from a list, draw a cumulative frequency diagram and use it to estimate the median, quartiles and other values.

Check what you already know

You need	Quick check	Answer
Ordering numbers	Put 0.5, −2, 3, 1.25 in order	−2, 0.5, 1.25, 3
Adding a list	4 + 7 + 9 + 12	32
Dividing	126 ÷ 8	15.75
Reading a table	Which value has frequency 9 in: 1→4, 2→9, 3→6?	2

32.1–32.2 The mode and the median

LO 1I can… find the mode and the median of a list of data and from a frequency table, including lists with an even number of values.

🔁 Quick recall before you start
Put 7, 2, 9, 4, 2 in order. (2, 2, 4, 7, 9.) Which number is in the middle? (4.)

📗 Collins: 32.1 the mode (Example 1, Exercise 32A), 32.2 the median (Examples 2–3, Exercise 32B).
📙 Pearson Book 2: no matching unit — Book 2 does not revisit averages (its Handling Data units cover histograms and probability).

🔑 Key idea · The mode
The mode (modal value) is the value that occurs most often. It is the only average that works for non-numerical data such as colours. A set can have no mode (everything equally common) or more than one.

🔑 Key idea · The median
The median is the middle value when the data are in order. For n values it is the [[n + 1||2]]th value. With an even number of values it is halfway between the two middle values.

✏️ Worked example · Mode (Collins Example 1)

Goals: 1, 2, 1, 0, 1, 0, 0, 1, 2, 1, 0, 2.
0 → 4 times, 1 → 5 times, 2 → 3 times. Mode = 1.

✏️ Worked example · Median of an even list

4, 9, 2, 7, 10, 5 → in order 2, 4, 5, 7, 9, 10.
Middle values 5 and 7 → median (5 + 7) ÷ 2 = 6.

✏️ Worked example · Median from a frequency table (Collins Example 3)

Values 2–7 with frequencies 2, 4, 6, 7, 8, 3 (total 30).
Median between the 15th and 16th values. Running totals 2, 6, 12, 19 → both are 5.
Median = 5.

⚠️ Common mistake · Forgetting to order
The median of 7, 1, 9 is not 1 (the middle as written). Order first: 1, 7, 9 → median 7.

⚠️ Common mistake · Value or frequency?
In a table, the mode is the value with the highest frequency, not the frequency itself.

✨ Tip · Tally big lists
With a long list, a tally chart stops you losing count when finding the mode.

🧠 Remember in 3 lines

Mode = most common value (can be words).
Median = middle value after ordering: the (n + 1) ÷ 2 th.
Even count → halfway between the two middle values.

Memory hook: MOde is MOst; MEDian is in the MIDDLE of the road.

32.3–32.4 The mean and the range

LO 2I can… calculate the mean and the range, use totals (mean × number) to solve problems, and compare data sets by average and spread.

🔁 Quick recall before you start
What is 48 ÷ 6? (8.) What is 9 − (−3)? (12.)

📗 Collins: 32.3 the mean (Example 4, Exercise 32C), 32.4 the range (Example 5, Exercise 32D).
📙 Pearson Book 2: no matching unit — Book 2 does not revisit averages (its Handling Data units cover histograms and probability).

🔑 Key idea · The mean
mean = [[sum of all values||number of values]]. It is what most people call 'the average'. It uses every value.

🔑 Key idea · The range
range = highest value − lowest value. It is not an average: it measures spread. A smaller range means more consistent data.

✏️ Worked example · Mean (Collins Example 4)

Ages 21, 23, 20, 27, 25, 24, 25, 30, 21, 22, 28 → sum 266.
Mean = 266 ÷ 11 = 24.18… = 24.2 years (1 d.p.).

✏️ Worked example · Comparing with mean and range (Collins Example 5)

Rachel: mean 6, range 9 − 4 = 5.
Adil: mean 6, range 8 − 5 = 3.
Same mean, but Adil is more consistent.

✏️ Worked example · Using totals

Mean of 8 walkers is 42 → total 336. With Joanne, 9 walkers, mean 40 → total 360.
Joanne is 360 − 336 = 24.

✨ Tip · Shift trick
Adding the same number to every value adds it to the mean (and median and mode) but does not change the range.

⚠️ Common mistake · Negative values in the range
Range of −4 to 3 is 3 − (−4) = 7, not 3 − 4.

⚠️ Common mistake · Rounding too early
Add everything first, divide last, and round only the final answer.

🧠 Remember in 3 lines

Mean = total ÷ how many.
Total = mean × how many — use it to find missing values.
Range = highest − lowest: it shows spread, not average.

Memory hook: Share it out equally — that's the mean.

32.5 Which average to use

LO 3I can… choose the most suitable average for a set of data and explain my choice, including the effect of extreme values.

🔁 Quick recall before you start
Find the mean and median of 1, 2, 3, 4, 90. (Mean 20, median 3 — which is more typical?)

📗 Collins: 32.5 which average to use (comparison table, Exercise 32E).
📙 Pearson Book 2: no matching unit — Book 2 does not revisit averages (its Handling Data units cover histograms and probability).

🔑 Key idea · Pick the average that represents the data
The right average gives a fair picture. The wrong one can be misleading — for example, a few very high salaries make the mean much bigger than what most people earn.

	Mode	Median	Mean
Advantages	Very easy to find · not affected by extreme values · works for non-numerical data	Easy to find for ungrouped data · not affected by extreme values	Uses all the values · totals can be found from it
Disadvantages	Does not use all the values · may not exist	Does not use all the values	Extreme values distort it · has to be calculated
Use for	Non-numerical data, the most likely value	Data with extreme values	Data spread in a balanced way

✏️ Worked example · Choosing (Collins Exercise 32E Q2)

6, 8, 6, 10, 6, 9, 6, 10, 6, 8 → 6 appears five times: the mode is useful.
7, 4, 5, 3, 28, 8, 2, 4, 10, 9 → 28 is extreme: use the median (6), not the mean (8).

✏️ Worked example · Reverse problem

Nine numbers have mean 7.6 (total 68.4). Add one number to make the mean 8 (total 80).
Number added = 80 − 68.4 = 11.6.

⚠️ Common mistake · Averaging averages
Mean of a group of 3 (mean 10) and a group of 7 (mean 20) is not 15. Use totals: (30 + 140) ÷ 10 = 17.

✨ Tip · Say why
Exam answers need a reason: 'the median, because it is not affected by the extreme value 28'.

🧠 Remember in 3 lines

Words or 'most popular' → mode.
Extreme values → median.
Balanced numbers → mean (uses every value).

Memory hook: Shoppers want the mode, statisticians check for outliers, accountants love the mean.

32.6 Averages from frequency tables

LO 4I can… find the mode, median and mean from a frequency table, using running totals and Σfx ÷ Σf.

🔁 Quick recall before you start
In the table 1→3, 2→5, 3→2, how many values are there? (10.) What is the total of all values? (3 + 10 + 6 = 19.)

📗 Collins: 32.6 using frequency tables (Example 6, Exercise 32F).
📙 Pearson Book 2: no matching unit — Book 2 does not revisit averages (its Handling Data units cover histograms and probability).

🔑 Key idea · Three averages from one table
Mode: the value with the highest frequency.
Median: find the position (n + 1) ÷ 2, then use running totals to find which value it falls on.
Mean: add an fx column; mean = Σfx ÷ Σf.

People in car (x)	Frequency (f)	f × x
1	45	45
2	198	396
3	121	363
4	76	304
5	52	260
6	13	78
Total	505	1446

✏️ Worked example · Collins Example 6

Mode = 2 (frequency 198).
Median position (505 + 1) ÷ 2 = 253rd. Running totals 45, 243, 364 → 3 people.
Mean = 1446 ÷ 505 = 2.9 people (1 d.p.).

⚠️ Common mistake · Dividing by the number of rows
Divide Σfx by the total frequency (505 cars), not by the number of rows (6).

✨ Tip · Sense check
The mean must lie between the smallest and largest values in the table. A mean of 241 people per car is clearly wrong.

🧠 Remember in 3 lines

Mode: biggest frequency → its value.
Median: (n + 1) ÷ 2 th value via running totals.
Mean: Σfx ÷ Σf.

Memory hook: Multiply across, add down, divide by the people counted.

32.7 Grouped data

LO 5I can… find the modal class and estimate the mean of grouped data using midpoints.

🔁 Quick recall before you start
What number is halfway between 20 and 30? (25.) Between 10 and 14? (12.)

📗 Collins: 32.7 grouped data (Example 7, Exercise 32G).
📙 Pearson Book 2: no matching unit — Book 2 does not revisit averages (its Handling Data units cover histograms and probability).

🔑 Key idea · Grouped tables
Large data sets — especially continuous data such as height, mass or time — are put into groups (classes). The modal class is the group with the highest frequency.

🔑 Key idea · Estimating the mean
Assume every value is at the midpoint of its group. Estimated mean = Σ(f × midpoint) ÷ Σf. It is only an estimate because the real values are unknown.

Pocket money p ($)	Frequency f	Midpoint m	f × m
0 < p ⩽ 1	2	0.50	1.00
1 < p ⩽ 2	5	1.50	7.50
2 < p ⩽ 3	5	2.50	12.50
3 < p ⩽ 4	9	3.50	31.50
4 < p ⩽ 5	15	4.50	67.50
Totals	36		120

✏️ Worked example · Collins Example 7

Modal class: $4 to $5.
Estimated mean = 120 ÷ 36 = $3.33.

⚠️ Common mistake · Midpoints of discrete groups
For heights to the nearest cm in 15–18, the midpoint is (15 + 18) ÷ 2 = 16.5 — use the end values given.

⚠️ Common mistake · Median and range from groups
You cannot find the exact median or range from a grouped table. Estimate the median with a cumulative frequency diagram (LO 6).

✨ Tip · Total from the table
Σ(f × midpoint) is also an estimate of the total — for example the total mass of all the pebbles.

🧠 Remember in 3 lines

Modal class = group with the biggest frequency.
Midpoint = (start + end) ÷ 2.
Estimated mean = Σ(f × midpoint) ÷ Σf.

Memory hook: Everyone in a group stands in the middle.

32.8–32.9 Quartiles and cumulative frequency

LO 6I can… find quartiles and the interquartile range from a list, draw a cumulative frequency diagram and use it to estimate the median, quartiles and other values.

🔁 Quick recall before you start
What is the median of 3, 5, 8, 10, 12? (8.) What is ¼ of 60? (15.)

📗 Collins: 32.8 measuring spread (Example 8, Exercise 32H), 32.9 cumulative frequency diagrams (Example 9, Exercise 32I).
📙 Pearson Book 2: no matching unit — Book 2 does not revisit averages (its Handling Data units cover histograms and probability).

🔑 Key idea · Quartiles and the IQR
The median splits ordered data into two halves. The lower quartile (LQ) is the median of the lower half and the upper quartile (UQ) the median of the upper half. Interquartile range IQR = UQ − LQ. It measures the spread of the middle half and is not affected by extreme values.

✏️ Worked example · Even number of values (Collins Example 8)

20 marks: median between the 10th and 11th = 20.5.
LQ between the 5th and 6th = 15; UQ between the 15th and 16th = 26.
IQR = 26 − 15 = 11.

🔑 Key idea · Cumulative frequency diagrams
Add a running-total (cumulative frequency) column. Plot each running total at the top value of its group and join the points. Read: median at ½n, LQ at ¼n, UQ at ¾n — go across to the line, then down.

✏️ Worked example · Collins Example 9

Cumulative frequencies 3, 12, 24, 39, 61, 77, 87, 95, 98, 100.
Median at 50 → about 65; LQ at 25 → about 51; UQ at 75 → about 79; IQR ≈ 28.
Below 44 marks: up from 44 to the curve, across → about 18 students.

⚠️ Common mistake · Plotting at the midpoint
Cumulative frequency counts everything up to the end of a group, so plot at the top of each group, not the midpoint.

✨ Tip · Percentiles
The 10th percentile is read at 10% of n; 'top 10%' starts at the 90th percentile.

🧠 Remember in 3 lines

LQ, median, UQ cut the ordered data into four quarters.
IQR = UQ − LQ: spread of the middle half.
CF diagram: plot running totals at group tops; read ¼n, ½n, ¾n.

Memory hook: Quarter, half, three-quarters — across, then down.