The Mean
Add all values then divide by how many there are
Mean Formula
Mean = Total of all values ÷ Number of values
✏️ Worked Example
Find the mean of: 5, 8, 3, 10, 4
- Add all values: 5+8+3+10+4 = 30
- Count values: 5 numbers
- Mean = 30 ÷ 5 = 6
Mean = 6
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Quick check before you practise
The mean of 4, 7, 9, 12, 8 is:
The mean of four numbers is 12. Three are 8, 14, 10. What is the fourth?
Find the mean of: 4, 7, 9, 2, 8
A student scored these marks in 4 tests: 12, 18, 15, 11. What is the mean score?
Daily temperatures (°C) for a week: 14, 17, 13, 19, 16, 18, 15. Calculate the mean temperature.
°C
Add all 7 temperatures first, then divide by 7.
Full Working
Total: 14+17+13+19+16+18+15 = 112Mean = 112 ÷ 7 = 16°C
The mean of 5 numbers is 8. Four of the numbers are 6, 10, 7, 9. What is the fifth number?
If mean = 8 and there are 5 numbers, total must be 5 × 8 = 40.
Full Working
Total needed: 5 × 8 = 40Sum of 4 known: 6+10+7+9 = 32
Fifth number: 40 − 32 = 8
🎉 Section 1 Complete!
The Mean mastered!
Median, Mode & Range
Find the middle value, most common value, and spread of data
Median
Order the data → middle value (mean of middle two if even count)
Mode & Range
Mode = most frequent value | Range = Highest − Lowest
✏️ Worked Example — Median (odd count)
Find the median of: 7, 2, 9, 4, 6, 1, 5
- Reorder: 1, 2, 4, 5, 6, 7, 9
- 7 values → middle is the 4th value
- Median = 5
Median = 5
✏️ Worked Example — Median (even count)
Find the median of: 3, 7, 2, 8, 5, 4
- Reorder: 2, 3, 4, 5, 7, 8
- 6 values → middle is between 3rd and 4th
- Median = (4+5) ÷ 2 = 4.5
Median = 4.5
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Quick check before you practise
What is the median of 3, 9, 1, 7, 5?
Range of 14, 7, 22, 5, 18:
Find the mode of: 3, 5, 3, 7, 5, 3, 8, 5, 3
Full Working
3 appears 4 times, 5 appears 3 timesMode = 3 (appears most)
Find the median of: 11, 4, 7, 9, 3, 6, 8
Find the range of these exam scores: 45, 72, 58, 91, 63, 37, 80
Range = Highest − Lowest. Find the biggest and smallest values first.
Full Working
Highest = 91, Lowest = 37Range = 91 − 37 = 54
Find the median of these 8 values: 5, 12, 3, 9, 7, 15, 6, 11
8 values (even) → median = (4th + 5th) ÷ 2 after ordering.
Full Working
Ordered: 3, 5, 6, 7, 9, 11, 12, 154th = 7, 5th = 9
Median = (7+9) ÷ 2 = 8
A class test gives scores: 31, 29, 20, 35, 32, 38, 32. Find the median score.
Order the 7 values first. The median is the 4th value.
Full Working
Ordered: 20, 29, 31, 32, 32, 35, 387 values → 4th value = 32
🎉 Section 2 Complete!
Median, Mode & Range mastered!
Mean from Frequency Tables
Use the x × f column to find the mean from discrete data
Mean from Frequency Table
Mean = Σ(x × f) ÷ Σf
✏️ Worked Example
Find the mean from this frequency table:
| Value (x) | Frequency (f) | x × f |
|---|---|---|
| 1 | 3 | 3 |
| 2 | 5 | 10 |
| 3 | 4 | 12 |
| 4 | 2 | 8 |
| Totals | 14 | 33 |
- Σ(xf) = 3+10+12+8 = 33
- Σf = 3+5+4+2 = 14
- Mean = 33 ÷ 14 ≈ 2.36
Mean ≈ 2.36
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Frequency table values 1→3, 2→5, 3→2. What is Σf?
Same table. Σ(x×f) to find mean numerator:
A shop records items customers buy. What is Σf (total frequency)?
| Items (x) | Customers (f) |
|---|---|
| 1 | 8 |
| 2 | 12 |
| 3 | 6 |
| 4 | 4 |
Calculate the mean items per customer (1 d.p.) from the table above.
| Items (x) | Customers (f) | x×f |
|---|---|---|
| 1 | 8 | 8 |
| 2 | 12 | 24 |
| 3 | 6 | 18 |
| 4 | 4 | 16 |
Add the x×f column: Σ(xf) = 66. Divide by Σf = 30.
Full Working
Σ(xf) = 8+24+18+16 = 66Mean = 66 ÷ 30 = 2.2
Robbie played 20 football matches. Calculate the mean goals per game.
| Goals (x) | Matches (f) |
|---|---|
| 0 | 3 |
| 1 | 9 |
| 2 | 5 |
| 3 | 2 |
| 4 | 1 |
Multiply goals × matches for each row, add up, divide by 20.
Full Working
xf: 0+9+10+6+4 = Σ(xf) = 29Mean = 29 ÷ 20 = 1.45
A shop records merchandise sales. What is the mean amount spent per customer?
| Product | Price (x) | Sales (f) |
|---|---|---|
| T-shirt | £10 | 25 |
| Key ring | £5 | 30 |
| Hoodie | £25 | 40 |
| CDs | £15 | 30 |
£
Multiply price × sales for each row, add up, divide by total sales = 125.
Full Working
xf: 250+150+1000+450 = Σ(xf) = 1850Σf = 125
Mean = 1850 ÷ 125 = £14.80
🎉 Section 3 Complete!
Mean from frequency tables mastered!
Mean from Continuous Data
Use midpoints to estimate the mean from grouped frequency tables
Midpoint Method
Midpoint = (Lower + Upper) ÷ 2 → use as x, then Mean = Σ(xf) ÷ Σf
✏️ Worked Example
Find the estimated mean height from this grouped frequency table:
| Height (cm) | Midpoint (x) | Frequency (f) | x × f |
|---|---|---|---|
| 150–160 | 155 | 4 | 620 |
| 160–170 | 165 | 9 | 1485 |
| 170–180 | 175 | 6 | 1050 |
| 180–190 | 185 | 1 | 185 |
| Totals | — | 20 | 3340 |
- Midpoints: (150+160)÷2=155, (160+170)÷2=165, etc.
- Σ(xf) = 620+1485+1050+185 = 3340
- Mean = 3340 ÷ 20 = 167 cm
Estimated Mean = 167 cm
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Class interval 20 ≤ x < 30. Which midpoint?
What is the midpoint of the class interval 20–30?
A photographer records photos ordered per customer. Find the estimated mean (1 d.p.).
| Photos | Midpoint (x) | Customers (f) | x×f |
|---|---|---|---|
| 1–10 | 5.5 | 26 | 143 |
| 11–20 | 15.5 | 14 | 217 |
| 21–30 | 25.5 | 6 | 153 |
| 31–40 | 35.5 | 4 | 142 |
| Totals | — | 50 | 655 |
Σ(xf) = 655 and Σf = 50. Divide to find the mean.
Full Working
Σ(xf) = 143+217+153+142 = 655Mean = 655 ÷ 50 = 13.1
A college records learner ages. Calculate the estimated mean age (1 d.p.).
| Age range | Midpoint (x) | Frequency (f) |
|---|---|---|
| 16–20 | 18 | 12 |
| 21–25 | 23 | 18 |
| 26–30 | 28 | 10 |
| 31–40 | 35.5 | 5 |
Multiply each midpoint × frequency, add up, divide by Σf = 45.
Full Working
xf: 216+414+280+177.5 = 1087.5Σf = 45
Mean = 1087.5 ÷ 45 = 24.2
Last week the mean photos ordered was 12. This week it is 13.1. What can we conclude about the price increase?
Explanation
Last week: 12, this week: 13.1 — difference of only 1.1No strong evidence the price rise reduced orders
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Mean & Range
Level 1 · Calculate the mean and range from a list of numbers
Mean & Range
Mean = Total ÷ Count | Range = Max − Min
✏️ Worked Example
James scores: 1, 4, 4, 2, 3, 4, 5, 1, 4, 1 — find the mean and range.
- Total: 1+4+4+2+3+4+5+1+4+1 = 29. Mean = 29 ÷ 10 = 2.9
- Range: Max=5, Min=1 → 5−1 = 4
Mean = 2.9, Range = 4
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Mean of 6, 10, 8, 4, 12:
Find the mean of: 6, 8, 4, 10, 7
Find the range of: 15, 3, 8, 22, 11, 6
A class scored: 31, 29, 20, 35, 32, 38, 32. Calculate the mean score.
Add all 7 scores together, then divide by 7.
Full Working
Total: 31+29+20+35+32+38+32 = 217Mean = 217 ÷ 7 = 31
Using the same scores (31, 29, 20, 35, 32, 38, 32), find the range.
Find the highest and lowest scores, then subtract.
Full Working
Max = 38, Min = 20Range = 38 − 20 = 18
A care worker records daily steps: 8200, 7500, 9100, 6800, 8800. Find the mean steps per day.
Add all 5 values together, then divide by 5.
Full Working
Total = 40400Mean = 40400 ÷ 5 = 8080
🎉 Section 1 Complete!
Mean & Range done!
Median & Mode
Level 1 · Find the middle value and the most common value
Median & Mode
Median: order then find middle | Mode: most frequent value
✏️ Worked Example
Scores: 1, 4, 4, 2, 3, 4, 5, 1, 4, 1 — find median and mode.
- Ordered: 1, 1, 1, 2, 3, 4, 4, 4, 4, 5
- Median (10 values): (5th+6th)÷2 = (3+4)÷2 = 3.5
- Mode = 4 (appears 4 times)
Median = 3.5, Mode = 4
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Mode of: 3, 5, 3, 7, 3, 5, 2:
Find the mode of: 2, 5, 3, 5, 7, 5, 2, 8
Full Working
5 appears 3 times — more than any other valueMode = 5
Find the median of: 9, 3, 7, 1, 5
A football team scored: 6, 0, 3, 2, 2, 5 in six games. Find the median goals scored.
6 values → median = mean of the 3rd and 4th values after ordering.
Full Working
Ordered: 0, 2, 2, 3, 5, 6Middle: (2+3) ÷ 2 = 2.5
Using the same football scores (6, 0, 3, 2, 2, 5), find the mean goals scored.
Add all 6 scores, then divide by 6.
Full Working
Total: 6+0+3+2+2+5 = 18Mean = 18 ÷ 6 = 3
Find the median of: 14, 9, 21, 6, 18, 3, 11, 16
8 values (even) → order them, mean of 4th and 5th.
Full Working
Ordered: 3, 6, 9, 11, 14, 16, 18, 214th=11, 5th=14 → (11+14)÷2 = 12.5
🎉 Section 2 Complete!
Median & Mode done!
Comparing Averages
Level 1 · Use averages and range to compare two sets of data
Key Principle
Higher mean = better average | Lower range = more consistent
✏️ Worked Example
Broadband A: 14,13,14,15,13 | Broadband B: 9,15,13,7,11. Which is better?
- Mean A = 13.8 | Mean B = 11
- Range A = 2 | Range B = 8
- A: higher mean AND lower range → faster AND more consistent
Broadband A is better
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Dataset A: mean 15, range 20. Dataset B: mean 15, range 5. What is true?
7 girls run 100m. Times (secs): 20, 24, 18, 19, 21, 26, 29. What is the mean time? (1 d.p.)
sec
Full Working
Total = 157Mean = 157 ÷ 7 = 22.4 seconds
Boys' mean time is 25 sec, girls' is 22.4 sec. Lower time = faster. Who is faster?
Explanation
Girls mean (22.4s) < Boys mean (25s) → Girls are fasterAfter 7 games the mean goals is 4. Previously 6 games with mean 3. How many goals in the 7th game?
New total = 7×4 = 28. Old total = 6×3 = 18. 7th game = 28 − 18.
Full Working
New total: 7×4 = 28 | Old: 6×3 = 187th game: 28 − 18 = 10
Worker A packs: 20, 22, 18, 24, 21 boxes/day. Worker B: 19, 25, 14, 28, 19. Who is more consistent?
Full Working
Range A: 24−18 = 6 | Range B: 28−14 = 14Worker A has lower range → more consistent
House prices on a street: £120k, £135k, £140k, £125k, £650k. Which average best represents a “typical” price?
Explanation
The £650k house distorts the mean upwardMedian is not affected by outliers → better “typical” measure
🎉 All Level 1 Sections Complete!
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Section Breakdown