Mean, Median & Mode
Mean, median, and mode form the foundation of data analysis in Year 10 GCSE maths, yet students often confuse these three measures of central tendency. When Charlotte's class calculates their average test score as 72%, finds the median height as 165cm, and identifies the most common shoe size as 7, they're using all three concepts that appear regularly in GCSE Foundation and Higher tier papers.
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Why it matters
These statistical measures appear everywhere in real life, from analysing football league tables to understanding house prices in your local area. Estate agents use the median house price (£285,000 in many UK regions) because it's less affected by extremely expensive properties than the mean. Sports statisticians calculate mean goals per match (2.8 in the Premier League) to compare team performance. Market researchers identify the modal age group (25-34) for targeting products. The Office for National Statistics uses all three measures when reporting data like average household income (mean £35,000), typical family size (median 2.4), and most common qualification level (mode: 5+ GCSEs). Understanding these concepts helps students interpret news articles, make informed decisions about purchases, and succeed in GCSE statistics questions worth up to 15 marks on foundation papers.
How to solve mean, median & mode
Mean, Median & Mode
- Mean = sum of all values ÷ count.
- Median = middle value when sorted (average of two middles if even count).
- Mode = value that appears most often.
Example: Data: 3, 5, 5, 7, 10. Mean=6, Median=5, Mode=5.
Worked examples
Here are 3 prices: 4, 7, 10. What is the mean?
Answer: 7.0
- Add all the numbers together → 4 + 7 + 10 = 21 — Line up all 3 values and add them one by one. Think of collecting all the prices into one big pile: the total is 21.
- Count how many numbers there are → n = 3 — Count each value in the list. We have 3 numbers. This is important because we'll divide by this count.
- Divide the total by the count → 21 / 3 = 7.0 — Mean = total / count = 21 / 3 = 7.0. If you put all the money together and shared it equally, each item would cost the mean price.
- Verify: does mean x count = total? → 7.0 x 3 = 21.0 (= 21 ✓) — Always check: multiply the mean by the count. If you get back the total (or very close due to rounding), you're correct!
Calculate the average of these heights in cm: 1, 2, 2, 7, 15
Answer: 5.4
- Add all the numbers together → 1 + 2 + 2 + 7 + 15 = 27 — Line up all 5 values and add them one by one. Think of collecting all the heights in cm into one big pile: the total is 27.
- Count how many numbers there are → n = 5 — Count each value in the list. We have 5 numbers. This is important because we'll divide by this count.
- Divide the total by the count → 27 / 5 = 5.4 — Mean = total / count = 27 / 5 = 5.4. If you could somehow even out everyone's height so they were all the same, that common height would be the mean.
- Verify: does mean x count = total? → 5.4 x 5 = 27.0 (= 27 ✓) — Always check: multiply the mean by the count. If you get back the total (or very close due to rounding), you're correct!
Find the median of these prices: 2, 4, 5, 9, 14, 28, 48
Answer: 9
- Put the numbers in order (smallest to largest) → 2, 4, 5, 9, 14, 28, 48 — The median is the middle value, so we need the numbers sorted. Like lining up kids by height to find the one in the middle.
- Count: 7 values (odd) → n = 7 (odd) — We have 7 values. This matters because: if odd, take the exact middle; if even, average the two middle values. With 7 values (odd), the middle position is 4.
- Find the middle value (position 4) → Median = 9 — Position 4 in the sorted list is 9. There are 3 values below it and 3 values above it -- it's right in the middle!
- Verify → Median = 9 ✓ — Check: 3 values below and 3 values above. The median sits right in the centre of the data.
Common mistakes
- Confusing mean and median calculations. Students often write the mean of 2, 5, 8, 12, 15 as 8 (the middle value) instead of calculating (2+5+8+12+15)÷5 = 8.4.
- Finding median without ordering the data first. For the set 15, 3, 7, 12, 9, students incorrectly choose 7 as the median instead of ordering to get 3, 7, 9, 12, 15 and selecting the middle value 9.
- Identifying mode incorrectly in sets with multiple frequencies. In the data 4, 5, 5, 6, 6, 7, students often claim there are two modes (5 and 6) when both appear twice, making this correct—it's bimodal.
- Calculating median with even numbers of values. For 2, 4, 6, 8, students pick either 4 or 6 instead of finding the mean of the middle two values: (4+6)÷2 = 5.