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§ Statistics

Mean, Median & Mode

§ Statistics

Mean, Median & Mode

CCSS.6.SP4 min read

Mean, median, and mode are three measures of central tendency that describe the typical value within a data set. The mean represents the arithmetic average, the median identifies the middle value when data is arranged in order, and the mode indicates the most frequently occurring value. These statistical measures appear in Year 10 GCSE mathematics and form the foundation for more advanced statistical analysis.

§ 01

Why it matters

Mean, median, and mode provide essential tools for interpreting real-world data across numerous fields. Market researchers use the median household income (£31,400 in the UK) rather than the mean because extreme wealth skews averages upward. Sports statisticians calculate mean goals per match to assess team performance, whilst fashion retailers identify the modal shoe size (size 6 for women) to optimise stock levels. Medical researchers compare median recovery times between treatments, and teachers analyse modal GCSE grades to understand class performance patterns. These measures help distinguish between different types of 'average' — the mean salary of £35,000 might seem typical, but if the median is £28,000, this reveals income inequality. Understanding when to use each measure becomes crucial for accurate data interpretation in economics, psychology, and scientific research.

§ 02

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.

§ 03

Worked examples

Beginner§ 01

The following heights in cm were collected: 3, 6, 9. Calculate the mean.

Answer: 6.0

  1. Add all the numbers together → 3 + 6 + 9 = 18 — Line up all 3 values and add them one by one. Think of collecting all the heights in cm into one big pile: the total is 18.
  2. 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.
  3. Divide the total by the count → 183 = 6.0 — Mean = total / count = 18 / 3 = 6.0. If you could somehow even out everyone's height so they were all the same, that common height would be the mean.
  4. Verify: does mean x count = total? → 6.0 x 3 = 18.0 (= 18 ✓) — Always check: multiply the mean by the count. If you get back the total (or very close due to rounding), you're correct!
Easy§ 02

The following test scores were collected: 8, 8, 9, 12, 14. Calculate the mean.

Answer: 10.2

  1. Add all the numbers together → 8 + 8 + 9 + 12 + 14 = 51 — Line up all 5 values and add them one by one. Think of collecting all the test scores into one big pile: the total is 51.
  2. 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.
  3. Divide the total by the count → 515 = 10.2 — Mean = total / count = 51 / 5 = 10.2. If the test was worth 100 points and everyone got the same score, they'd all have the mean.
  4. Verify: does mean x count = total? → 10.2 x 5 = 51.0 (= 51 ✓) — Always check: multiply the mean by the count. If you get back the total (or very close due to rounding), you're correct!
Medium§ 03

A survey collected these prices: 2, 7, 11, 12, 12, 20, 49. Calculate the median.

Answer: 12

  1. Put the numbers in order (smallest to largest) → 2, 7, 11, 12, 12, 20, 49 — 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.
  2. 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.
  3. Find the middle value (position 4) → Median = 12 — Position 4 in the sorted list is 12. There are 3 values below it and 3 values above it -- it's right in the middle!
  4. Verify → Median = 12 ✓ — Check: 3 values below and 3 values above. The median sits right in the centre of the data.
§ 04

Common mistakes

  • Confusing mean and median calculations, such as finding the mean of 2, 4, 6, 8, 10 as 6 (the median) instead of 6 (which happens to be correct, but through wrong reasoning).
  • Forgetting to arrange data in ascending order before finding the median, leading to selecting 15 as the median of 10, 15, 12, 18, 20 instead of the correct answer of 15.
  • Assuming every data set has a mode, when sets like 1, 2, 3, 4, 5 have no repeated values and therefore no mode.
  • Averaging the two middle values incorrectly for even-numbered data sets, such as finding the median of 3, 5, 7, 9 as 6.5 instead of 6.
  • Including the mode frequency in calculations, mistakenly writing the mode of 2, 2, 2, 5, 8 as 6 (counting three 2s) rather than simply 2.
§ 05

Frequently asked questions

What is the difference between mean, median, and mode?
Mean is the arithmetic average (sum divided by count), median is the middle value when data is ordered, and mode is the most frequent value. For the data set 2, 3, 3, 4, 8: mean = 4, median = 3, mode = 3. Each measure reveals different aspects of the data's centre.
When should you use median instead of mean?
Use median when data contains extreme values (outliers) that skew the mean. House prices are typically reported as median values because a few expensive properties inflate the mean. For example, if five houses cost £200k, £220k, £240k, £260k, and £800k, the median (£240k) better represents typical prices than the mean (£344k).
Can a data set have more than one mode?
Yes, data sets can be bimodal (two modes) or multimodal (several modes). The set 1, 2, 2, 3, 4, 4, 5 has two modes: 2 and 4, since both appear twice. Some data sets have no mode when all values appear with equal frequency, such as 10, 20, 30, 40.
How do you find the median with an even number of values?
With an even count, calculate the mean of the two middle values. For 4, 6, 8, 12, the two middle positions are 6 and 8, so the median is (6 + 8) ÷ 2 = 7. This ensures the median represents the true centre point of the ordered data.
Do mean, median, and mode always have the same units as the original data?
Yes, all three measures maintain the same units as the original data. If measuring weights in kilograms, the mean, median, and mode are also expressed in kilograms. If calculating test scores as percentages, these measures remain as percentages. The units never change during calculation.
§ 06

See also

MeanMedianModeSetInequality
§ 06

Where to next?

Prerequisites
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