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Functional Skills Level 2 — Statistics

The Median

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What Is the Median?

Sort → Find position \(\dfrac{n+1}{2}\) → Read off the value

The median is the middle value of a dataset once every value has been arranged in order, from smallest to largest. For \(n\) values, the median sits at position \(\frac{n+1}{2}\). Unlike the mean, it isn't affected much by extreme values, which makes it useful for skewed data.

7 values sorted — position 4 is the median 2 4 5 7 9 11 15 1st 2nd 3rd 4th ★ 5th 6th 7th Median = 7 (position = (7+1)÷2 = 4) Even count (6 values) — average the two middle values 3 5 8 10 13 16 1st 2nd 3rd 4th 5th 6th Median = (8 + 10) ÷ 2 = 9

Example 1 — Median of 3, 9, 1, 7, 5

Step 1
Sort: \(1,\; 3,\; 5,\; 7,\; 9\)
Step 2
Find middle position: \(\frac{5+1}{2} = 3\text{rd}\)
Step 3
3rd value = \(\mathbf{5}\)

Example 2 — Median of 12, 4, 8, 20, 6, 16

Step 1
Sort: \(4,\; 6,\; 8,\; 12,\; 16,\; 20\)
Step 2
Even count (6): middle positions are 3rd and 4th
Step 3
Average: \((8 + 12) \div 2 = \mathbf{10}\)
⚠ Most common mistake: Forgetting to sort first. Always sort before identifying the middle value.

✓ Quick Check 1 — Finding the median (odd and even counts)

Question 1 of 10

Even Count — Averaging the Two Middle Values

Even \(n\): Median = \(\dfrac{\text{value at } \frac{n}{2}\text{th position} + \text{value at } (\frac{n}{2}+1)\text{th position}}{2}\)

When there is an even number of values, there is no single middle. Take the mean of the two central values.

6 values — average positions 3 and 4 7 11 14 18 22 30 Median = (14 + 18) ÷ 2 = 16 For n = 6: positions are n/2 = 3rd and n/2+1 = 4th Median = (3rd value + 4th value) ÷ 2

Example 3 — Median of 17, 5, 23, 9, 14, 11

Step 1
Sort: \(5,\; 9,\; 11,\; 14,\; 17,\; 23\)
Step 2
\(n = 6\): middle positions are 3rd and 4th → 11 and 14
Step 3
Median = \((11 + 14) \div 2 = \mathbf{12.5}\)

Example 4 — Median of 30, 10, 50, 20, 40, 60, 70, 80

Step 1
Sort: \(10,\; 20,\; 30,\; 40,\; 50,\; 60,\; 70,\; 80\)
Step 2
\(n = 8\): 4th and 5th values → 40 and 50
Step 3
Median = \((40 + 50) \div 2 = \mathbf{45}\)
Quick rule: Odd \(n\) → one middle. Even \(n\) → average the two middles.

✓ Quick Check 2 — Even count and the two-middle rule

Question 1 of 10

Median vs Mean — When Outliers Matter

An outlier is a value much larger or smaller than the rest. It pulls the mean away from the typical value but does not affect the median as much.

House Prices — One Outlier Distorts the Mean £180k £200k £220k £240k £260k £1.2M OUTLIER Mean ≈ £383k Median = £230k Mean vs Median — Outlier Impact Average type Without outlier With outlier Mean £220k £383k ⚠ Median £220k £230k ✓ Median is more representative when outliers are present.

Example 5 — Wages: 5 workers earn £18k, £20k, £22k, £24k, £26k and the manager earns £120k

Step 1
Mean: \((18+20+22+24+26+120) \div 6 = 230 \div 6 \approx £38\text{k}\)
Step 2
Median: sort (already done), \(n=6\), average 3rd & 4th: \((22+24) \div 2 = £23\text{k}\)
Step 3
Most workers earn around £23k — median is more representative here.

Example 6 — When to use the mean

Step 1
If data has no outliers and is roughly symmetrical, use the mean — it uses every value.
Step 2
If data has outliers or is skewed, use the median — it is resistant to extreme values.
Key rule: Use the median for house prices, income data, or any dataset with extreme values. Use the mean when all values are similar.

✓ Quick Check 3 — Median vs mean, outliers, real-life context

Question 1 of 10

Interactive — Median Sorter

Drag the cards into ascending order (smallest → largest).

Practice: Finding the Median

Sort each list and find the median. Type your answer, then click Check.

#QuestionAnswerResult
1Find the median of 3, 9, 1, 7, 5
2Find the median of 12, 4, 8, 20, 6, 16
3Find the median of 11, 3, 7, 15, 9
4Find the median of 10, 4, 2, 8
5Find the median of 20, 5, 15, 10
6Find the median of 17, 5, 23, 9, 14, 11
7Find the median of 30, 10, 50, 20, 40, 60, 70, 80
8House prices: £150k, £160k, £170k, £180k, £1.5M. Find the median (in £k).
9Find the median of 2, 2, 4, 6, 8, 8
10Find the median of 100, 200, 300, 400, 500, 600

⚠ Things to Watch Out For

  • Forgetting to sort first: The median is the middle value of the SORTED list. Picking the middle number from the original, unsorted order gives the wrong answer.
  • Even count — forgetting to average the two middles: With an even number of values there is no single middle. Add the two middle values and divide by 2.
  • Getting the position formula wrong: The median sits at position \((n+1) \div 2\) for odd \(n\). Miscounting the position is a common source of errors.
  • Confusing median with mean: The median does not involve adding all the values — it only needs the middle position(s) once sorted.
  • Not checking the list length: Always count how many values there are before deciding whether you need one middle value (odd) or two averaged together (even).
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