Mathematics • Year 7 • Unit 4 • Lesson 9

The Median

Build fluency with the three-step recipe: Sort → Find position (n+1)/2 → Read the middle. For odd n: one middle value. For even n: average the two middle values. The median is RESISTANT to outliers, which is why house prices and incomes are usually reported as medians.

Build · I Do / We Do / You Do

1. I do — fully worked example

Read every line. Each step shows the question to ask and the reason for the answer.

Problem. Find the median of: 23, 15, 41, 8, 37, 29, 12.

Step 1 — Sort the data in ascending order.

8, 12, 15, 23, 29, 37, 41.

Reason: NEVER skip this step. The median is the middle of the ORDERED list, not the middle of however you wrote the data down.

Step 2 — Find the median position with (n + 1) ÷ 2.

n = 7 (odd). Position = (7 + 1) ÷ 2 = 4th.

Reason: odd n gives a whole-number position, so there's exactly one middle value.

Step 3 — Read the value at that position.

Counting along: 1st=8, 2nd=12, 3rd=15, 4th=23.

Reason: 3 values sit below (8, 12, 15) and 3 above (29, 37, 41) — perfect balance. Median = 23.

Answer: Median = 23.

Stuck? Revisit lesson § "Median for Odd n" — three steps: Sort → Position → Value.

2. We do — fill in the missing steps

Find the median of 6 values: 14, 22, 18, 25, 11, 19. Fill in each blank. 5 marks

Step 1 — Sort ascending: _____, _____, _____, _____, _____, _____.

Step 2 — Count n: n = ___. This is an _______ number, so we use the _______ -n rule.

Step 3 — Find the two middle positions: Position 1 = n ÷ 2 = ___. Position 2 = n ÷ 2 + 1 = ___.

Step 4 — Read the two middle values: ___ rd value = ___. ___ th value = ___.

Step 5 — Average the two middle values: Median = (___ + ___) ÷ 2 = _______.

Stuck? Revisit lesson § "Median for Even n" — even n means there is NO single middle value, so average the two that sit closest to the middle.

3. You do — independent practice

Eight graduated problems. Show working — final-answer-only earns half marks.

Foundation — odd n, already sorted

3.1 Find the median of: 3, 7, 9, 12, 15. 1 mark

3.2 Find the median of: 1, 2, 4, 6, 8, 9, 10. 1 mark

3.3 Find the median of: 7, 3, 9, 1, 5 (note: UNSORTED!). 1 mark

3.4 Find the median of: 4, 6, 8, 10 (even n). 1 mark

Standard — full method needed

3.5 Find the median of these 8 reaction times (ms): 215, 240, 219, 224, 247, 220, 215, 231. Show your sort first. 2 marks

3.6 A dataset has 11 values. Use the position formula to find which position contains the median. Then write a brief example of an 11-value dataset and read its median. 2 marks

Extension — push your thinking

3.7 Data: 2, 3, 4, 5, 100. (a) Find the mean. (b) Find the median. (c) Which one would you report as "typical" of this dataset, and why? 3 marks

3.8 Five numbers have a median of 12. Four of the numbers are 5, 8, 14 and 20. What is the fifth number? Find ALL possible answers. (Hint: think about where the unknown could sit when sorted.) 3 marks

Stuck on 3.8? With n=5 the median is the 3rd value. Sort the four known values and figure out where the unknown could go.

How did this worksheet feel?

Got it Partly Lost

What I'll revisit before next class:

Answers — Do not peek before attempting

Section 2 — Even-n median (We do)

Step 1: 11, 14, 18, 19, 22, 25.
Step 2: n = 6. Even number → even-n rule.
Step 3: Position 1 = 6 ÷ 2 = 3. Position 2 = 4.
Step 4: 3rd value = 18. 4th value = 19.
Step 5: Median = (18 + 19) ÷ 2 = 18.5.

3.1 — Median of {3,7,9,12,15}

Already sorted. n = 5, position = 3rd = 9.

3.2 — Median of {1,2,4,6,8,9,10}

Already sorted. n = 7, position = (7+1)÷2 = 4th = 6.

3.3 — Median of unsorted {7,3,9,1,5}

Sort: 1, 3, 5, 7, 9. n = 5, position = 3rd = 5. (NOT 9 — that was the middle of the UNSORTED list.)

3.4 — Median of {4,6,8,10}

n = 4 (even). Middle positions 2 and 3 = 6 and 8. Median = (6 + 8) ÷ 2 = 7.

3.5 — Median reaction time

Sort: 215, 215, 219, 220, 224, 231, 240, 247. n = 8 (even). Middle = 4th and 5th = 220 and 224. Median = (220 + 224) ÷ 2 = 222 ms.

3.6 — Position for n = 11

n = 11 (odd). Position = (11+1)÷2 = 6th. Example dataset: 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22 — the 6th value is 12, so median = 12.

3.7 — Mean vs median with outlier

(a) Mean = (2+3+4+5+100) ÷ 5 = 114 ÷ 5 = 22.8.
(b) Sorted, n = 5, position 3rd = 4. Median = 4.
(c) Report the median (4) — it represents the cluster of small values (2, 3, 4, 5) honestly. The mean (22.8) is distorted by the outlier 100 and is not "typical" of any actual value in the data.

3.8 — Finding the missing number

Sort the four known values: 5, 8, 14, 20. With 5 values and median = 12, the 3rd value (after sorting all 5) must be 12. But 12 isn't yet in the list, so it must BE the unknown — and the unknown must sit at position 3 when inserted.
Check: inserting 12 → 5, 8, 12, 14, 20 → 3rd value = 12 ✓. The fifth number is 12. (Any value strictly between 8 and 14 placed as the 3rd would still need to BE 12 to make the median 12, so 12 is the unique answer.)