Learn9FM0Option · Further Statistics 2Continuous probability distributionsMedian and percentiles

Continuous probability distributions · Subtopic 4 of 4

Median and Percentiles

The median cuts the area under the pdf in half: integrate from the lower limit up to $m$ and set the answer equal to $\tfrac12$ . Quartiles and percentiles work the same way with $\tfrac14$ , $\tfrac34$ or any other fraction.

Cambridge 9709 Paper 6Cambridge 9231 Paper 4Edexcel YMA01 WST02Edexcel 9FM0 OptionAQA 7367 Paper 3≈ 30 minRevision 6 · updated 2026-10-27

Section 1 of 14

Learning objectives

Find the median $m$ from $\int_a^mf(x)\,dx=\tfrac12$ .
Find quartiles and percentiles in the same way.
Show that a median lies between two given values.
Compare mean, median and mode to describe skew.

Pause and think

For $f(x)=\tfrac x8$ on $[0,4]$ , the mean is $\tfrac83\approx2.67$ . Is the median bigger or smaller than $2$ ? Than the mean?

Section 2 of 14

Core idea

Start hereSplitting the area in half

The median waiting time is the time you'll beat half the time. For a pdf, it's where a vertical line cuts the area into two equal halves.

The big idea

Median $m$ : $\int_{\text{start}}^mf(x)\,dx=0.5$ . Quartiles: areas $0.25$ and $0.75$ . Mode: where $f$ is greatest.

1. Solving for the median

f(x) = x/2 on [0, 2]: half the area lies left of √2

\displaystyle\int_0^m\frac x2\,dx=\frac{m^2}{4}=0.5\Rightarrow m=\sqrt2

. The median (

1.414

) is a little above the mean (

\tfrac43\approx1.333

), because the thin tail of the area is on the left.

Worked example $f(x)=\tfrac x2$ on $[0,2]$

Find the median.

Goal: $m$ .
Strategy: Area to the left $=0.5$ .

$\int_0^m\tfrac x2dx=\tfrac{m^2}4$
Integrate.
$\tfrac{m^2}4=0.5\Rightarrow m=\sqrt2$
Positive root (in range).

Answer

\sqrt2\approx1.414

✓ CheckMedian

1.414

> mean

1.333

: tail to the left (negative skew) ✓.

Worked example $f(x)=\tfrac38x^2$ on $[0,2]$

Find the median and lower quartile.

Goal: Two values.
Strategy: $F(x)=\frac{x^3}8$ .

$\tfrac{m^3}8=0.5\Rightarrow m=\sqrt[3]4=1.587$
Median.
$\tfrac{q^3}8=0.25\Rightarrow q=\sqrt[3]2=1.260$
Lower quartile.

Answer

1.587

;

1.260

✓ Check

F(2)=1

✓.

Edge caseMode at an end If

f

increases throughout its range, the mode is the right-hand end point.

Pause and think

Symmetric pdf: mean vs median?

Summary

Median: area $0.5$ .
Mode: peak.

Median: Half the probability each side.

HistoryGalton's median. The word "median" in its statistical sense was popularised by Francis Galton in the 1880s, along with "quartile" and "percentile". Galton liked the median because it could be found by simply lining up people in order, with no arithmetic at all.

Fun factHalf-life is a median. The half-life of a radioactive atom is the median of its lifetime: the time by which it has a

50\%

chance of having decayed. For carbon-14 that median is about

5730

years. A single atom might decay in a second or survive for

50\,000

years.

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