Quartiles and percentiles from a curve A cumulative frequency curve for n values treats the data as spread continuously along the horizontal axis. So the value with half of the data at or below it is read at height n2, not n+12. In the same way, read the lower quartile Q_1 at n4, the upper quartile Q_3 at 3n4 and the kth percentile P_k at k100n. The curve shows the heights of 120 seedlings. Here n4=30, n2=60, 3n4=90 and 90100 x 120=108. Readings: median 12 cm, Q_1=8 cm, Q_3=16 cm, so the interquartile range is 16-8=8 cm. The 90th percentile is 20 cm: about 10% of the seedlings are taller than 20 cm. Method - Find the total n from the top of this curve. - Work out the height needed: n4, n2, 3n4 or k100n. - Go across from that height to the curve, then down to the horizontal axis, and read the value. - For “the top 20%” or “a probability of more than t”, first change it into “at or below”: the top 20% starts at P_80, read at 0.8n. Two conventions side by side **Measure** | **Discrete data (an ordered list of n values)** | **Cumulative frequency curve (n values)** Median | middle value, or the mean of the two middle values (position n+12) | read at height n2 Lower quartile Q_1 | median of the lower half; leave out the median when n is odd | read at n4 Upper quartile Q_3 | median of the upper half; leave out the median when n is odd | read at 3n4 kth percentile | not used for short lists | read at k100n Interquartile range | Q_3-Q_1 | Q_3-Q_1 (an estimate) Range | largest value minus smallest value | end of the curve minus start of the curve: only an estimate, because the class boundaries need not be actual values Checks and common errors - Use this curve’s own total: for n=120 the median is read at height 60. If the vertical axis goes above the total, do not read halfway up the axis. - P_25=Q_1, P_50 is the median and P_75=Q_3. - A count “more than” a value is n minus the reading at that value. - The interquartile range is the spread of the middle half of the data, between the lower quartile and the upper quartile.

Quartiles and percentiles from a cumulative frequency curve

Secondary 4

Quartiles and percentiles from a curve

A cumulative frequency curve for $n$ values treats the data as spread continuously along the horizontal axis. So the value with half of the data at or below it is read at height $\dfrac n2$, not $\dfrac{n+1}2$. In the same way, read the lower quartile $Q_1$ at $\dfrac n4$, the upper quartile $Q_3$ at $\dfrac{3n}4$ and the $k$th percentile $P_k$ at $\dfrac{k}{100}n$.

The curve shows the heights of $120$ seedlings. Here $\dfrac n4=30$, $\dfrac n2=60$, $\dfrac{3n}4=90$ and $\dfrac{90}{100}\times120=108$.

Readings: median $12$ cm, $Q_1=8$ cm, $Q_3=16$ cm, so the interquartile range is $16-8=8$ cm. The $90$th percentile is $20$ cm: about $10\%$ of the seedlings are taller than $20$ cm.

Method

Find the total $n$ from the top of this curve.
Work out the height needed: $\dfrac n4$, $\dfrac n2$, $\dfrac{3n}4$ or $\dfrac k{100}n$.
Go across from that height to the curve, then down to the horizontal axis, and read the value.
For “the top $20\%$” or “a probability of more than $t$”, first change it into “at or below”: the top $20\%$ starts at $P_{80}$, read at $0.8n$.

Two conventions side by side

Measure	Discrete data (an ordered list of $n$ values)	Cumulative frequency curve ($n$ values)
Median	middle value, or the mean of the two middle values (position $\tfrac{n+1}2$)	read at height $\tfrac n2$
Lower quartile $Q_1$	median of the lower half; leave out the median when $n$ is odd	read at $\tfrac n4$
Upper quartile $Q_3$	median of the upper half; leave out the median when $n$ is odd	read at $\tfrac{3n}4$
$k$th percentile	not used for short lists	read at $\tfrac k{100}n$
Interquartile range	$Q_3-Q_1$	$Q_3-Q_1$ (an estimate)
Range	largest value minus smallest value	end of the curve minus start of the curve: only an estimate, because the class boundaries need not be actual values

Checks and common errors

Use this curve’s own total: for $n=120$ the median is read at height $60$. If the vertical axis goes above the total, do not read halfway up the axis.
$P_{25}=Q_1$, $P_{50}$ is the median and $P_{75}=Q_3$.
A count “more than” a value is $n$ minus the reading at that value.
The interquartile range is the spread of the middle half of the data, between the lower quartile and the upper quartile.

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Measure	Discrete data (an ordered list of \(n\) values)	Cumulative frequency curve (\(n\) values)
Median	middle value, or the mean of the two middle values (position \(\tfrac{n+1}2\))	read at height \(\tfrac n2\)
Lower quartile \(Q_1\)	median of the lower half; leave out the median when \(n\) is odd	read at \(\tfrac n4\)
Upper quartile \(Q_3\)	median of the upper half; leave out the median when \(n\) is odd	read at \(\tfrac{3n}4\)
\(k\)th percentile	not used for short lists	read at \(\tfrac k{100}n\)
Interquartile range	\(Q_3-Q_1\)	\(Q_3-Q_1\) (an estimate)
Range	largest value minus smallest value	end of the curve minus start of the curve: only an estimate, because the class boundaries need not be actual values