Other Measures of Location
The median splits a dataset into two equal halves. Quartiles and percentiles cut it into finer pieces still — quarters and hundredths — giving a much more detailed map of where the data sits.
Quartiles and percentiles
The median divides an ordered dataset into two equal halves. Quartiles divide it into four equal quarters: the lower quartile $Q_1$ sits one-quarter of the way through the data, the median $Q_2$ sits halfway through, and the upper quartile $Q_3$ sits three-quarters of the way through. Percentiles divide the data into 100 equal parts — the $k$th percentile, written $P_k$, has $k\%$ of the data below it.
Finding quartiles for discrete data
For an ordered dataset of $n$ discrete values:
When $\frac{n}{4}$ lands exactly on a whole number, that position sits precisely on the boundary between two data points, so $Q_1$ is taken as their average. When $\frac{n}{4}$ isn't whole, the true quarter-way position falls strictly inside a single data point's "slot" in the order, so rounding up picks out that value directly — no averaging needed.
Percentiles for discrete data
The same logic extends to any percentile: to find $P_k$, calculate $\dfrac{k}{100}\times n$ and apply the identical rule — average two points if the result is whole, otherwise round up and read off that data point.
Interpolation for grouped or continuous data
When data is grouped into classes, individual positions can't be read off directly — instead, interpolation is used, assuming the data values are spread evenly across each class. The positions used are:
Once the class containing that position is identified from the cumulative frequencies, the estimate is found by proportion: if the required position lies a fraction of the way from the lower to the upper cumulative frequency of that class, the estimate lies the same fraction of the way between the class's boundaries.
The "round up if not whole" rule is only for discrete, ungrouped data, where positions must land on an actual data point. For grouped or continuous data using interpolation, $\frac{n}{4}$, $\frac{n}{2}$ and $\frac{3n}{4}$ are used exactly as calculated, with no rounding — interpolation can estimate a value at any fractional position within a class.
Worked example 1
$12, 14, 15, 15, 17, 18, 18, 19, 20, 21, 22, 23, 24, 25, 27, 29$
Worked example 2
$5, 6, 8, 9, 9, 10, 11, 12, 12, 13, 14, 14, 15, 16, 17, 18, 19, 21$
Round up, never down or to the nearest — $4.5$ rounds up to $5$, not down to $4$. Rounding the wrong way is the most common error with this rule.
Worked example 3
Quartiles are just specific, commonly-used percentiles: $Q_1=P_{25}$, $Q_2=P_{50}$ (the median), and $Q_3=P_{75}$. Learning the percentile method means you already know how to find any quartile too.
Worked example 4
| Time, $t$ (min) | Frequency |
|---|---|
| $20\le t<30$ | 8 |
| $30\le t<40$ | 19 |
| $40\le t<50$ | 22 |
| $50\le t<70$ | 11 |
Find estimates for (a) the median, (b) the lower quartile.
Interpolation only ever gives an estimate, since it assumes runners are spread perfectly evenly across each class — real data rarely is. Still, it's the best estimate available without the raw, ungrouped times.
Don't apply the discrete "round up" rule to grouped, continuous data, and don't use raw interpolation positions on a small, ungrouped dataset — matching the method to the type of data is the first decision to make, before any calculation begins.
Check yourself
An ordered dataset has $n=24$ values. Find the positions used to locate $Q_1$ and $Q_3$, and state whether each requires averaging or rounding up.
(Hint: calculate n/4 and 3n/4 first.)