Medium

Reading the Median and Quartiles from an Ogive

Simple Explanation

To estimate the median from an ogive, find n/2 on the cumulative frequency axis, draw a horizontal line to the curve, then a vertical line down to the data axis — that x-value is the estimated median. Q₁ and Q₃ are found the same way, using n/4 and 3n/4 instead.

Why Do We Need It?

This gives a fast visual estimate of the median and quartiles directly from grouped data, without needing the individual raw data values.

See It

Reading the median from an ogive (n=50, so read at 25)

Worked Example

Estimate the median from cumulative frequencies

Using the cumulative frequency table (boundaries 10,20,30,40,50 → cumulative 5,18,35,45,50, total n=50), estimate the median.

Why Does This Work?

The ogive shows exactly how many data values fall below each point — since the median is, by definition, the value with exactly half the data below it, finding where the curve reaches cumulative frequency n/2 finds exactly that value (using straight-line interpolation within the class where the curve doesn't land exactly on a plotted point).

Real-Life Example

Estimating the median wait time from grouped data

A clinic records patient wait times only in grouped ranges (e.g. 0-10 min, 10-20 min), not exact individual times.

The ogive method estimates the median wait time directly from this grouped data, even though the exact individual wait times were never recorded.

Practice

For a dataset with n=40, what cumulative frequency value corresponds to the median?

Medium

Common mistake

Using n instead of n/2 (or n/4, 3n/4 for quartiles) when locating the median on the cumulative frequency axis — always divide n by the appropriate fraction first.

Quick Review

Median: locate n/2 on the cumulative frequency axis, read across then down.
Q₁: use n/4. Q₃: use 3n/4.
Interpolate within the class where the target cumulative frequency falls.