Procedure

Locating the median graphically from an ogive

Statistics • Median

Statement

To locate the median of a distribution of

N

observations graphically, mark

\frac{N}{2}

on the cumulative frequency axis of the 'less than' ogive, read across to the curve and drop a perpendicular to the variable axis; the foot of that perpendicular is the median, and it is also the abscissa of the point where the 'less than' and 'more than' ogives cross. The mode has a graphical location too, but from the modal bar of the histogram, and the arithmetic mean has no graphical construction at all.

How to use it

1Draw the 'less than' ogive from the cumulative frequency column, plotting each cumulative total against the upper limit of its class.
2Enter the curve at $\frac{N}{2}$ — not at $\frac{N + 1}{2}$ , which belongs to the ungrouped positional formula.
3Drawing both ogives is the quicker route when only the median is wanted: their intersection gives it without any scale reading on the frequency axis.

Worked example

A 'less than' ogive is drawn for

50

observations. The curve reaches a cumulative frequency of

25

directly above the value

34

. What is the median?

Step-by-step

1
$\frac{N}{2} = \frac{50}{2} = 25$
Why: Half the total frequency is the height at which the curve is entered.
2
$median = 34$
Why: Reading across at $25$ and dropping a perpendicular lands on the variable axis at $34$ .

Answer

34

Watch out

The ogive locates the median and the other partition values — quartiles, deciles, percentiles — and nothing else: pairing it with the mode or with the geometric mean is the standard trap. Read the median off a 'less than' ogive against upper class limits; plotting against class marks gives the frequency polygon instead.

Builds on

Median of a data set Cumulative frequency

Also known as

graphical location of the median, median from ogive, ogive median, cumulative frequency curve median, graphical determination of median and mode, intersection of the two ogives

Appears in past papers

2019 (I)

Q69