Formula

Mode of a grouped frequency distribution

Statistics • Mode

Formula

Mode = L + \frac{f _{1} - f _{0}}{2 f _{1} - f _{0} - f _{2}} \times h

For a continuous frequency distribution with equal class width

h

, let the modal class be the one of greatest frequency

f_{1}

, with lower boundary

L

, preceding frequency

f_{0}

and following frequency

f_{2}

. Then

Mode = L + \frac{f _{1} - f _{0}}{2 f _{1} - f _{0} - f _{2}} \times h

How to use it

1Identify the modal class first; the formula then only needs the two frequencies either side of it.
2The mode leans towards whichever neighbour is heavier, so a quick sanity check is whether the answer sits on the correct side of the class mid-point.
3When only the modal class is wanted, stop at the greatest frequency — many questions are decided by that alone.
4With unequal class widths, compare frequency densities rather than frequencies before choosing the modal class.

A distribution has classes

0

10

10

20

and

20

30

with frequencies

5

12

and

8

. Find the mode.

Step-by-step

1
$L = 10, f_{1} = 12, f_{0} = 5, f_{2} = 8, h = 10$
Why: The greatest frequency $12$ makes $10$ - $20$ the modal class.
2
$Mode = 10 + \frac{12 - 5}{24 - 5 - 8} \times 10 = 10 + \frac{70}{11}$
3
$= 10 + 6.36 = 16.36$

Answer

16.36

Watch out

The formula needs exclusive (continuous) class boundaries — convert an inclusive table such as

10

19

20

29

first, or

L

is wrong by half a unit. It also assumes equal class widths. It is a different quantity from the empirical estimate

Mode = 3 Median - 2 Mean

, which will not agree with it.

Builds on

Also known as

modal class formula, grouped mode formula, mode by interpolation, mode of continuous distribution