Measures of location — mean, median, mode and quartiles

The three averages (mean, median, mode) and the other measures of location (quartiles and percentiles); how to find them from a list and from a frequency table; and how to estimate the median and quartiles of grouped continuous data by linear interpolation.

Statistics · Data presentation

Measures of location — mean, median, mode and quartiles

Why it works

A measure of location is a single value that stands in for "where the data sits". There are three averages, each answering a slightly different question.

The mean adds everything up and shares it out equally:

\bar{x} = \frac{\sum x}{n}, \qquad \text{or from a frequency table} \quad \bar{x} = \frac{\sum f x}{\sum f}.

It uses every value, so it's the most informative — but it's pulled towards outliers and skew.

The median is the middle value once the data is in order. Half the data is below it, half above, so it's barely affected by a few extreme values — the average to trust for skewed data.

The mode is the most common value (or the modal class for grouped data). It's the only average that works for qualitative data — you can have a modal eye colour, but not a mean one.

Finding the median and quartiles of a list (discrete data). The quartiles cut ordered data into four equal parts: the lower quartile

Q_1

, the median

Q_2

, and the upper quartile

Q_3

. For a list of

n

values, Edexcel's rule is:

For $Q_2$ (median) use position $\tfrac{n}{2}$ ; for $Q_1$ use $\tfrac{n}{4}$ ; for

Q_3

use

\tfrac{3n}{4}

If that position is not a whole number, round up and take that value.
If it is a whole number, take the value halfway between that term and the

next.

So for nine ordered values,

\tfrac{n}{4} = 2.25 \to

round up

\to

the

3

rd value is

Q_1

. For eight values,

\tfrac{n}{4} = 2 \to

whole

\to Q_1

is halfway between the

2

nd and

3

rd values.

Estimating from grouped continuous data — linear interpolation. Once data is grouped you've lost the individual values, so you estimate the median/quartiles by assuming the data is spread evenly across the class that contains them. Find the position (

\tfrac{n}{2}

for the median, etc.), locate which class it falls in using the cumulative frequency, then step the right fraction of the way into that class:

Q \approx L + \frac{\,\text{position} - F\,}{f}\times w,

where

L

is the lower boundary of that class,

F

the cumulative frequency before it,

f

the class frequency and

w

the class width. (For grouped data we use

\tfrac{n}{2}

etc. directly — no rounding and no "

+1

".)

Which average? Use the mean when data is roughly symmetric and you want to use every value; the median when there are outliers or skew; the mode for the most popular category or for qualitative data.