Year 7AC9M7ST01

Choosing the Best Measure of Centre

Acquire data sets for discrete and continuous numerical variables and calculate the range, median, mean and mode; make and justify decisions about which measures of central tendency provide useful insights into the nature of the distribution of data.

The mean is not always the best answer. When one value is far from the rest it drags the mean with it, and the median describes the group better.

Builds onYr 7 · Mean, Median, Mode and Range

Method

An outlier pulls the mean

One extreme value can move the mean a long way. It barely moves the median.

Median for skewed data

House prices and salaries are almost always reported as medians.

Mode for categories

The only measure that works when the data is not numbers.

Worked example

e.g.

\text{Wages: } 40,\ 42,\ 44,\ 45,\ 329 \text{ thousand. Which measure is fairest?}

Find the mean. $\frac{500}{5} = 100$
Find the median. $44$
Compare them with the actual data. $\text{four of the five earn under } 46$
Decide which describes a typical worker. $\text{the median, } 44$
Say why. $\text{The } 329 \text{ is an outlier and drags the mean upward.}$

Practice

Say which measure you would use, and why.

\text{House prices in a suburb}

Answer

\text{Median — a few mansions distort the mean.}

\text{Favourite colour of a class}

Answer

\text{Mode — the data is not numerical.}

\text{Test scores with no extreme values}

Answer

\text{Mean — it uses every value.}

\text{Data: } 5,6,6,7,100. \text{ Mean or median?}

Answer

\text{Median} = 6. \text{ The mean of } 24{,}8 \text{ describes nobody.}

\text{Does removing an outlier change the range a lot?}

Answer

\text{Yes — the range depends only on the extremes.}