Year 10AC9M10ST02

Boxplots and Comparing Distributions

Compare data distributions for continuous numerical variables using appropriate data displays including boxplots; discuss the shapes of these distributions in terms of centre, spread, shape and outliers.

A boxplot throws away almost all the data and keeps five numbers. That is the point — it makes two groups comparable at a glance, in a way a list of results never can.

Method

The five-number summary

\text{min} \;\; Q_1 \;\; \text{median} \;\; Q_3 \;\; \text{max}

Every boxplot is built from these five and nothing else.

The box holds the middle half

IQR = Q_3 - Q_1

Half the data sits inside the box. The interquartile range is the spread of that middle half.

Each section is a quarter

Each whisker and each half of the box holds 25% of the data — however long or short it looks.

Long section means spread out

A long whisker does not mean more data. It means the same quarter of the data is stretched over a wider range.

The outlier rule

\text{outlier} < Q_1 - 1{,}5 \times IQR \quad\text{or}\quad > Q_3 + 1{,}5 \times IQR

Anything past those fences is plotted as a separate point.

Worked example

e.g.

\text{Test marks: } 12,\ 18,\ 19,\ 22,\ 24,\ 26,\ 29,\ 31,\ 34,\ 40

Put the data in order. Nothing works until you do. $12,\ 18,\ 19,\ 22,\ 24,\ 26,\ 29,\ 31,\ 34,\ 40$
Find the median. With 10 values it sits between the 5th and 6th. $\text{med} = \frac{24+26}{2} = 25$
Split the data in half. $Q_1$ is the median of the lower half. $Q_1 = 19$
$Q_3$ is the median of the upper half. $Q_3 = 31$
Write the five-number summary and work out the IQR. $12,\ 19,\ 25,\ 31,\ 40 \qquad IQR = 31 - 19 = 12$
Check for outliers using the fences before you draw. $19 - 18 = 1 \quad\text{and}\quad 31 + 18 = 49 \quad\therefore\quad \text{none}$
Draw the box from $Q_1$ to $Q_3$ , the line at the median, whiskers to min and max.
To compare two groups, draw them on the same scale and talk about centre, spread and shape.

Practice

Find the five-number summary first, then answer.

\text{Find the five-number summary of } 4,\ 7,\ 8,\ 11,\ 13,\ 15,\ 21

Answer

4,\ 7,\ 11,\ 15,\ 21 \qquad IQR = 8

\text{A boxplot has } Q_1 = 30,\ Q_3 = 54. \text{ Find the fences.}

Answer

IQR = 24 \quad\therefore\quad 30 - 36 = -6 \ \text{ and } \ 54 + 36 = 90

\text{Two classes have the same median but Class B has a much smaller IQR. What does that mean?}

Answer

\text{Same typical mark, but Class B is far more consistent.}

\text{The right whisker is three times the length of the left. What does that tell you?}

Answer

\text{The distribution is skewed right — the top 25\% is stretched over a much wider range.}

\text{What percentage of the data lies between } Q_1 \text{ and the median?}

Answer

25\%

\text{Marks: } 3,\ 21,\ 23,\ 24,\ 25,\ 26,\ 28. \text{ Is 3 an outlier?}

Answer

Q_1 = 21,\ Q_3 = 26,\ IQR = 5 \quad\therefore\quad 21 - 7{,}5 = 13{,}5 \quad\therefore\quad \text{yes}