Box Plots (GCSE Maths)

• A box plot is a time-consuming method for representing such a small set of data.
• Because of the small set of data, the box plot won't be useful for comparing with other sets of data.

• It is easy to draw the box plot, because each result is one of the 5 parts required.
• A box plot allows you to easily identify outliers.

This value seems to be an outlier in the data set:

To draw the box plot, you must first find the five-number summary from the ordered data set of $15$ times. Identify the minimum and maximum values directly from the ordered list:

Since there are $15$ values, the median is the 8th value.

The lower quartile ($\text{LQ}$) is the median of the first $7$ values, which is the 4th value.

The upper quartile ($\text{UQ}$) is the median of the last $7$ values, which is the 12th value.

The distance between $50$ and $55$ covers $10$ small squares, meaning each small square represents $0.5$ units.

• Plot short vertical lines at the minimum ($52$) and maximum ($75$).
• Plot longer vertical lines at the $\text{LQ}$ ($55$), median ($60$), and $\text{UQ}$ ($64$).
• Connect the tops and bottoms of the lines at $55$ and $64$ to form a box, and draw horizontal whiskers from the middle of the box to the minimum and maximum points.

To draw the box plot, first calculate the minimum value by subtracting the range from the maximum:

Between $80$ and $100$ there are $10$ small squares, which means each small square represents $2$ units.

Draw a short vertical line at $x = 84$ ($2$ small squares after $80$).

Draw a longer vertical line at $x = 110$ (exactly halfway between $100$ and $120$).

Draw a longer vertical line at $x = 132$ ($6$ small squares after $120$).

Draw a longer vertical line at $x = 162$ ($1$ small square after $160$).

Draw a short vertical line at $x = 185$ ($2.5$ small squares after $180$).

• Draw horizontal lines connecting the tops and bottoms of the lines at $110$ and $162$ to create the central box.
• Draw horizontal whiskers from the middle of the box out to the minimum ($84$) and the maximum ($185$).

Use the x-axis scale where each small square represents $1$ unit.

• Minimum: Draw a short vertical line at $x = 11$.
• Lower Quartile: Draw a longer vertical line at $x = 28$.
• Median: Draw a longer vertical line at $x = 37$.
• Upper Quartile: Draw a longer vertical line at $x = 42$.
• Maximum: Draw a short vertical line at $x = 51$.

• Draw horizontal lines connecting the tops and bottoms of the lines at $28$ and $42$ to create the central box.
• Draw a horizontal line right through the middle connecting the minimum ($11$) to the lower quartile ($28$).
• Draw another horizontal line connecting the upper quartile ($42$) to the maximum ($51$).

To draw the box plot, you first need to find the five key values (minimum, lower quartile, median, upper quartile, and maximum) from the ordered data of 12 test results.

Identify the minimum and maximum values directly from the table.

Since there are 12 data points, the median is the average of the 6th and 7th values ($29$ and $31$).

This is the median of the lower half of the data (the first 6 values: $15$, $17$, $17$, $21$, $26$, $29$). It lies halfway between the 3rd and 4th values ($17$ and $21$).

This is the median of the upper half of the data (the last 6 values: $31$, $35$, $40$, $46$, $49$, $57$). It lies halfway between the 3rd and 4th values of this upper subset ($40$ and $46$).

There are 10 small squares between $0$ and $10$, meaning each small square represents $1$ unit.

• Plot short vertical lines for the lower bound at $15$ and the upper bound at $57$.
• Plot longer vertical lines at the LQ ($19$), median ($30$), and UQ ($43$).
• Connect the tops and bottoms of the lines at $19$ and $43$ to form a rectangular box.
• Finally, draw horizontal whiskers connecting the middle of the box to the bounds at $15$ and $57$.

To complete the box plot, you need all five key data points: minimum, lower quartile, median, upper quartile, and maximum. You must calculate the missing lower quartile using the given upper quartile and interquartile range (IQR):

There are $10$ small squares for every $10$ units, so each small square represents $1$ unit.

Draw a short vertical line for the lowest mark at $12$, and another short vertical line for the highest mark at $55$.

• Draw longer vertical lines at the lower quartile ($19$), the median ($32$), and the upper quartile ($41$).
• Connect the tops and bottoms of the lines at $19$ and $41$ to form the central rectangular box.

Draw horizontal lines (whiskers) connecting the middle of the box to the lowest mark ($12$) and the highest mark ($55$).

The inter-quartile range (IQR) is visually represented by the width of the central rectangular box in a box plot, and is defined as:

By looking at the two plots, the box for data set A is much narrower than the box for data set B. Therefore, data set A has the smallest inter-quartile range.

• A standard box plot features a left whisker that connects the lower quartile to the minimum value (the smallest piece of data).
• Because data set B is missing this left whisker, it indicates that the minimum value and the lower quartile fall on the exact same value.
• Thus, the lower quartile is the same as the smallest piece of data.