Cumulative frequency and box plotsIB MYP Maths Standard: Revision notes

Cumulative frequency is a running total of the frequencies. It tells you how many values are less than or equal to the upper end of each class. For the masses $0<m\le2$: 4, $2<m\le4$: 11, $4<m\le6$: 17 the cumulative frequencies are $4$, $4+11=15$ and $15+17=32$. To draw a cumulative frequency graph, plot each total against the upper class boundary, giving points such as $(2,4)$, $(4,15)$, $(6,32)$. Start at the lowest value with cumulative frequency $0$, then join the points with a smooth curve (or straight lines if you are told to). The last point has cumulative frequency equal to the total frequency.

For $n$ values, read the graph at these heights on the cumulative frequency axis:

• lower quartile $Q_1$ at $\frac{n}{4}$
• median at $\frac{n}{2}$
• upper quartile $Q_3$ at $\frac{3n}{4}$ Go across from the height to the curve, then down to the horizontal axis and read the value. For $n=40$ the median is read at $20$, $Q_1$ at $10$ and $Q_3$ at $30$. The quartiles split the data into four equal parts.

The range is the largest value minus the smallest value. The interquartile range (IQR) is $Q_3-Q_1$ and measures the spread of the middle half of the data. Worked example: $Q_1=30$ and $Q_3=50$ minutes, so IQR $=50-30=20$ minutes. The IQR is not affected by extreme values, so it is often a better measure of spread than the range. A smaller IQR means the data is more consistent.

A box-and-whisker plot uses a number line and the five-number summary: minimum, lower quartile, median, upper quartile, maximum.

1. Draw a scale that covers all values.
2. Draw a box from $Q_1$ to $Q_3$.
3. Draw a vertical line inside the box at the median.
4. Draw whiskers from the box out to the minimum and the maximum. The box contains the middle 50% of the data. Each whisker and each half of the box holds about 25% of the data.

Compare two data sets using one measure of average and one of spread, always in context.

• Average: the median. A higher median means higher typical values.
• Spread: the IQR or the range. A smaller spread means more consistent values. Example: restaurant X has median 25 min and IQR 16 min. Restaurant Y has median 26 min and IQR 8 min. Y is slightly slower on average, but its deliveries are much more consistent. About 75% of values lie below $Q_3$, and about 25% lie above it.