How Do You Construct a Histogram or a Cumulative Frequency Graph?

For IB Mathematics: Applications and Interpretation SL 4.2, construct a histogram by plotting class intervals against their frequencies, with touching bars. Construct a cumulative frequency graph by plotting each upper class boundary against the running total of frequencies.

Construction Methods

For example, suppose the frequencies for intervals $0 \leq x < 10$, $10 \leq x < 20$, and $20 \leq x < 30$ are $4$, $7$, and $9$.

The cumulative frequencies are:

$$4,\quad 4+7=11,\quad 4+7+9=20.$$

For the histogram, draw touching bars of heights $4$, $7$, and $9$. For the cumulative frequency graph, plot $(0,0)$, $(10,4)$, $$, and , then join them with a smooth increasing curve.

The final cumulative frequency must equal the total frequency, here $20$. The graph can then estimate the median at cumulative frequency $\frac{20}{2}=10$, and the lower and upper quartiles at $5$ and $15$.

Exam Technique

A common misconception is that an AI SL histogram uses frequency density. It does not: SL 4.2 examination questions use equal class intervals and frequency histograms; frequency density is not required. Label both axes, use a consistent scale, make histogram bars touch, and plot cumulative frequencies against upper boundaries rather than class midpoints.

When reading a cumulative frequency graph, move horizontally from the chosen cumulative frequency to the curve, then vertically to the horizontal axis. This gives an estimate, so quote an appropriate level of accuracy. If asked to construct rather than sketch, plot points accurately and use graph paper scales carefully.