ned} 7 + 11 &= 18\\ 18 + 13 &= 31\\ 31 + 8 &= 39\\ 39 + 3 &= 42 \end{aligned} $$ 4. The cumulative frequencies are: | Time up to | Cumulative frequency | |---|---:| | 5 seconds | 7 | | 10 seconds | 18 | | 15 seconds | 31 | | 20 seconds | 39 | | 25 seconds | 42 | 5. Check the final cumulative frequency is 42, the same as the total number of people. The last cumulative frequency should equal the total frequency. If it does not, one of your additions is wrong. ## 2. Plotting a Cumulative Frequency Graph A **cumulative frequency graph** shows the running total on the vertical axis. An **axis** is one of the numbered lines on a graph. The upper boundary, or upper bound, is the largest value included in a class interval. For `$5 < t \le 10$`, the upper boundary is 10. For cumulative frequency graphs, plot the upper boundary against the cumulative frequency. Also include the starting point at the lower boundary with cumulative frequency 0. 1. Use the cumulative frequencies from the previous example. 2. Put time in seconds on the horizontal axis and cumulative frequency on the vertical axis. 3. Plot the starting point `(0, 0)` because up to 0 seconds, nobody has been counted. 4. Plot the cumulative frequency points: `(5, 7)`, `(10, 18)`, `(15, 31)`, `(20, 39)`, `(25, 42)`. 5. Join the points with a smooth increasing curve. Do not plot the middle of each class interval. For cumulative frequency, you plot the upper boundary, such as 5, 10, 15 and so on. ## 3. Estimating the Median and Quartiles The graph helps you estimate key positions in the data. - The **median** is the middle value. - The **lower quartile**, written $Q_1$, is one quarter of the way through the data. - The **upper quartile**, written $Q_3$, is three quarters of the way through the data. - The **interquartile range**, or IQR, is $Q_3 - Q_1$. It measures the spread of the middle 50% of the data. To find these from a cumulative frequency graph, use the total frequency, usually called $N$. - Lower quartile position: $\frac{N}{4}$ - Median position: $\frac{N}{2}$ - Upper quartile position: $\frac{3N}{4}$ 1. A cumulative frequency graph represents 60 batteries and their lifetimes. So $N=60$. 2. Find the positions for $Q_1$, the median and $Q_3$: $$ \begin{aligned} Q_1 &: \frac{60}{4} = 15\\ \text{median} &: \frac{60}{2} = 30\\ Q_3 &: \frac{3 \times 60}{4} = 45 \end{aligned} $$ 3. On the vertical cumulative frequency axis, start at 15, 30 and 45. For each one, read across to the curve, then down to the horizontal axis. 4. Suppose the graph gives these estimates: $Q_1=22$, median 31, and $Q_3=41$ minutes. 5. Use the quartile estimates to find the IQR in minutes: $$ \text{IQR} = 41 - 22 = 19 $$ Because the data is grouped and the curve is drawn by hand, medians and quartiles read from a cumulative frequency graph are usually estimates. ## 4. From Cumulative Frequency to Box Plots A **box plot** is a diagram that summarises a set of data using five important values. A box plot shows the five-number summary: lowest value, lower quartile, median, upper quartile and highest value. The **range** is the highest value minus the lowest value. It shows the overall spread. 1. Suppose a set of times has lowest value 4 seconds, $Q_1=10$ seconds, median 14 seconds, $Q_3=21$ seconds and highest value 30 seconds. 2. Draw a horizontal number line with a sensible scale, such as 0 to 30 seconds. 3. Draw a box from 10 to 21. 4. Put a vert