Interquartile Range from a Cumulative Frequency Graph

Grades 9-10
Student math problem: Interquartile Range from a Cumulative Frequency Graph
Original question
Interactive Interquartile Range from a Cumulative Frequency Graph graph — click to explore
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Question: (b) Use your graph to find an estimate for the interquartile range of the times. .............................................................. hours (2) 25 workers spent more than W hours working from home. (c) Use your graph to find an estimate for the value of W W = .............................................................. (2) One of the 60 workers is chosen at random. This worker spent H hours working from home. (d) Find the probability that 5 < H eaqual and smaller 10
To solve these problems using the cumulative frequency graph, we follow the distribution of the 60 workers across the working hours.

(b) Estimating the Interquartile Range (IQR)

The Interquartile Range is the difference between the Upper Quartile ( Q_3) and the Lower Quartile ( Q_1).
  1. Lower Quartile ( Q_1): This represents the value at 25\% of the total frequency.
    • Calculation: 0.25 \times 60 = 15.
    • From the y-axis at 15 (marked with a blue horizontal line), move horizontally to the curve and then vertically down to the x-axis. This gives Q_1 \approx 9.1 hours.
  2. Upper Quartile ( Q_3): This represents the value at 75\% of the total frequency.
    • Calculation: 0.75 \times 60 = 45.
    • From the y-axis at 45 (marked with a blue horizontal line), move horizontally to the curve and then vertically down to the x-axis. This gives Q_3 \approx 21.4 hours.
  3. IQR Calculation: IQR = Q_3 - Q_1 \approx 21.4 - 9.1 = \mathbf{12.3} hours.

(c) Finding the value of W

We are told that 25 workers spent more than W hours working from home.
  • Since the total number of workers is 60, "more than 25" corresponds to the bottom 60 - 25 = 35 workers on the cumulative frequency scale.
  • Locate 35 on the y-axis (marked with a green horizontal line). Move horizontally to the curve and then vertically down to the x-axis.
  • This indicates W \approx \mathbf{17.7} hours.

(d) Probability for 5 < H \le 10

To find the probability that a randomly chosen worker spent between 5 and 10 hours working:
  1. Find the cumulative frequency at t = 10: CF(10) = 17 (from the table/graph).
  2. Find the cumulative frequency at t = 5: CF(5) = 6 (from the table/graph).
  3. The number of workers in this interval is 17 - 6 = 11.
  4. The probability is \frac{\text{number of workers in range}}{\text{total workers}} = \frac{11}{60} \approx \mathbf{0.183}.
The diagram below visualizes these readings: the points from the table are plotted in black, the cumulative curve is red, and the dashed lines show how to extract the quartiles and W.
statisticscumulative-frequencyinterquartile-range