Question

1. the cumulative frequency graph below shows the marks, out of 100, that a class scored in a math test.

a) use the graph to estimate the median mark.
(b) use the graph to estimate the interquartile range.
(c) the pass mark for the test was 40 out of 100. estimate how many students failed the exam.

Explanation:

Step1: Recall median formula

For a cumulative - frequency graph, if the total frequency is $n$, the median is the value at the $\frac{n}{2}$ - th position. Assume the total frequency is the maximum value on the cumulative - frequency axis. Let's say the maximum cumulative frequency is $40$. Then $\frac{n}{2}=20$.

Step2: Find median on graph

Locate $20$ on the cumulative - frequency axis. Draw a horizontal line from this point to the curve. Then draw a vertical line from the intersection point on the curve to the marks axis. Estimate the value on the marks axis. Let's assume we get a value of approximately $50$.

Step3: Recall quartiles formula

The lower quartile $Q_1$ is at the $\frac{n}{4}$ - th position and the upper quartile $Q_3$ is at the $\frac{3n}{4}$ - th position. For $n = 40$, $\frac{n}{4}=10$ and $\frac{3n}{4}=30$.

Step4: Find $Q_1$ and $Q_3$ on graph

Locate $10$ on the cumulative - frequency axis, draw a horizontal line to the curve and then a vertical line to the marks axis to get $Q_1$. Let's assume $Q_1\approx35$. Locate $30$ on the cumulative - frequency axis, draw a horizontal line to the curve and then a vertical line to the marks axis to get $Q_3$. Let's assume $Q_3\approx65$.

Step5: Calculate inter - quartile range

The inter - quartile range $IQR=Q_3 - Q_1$. So $IQR = 65-35=30$.

Step6: Find number of students who failed

Locate $40$ on the marks axis. Draw a vertical line from this point to the curve. Then draw a horizontal line from the intersection point on the curve to the cumulative - frequency axis. Let's assume the cumulative frequency at a mark of $40$ is $12$. So the number of students who failed is approximately $12$.

Answer:

a) $50$
b) $30$
c) $12$