The cumulative frequency graph shows the distribution of times of the first 120 runners to finish a marathon in 2022. This box-and-whisker plot represents the distribution of the times of the first 120 runners to finish the marathon in 2023. Use the two diagrams to complete this table for the two marathons. Simon says that the runners in 2022 were quicker on average. Do you agree? Give a reason for your answer. Complete the frequency table below for the runners in 2023.

1. Compare the average finishing times using the medians

For 2022, the median time is $154.5$ minutes.

From the 2023 box-and-whisker plot, the median time is about $150$ minutes.

Since $154.5>150$, the typical runner in 2022 took longer, so the runners in 2022 were not quicker on average.

1. Find the frequency for $140<t\leq145$ using the lower quartile

There are $120$ runners, so the lower quartile is the $\dfrac{120}{4}=30$th value.

For 2023, the lower quartile is $145$ minutes, so $30$ runners have times $t\leq145$.

Given $4$ runners have $135<t\leq140$,
$\text{frequency for }140<t\leq145=30-4=26.$

2. Find the frequency for $155<t\leq160$ using the upper quartile and total frequency

The upper quartile is the $\dfrac{3}{4}\times120=90$th value.

For 2023, the upper quartile is $155$ minutes, so $90$ runners have times $t\leq155$.

Hence the number of runners with $t>155$ is
$120-90=30.$

Given $9$ runners have $160<t\leq165$,
$\text{frequency for }155<t\leq160=30-9=21.$