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# Distribution of Runners in a Race

Analyzing the distribution of runners based on their finishing times and showing the results on charts for a half-marathon and a marathon.

Have you ever noticed that the majority of the runners at a race are in the mid-pack? Did you ever wonder what is the distribution of these runners? Well, I did and created two charts to see the data better about a half-marathon and a marathon.

## Normal Distribution and the Gaussian Curve

I am not going into details about the mathematics. I just want to mention that I expect the runners finishing a race are normally distributed and can be shown on a chart with a Gaussian Curve — also known as the bell curve.

## Half-Marathon

To create the chart of runners’ finishing times for a half-marathon I picked the Budapest Vivicittá Half-Marathon 2019.

This race had 7359 finishers, with times between 1:08:35 and 3:08:08.

The X-axis of the chart shows the time ranges. The times under the bars show the beginning of that range and the size of the range is 5 minutes. So, if you see 1:50:00 under a bar, it means that bar represents the time range of 1:50:00-1:54:59.

The Y-axis shows the number of runners.

The chart looks how I expected it will be, being fairly close to a perfect Gauss curve.

At this race there were only five runners under 1:15:00, a number so small that it is not even visible on the chart. From there the number of runners climbs constantly until it reaches its peak.

It is visible that most runners finished in the 2:05:00-2:34:59 time range, exactly 3545 runners. That is 48% of all the finishers, so almost half the runners finished in this 30 minutes long time range. No wonder it feels crowded when you are running with the mid-pack during a race!

The cut-off time for this race is 3 hours, counted from the last runner crossing the start line. This gives us a cut-off time of about 3:10 hours in practice. The effect of this cut-off is not noticeable, the majority of runners managed to finish their races before the “sweeper bus” caught them.

## Marathon

To create the chart of runners’ finishing times for a marathon I picked the Budapest Marathon 2019.

This race had 4533 finishers with times between 2:23:22 and 6:00:00.

The X-axis of the chart shows the time ranges. The times under the bars show the beginning of that range and the size of the range is 10 minutes. So, if you see 2:50:00 under a bar, it means that bar represents the time range of 2:50:00-2:59:59.

The Y-axis shows the number of runners.

With only eleven runners below 2:50:00 this was not a very fast race. I wonder if there was some issue with the weather or the route was too hard.

This chart is not as smooth as the half-marathon one we saw previously. There are a few spikes, but the one that really stands out is for the 4:00:00-4:04:59. I am sure those runners wanted to finish under 4 hours, but they could not reach their goals.

As we have seen previously, the majority of runners are in the mid-pack, mostly between 4:00:00 and 5:09:59, precisely 2590 runners. That is 57% of the finishers!

Contrary to the half-marathon, in this case the cut-off time had an effect of the finishers and therefore on the chart. This race had a cut-off time of 5 and a half hours, counted from the last runner crossing the start line, so practically around 5:45 hours.

There is a sudden drop in the chart after 5:50 which suggests that the “sweeper bus” caught quite a lot of runners before they reached the finish line.

## Summary

To sum up it can be said that runners finishing times are normally distributed. Seeing this on charts also confirms that the majority of runners are in the mid-pack.