Find the Mean for Grouped Data

Do you like to run?

 

 

At EdPlace's Sports Day, students participated in the 100-metre dash event.

Their times were recorded and organised into the following table:

 

Time t [seconds]	Frequency
0 ≤ t < 11	5
11 ≤ t < 13	45
13 ≤ t < 15	110
15 ≤ t < 17	90
17 ≤ t < 20	55

 

Do you remember how to calculate the mean from a frequency table like this?

 

 

We have five steps to this calculation:

 

1) Add all the numbers in the frequency column to get the total number of data points.

 

Time t [seconds]	Frequency
0 ≤ t < 11	5
11 ≤ t < 13	45
13 ≤ t < 15	110
15 ≤ t < 17	90
17 ≤ t < 20	55

 

5 + 45 + 110 + 90 + 55 = 305

 

2) Find the midpoints for all the intervals and organise them in a third column.

 

Time t [seconds]	Frequency	Midpoint
0 ≤ t < 11	5	5.5
11 ≤ t < 13	45	12
13 ≤ t < 15	110	14
15 ≤ t < 17	90	16
17 ≤ t < 20	55	18.5

 

If we can't spot what the number in the middle is, we add the two numbers at the ends of the interval and divide that by 2 - that always works!

e.g. (0 + 11) ÷  2 = 11 ÷  2 = 5.5

 

3) Multiply frequency with the midpoint for each line in the table - make this the fourth column of the table.

 

Time t [seconds]	Frequency	Midpoint	Frequency × Midpoint
0 ≤ t < 11	5	5.5	5 x 5.5 = 27.5
11 ≤ t < 13	45	12	45 x 12 = 540
13 ≤ t < 15	110	14	110 x 14 = 1540
15 ≤ t < 17	90	16	90 x 16 = 1440
17 ≤ t < 20	55	18.5	55 x 18.5 = 1017.5

 

4) Add all of the results in the fourth column (i.e. the frequency x midpoint column) to get the sum of all the data points.

 

27.5 + 540 + 1540 + 1440 + 1017.5 = 4,565

 

5) Finally, divide the sum of all the data points by the total number of data points to get the estimate of the mean.

 

4,565 ÷  305 = 14.97 s

 

Phew - that was a lot of work!

 

 

Why is it only an estimate?

Well, we used midpoints since we don't have the actual data!

So, e.g. 5 times between 0 and 11 s could all be 9's!

 

Ready to grab your calculator and get some practice in?