In a GCSE maths exam, once you have drawn a histogram, you will be expected to analyse it. Luckily there are only a limited number of things they can ask you to do.

- Find the total frequency
- Find the Median and IQR
- Find the total that are greater than, or less than, a specific amount.

**Example: A histogram showing the how late trains were in a month is shown. Find the…**

- Total number of trains

All we have to do for this is to find the value of each rectangle in the histogram.

Remember that the frequency in a histogram is represented by the area of the rectangle.

Class | Frequency |

0 < x ≤ 5 | 125 |

5 < x ≤ 10 | 150 |

10 < x ≤ 20 | 200 |

20 < x ≤ 30 | 100 |

30 < x ≤ 50 | 100 |

50 < x ≤ 60 | 100 |

The total number of trains is therefore 775

- Estimate the median and interquartile range (IQR) for how late the trains were.

We know there are 775 trains in the survey. This means the median would be at the 388^{th} position, the lower quartile would be at 194^{th} number and the upper quartile at 582^{nd} position.

Finding an estimate of the Median (388^{th} position).

We can count through the frequencies and find that the 388^{th} position is in the group 10 < x ≤ 20. The question is how far into this group?

The first two bars use 125 + 150 = 275 of the positions so we need to go 113 (388 – 275) positions into this group.

As the group represents 200 trains, we need to go 113/200 into the group which is 10 wide.

If we calculate 113/200 x 10 we get 5.65

Adding this onto the start of the group gives 10 + 5.65

So the estimated median is 15.65 minutes.

If we repeat this process for the lower and upper quartiles, we get LQ = 7.3 and UQ of 31.4 minutes.

This gives an IQR of 31.4 – 7.3 = 24.1 minutes

- Find the percentage of trains that were more than 45 minutes late.

If we draw a line showing the point we are looking at we get…

As we’re looking for the percentage that are greater than 45 minutes, we need to find the frequency represented by the bars to the right of the line.

The first one is ¼ of the group 30 < x ≤ 50 which will be 25 and the second group is all of the group 50 < x ≤ 60 (100 trains)

With our total of 125 / 775, this is 16.13% (to 2 dp)