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 number that are greater than, or less than, a specific amount.

**Example:**

Below is a histogram showing how late trains were in a month.

**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.

To find the frequency, we need to use the formula:** frequency = class width x frequency density**

For the first rectangle, the class width is 5 and the frequency density is 25.

If we do the same for the other rectangles, we get the following frequencies:

5 x 25 = 125

5 x 30 = 150

10 x 20 = 200

10 x 10 = 100

20 x 5 = 100

10 x 10 = 100

These can be recorded in the frequency table below:

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 that there are 775 trains in the survey. This means that the median would be at the 388^{th} position, the lower quartile would be at the 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 that we are looking at, we get:

As we’re looking for the percentage that is 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), making a total of 125.

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

That's a lot to remember, so let's move straight on to try out some questions.