A histograms is a form of bar chart; however, there are two main differences. In a histogram,

So, a histogram must be constructed so that the area of a bar is proportional to the frequency.


Marks obtained for maths in a certain class is as follows:

Classes of MarksFrequencyArea of BarArea of Bar / Frequency
0 - 20510020
21 - 40612020
41 - 60816020
61 - 8048020
81 - 10024020

Since the area is proportional to the frequency - shown by the last column - this data can make a proper histogram as follows:




Now look at the following set of data


Marks obtained for maths in a certain class is as follows:

Classes of MarksFrequencyArea of BarArea of Bar / Frequency
0 - 1055010
11 - 40618030
41 - 5088010
51 - 654205
66 - 10027035

It is obvious that the area of a bar is not proportional to the frequency; therefore, this data, as it is, will not make a proper histogram - it is just a bar chart, that is misleading. The following diagram shows it.




Check the tallest column - it has not got the highest frequency. That is why this graph, when considered as shown, is misleading.

To get round this problem - area being not proportional to the frequency, when class widths are not the same - we define a new term. It is frequency density

Frequency Density = Frequency / Class Width

Now, let's change the data table in the above example with frequency density.

Classes of MarksFrequencyFrequency DensityArea of BarArea of Bar / Frequency
0 - 1050.551
11 - 4060.261
41 - 5080.881
51 - 6540.341
66 - 10020.0621

The introduction of the term, frequency density, has got round the problem; therefore, if the widths of classes are not equal, we must use frequency density, instead of frequency in dealing with histograms. The following histograms shows all that.




Check the tallest column - it now represents the highest frequency; this graph, when considered as shown, is the right one.


Look at the following histogram and then fill in the blanks of the table. It is about the performance ratings of a group of professionals.




Classes of MarksFrequency
2 - 4thgie
5 - 8neethgie
9 - 10thgie
11 - 12rouf

Interactive Practice

The daily rainfall in a certain tropical region is shown in the following histogram: the data has been collected for 25 days; you can see the frequency, frequency density and raw data in the following applet.


The data is confined to five classes or bars. It also shows that Frequency density = Frequency / class width.

Work out the following questions:


Draw histograms for the following:

  1. Waiting time for a cashier at a bank in minutesFrequency
    2 - 45
    5 - 76
    7 - 98
    10 - 124
  2. Price of a crisp packet in penceFrequency
    20 - 5015
    61 - 706
    71 - 7538
    76 - 10014




Recommended Reading


Maths is challenging; so is finding the right book. K A Stroud, in this book, cleverly managed to make all the major topics crystal clear with plenty of examples; popularity of the book speak for itself - 7th edition in print.

Recommended - GCSE & iGCSE


This is the best book available for the new GCSE(9-1) specification and iGCSE: there are plenty of worked examples; a really good collection of problems for practising; every single topic is adequately covered; the topics are organized in a logical order.

Recommended for A Level


This is the best book that can be recommended for the new A Level - Edexcel board: it covers every single topic in detail;lots of worked examples; ample problems for practising; beautifully and clearly presented.