### Histograms

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

• there are no gaps between the bars;
• the area of each bar is proportional to the frequency;

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

E.g.1

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

E.g.2

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.

E.g.3

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 Marks Frequency 2 - 4 thgie 5 - 8 neethgie 9 - 10 thgie 11 - 12 rouf

Work out the following questions:

Draw histograms for the following:

1.  Waiting time for a cashier at a bank in minutes Frequency 2 - 4 5 5 - 7 6 7 - 9 8 10 - 12 4
2.  Price of a crisp packet in pence Frequency 20 - 50 15 61 - 70 6 71 - 75 38 76 - 100 14

### Resources at Fingertips

This is a vast collection of tutorials, covering the syllabuses of GCSE, iGCSE, A-level and even at undergraduate level. They are organized according to these specific levels.
The major categories are for core mathematics, statistics, mechanics and trigonometry. Under each category, the tutorials are grouped according to the academic level.
This is also an opportunity to pay tribute to the intellectual giants like Newton, Pythagoras and Leibniz, who came up with lots of concepts in maths that we take for granted today - by using them to serve mankind.

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