ICSE Class 8 Maths Chapter 30 Statistics

Chapter 30: Statistics

Statistics means collection of information in the form of numerical data, organisation, summarisation and presentation of data by tables and graphs (charts), analysing the data and drawing inferences from the data.

Raw data. The numerical data recorded in its original form as it is measured or received is called raw (or ungrouped) data.

Variable. A quantity which is being measured in an experiment (or survey) is called a variable.

Height, age and weight of people, income and expenditure of people, number of members in a family, marks obtained by students in a test, number of goals scored in a football match etc. are examples of variables.

Range. The difference between the maximum and minimum values of a variable is called its range.

Variate. A particular value of a variable is called variate.

Frequency. The number of times a variates occurs in a given data is called frequency of that variate.

Frequency distribution. A tabular arrangement of given numerical data showing the frequency of the different variates is called frequency distribution, and the table itself is called frequency distribution table.

Presentation Of Data

Suppose there are 47 employees in a Government office. They were asked how many children they have. The results were:

1, 2, 3, 1, 0, 2, 0, 1, 2, 2, 1, 3, 5, 2, 4, 0, 0, 2, 4, 1, 1, 2, 2, 0, 3, 0, 0, 2, 1, 3, 6, 0, 2, 1, 0, 3, 2, 2, 2, 1, 0, 0, 1, 1, 3, 1, 4.

Each entry in the above list is an observation and the collection of these observations is the raw data. Here the number of children is the variable and the numbers 0, 1, 2, 3, 4, 5, 6 are variates.

The data in the above form gives very little information and it is very difficult to interpret it. Let us arrange the above data in ascending or descending order of size. The above data can be written in the ascending order of size as:

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 4, 4, 4, 5, 6

The data arranged in this form is called data array or arrayed data. The presentation of the data in this form gives much more information:

(i) the number of children varies from 0 to 6.

(ii) most of the employees have 0, 1 or 2 children.

(iii) only two employees have more than 4 children.

However, the presentation of data in this form is quite tedious and time consuming, particularly when the number of observations is large.

To make the above (raw) data easily understandable, we present it in the form of a table called frequency distribution table. To prepare this, write the variates in the extreme left column, then we take each observation from the raw data, one at a time, and mark a stroke (|) called tally mark in the next column opposite to the variate. For convenience, we write tally marks in bunches of five, the fifth one crossing the four diagonally. The number of tally marks opposite to a variate is its frequency and it is written in the next column opposite to tally marks of the variate.

The frequency distribution table for the above (raw) data is:

The above table is called simple frequency distribution table.

Teacher's Note

Frequency distribution tables help organize messy data into meaningful patterns, much like how a teacher organizes student test scores to identify which concepts need more teaching.

Example 1

The number of goals scored by a football team in different matches is given below:

3, 1, 0, 4, 6, 0, 0, 1, 1, 2, 2, 3, 5, 1, 2, 0, 1, 0, 2, 3, 9, 2, 0, 1, 0, 1, 4, 1, 0, 2, 5, 1, 2, 2, 3, 1, 0, 0, 0, 1, 1, 0, 2, 3, 0, 1, 5, 2, 0

(i) Construct data array.

(ii) Construct simple frequency distribution table.

Solution

(i) Arranging the given data in ascending order, we get

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 4, 4, 5, 5, 5, 5, 6, 9

(ii) The simple frequency distribution table for the given data is:

Teacher's Note

Creating a frequency table transforms raw sports statistics into actionable insights, helping coaches identify which scoring patterns are most common in their team's performance.

Column (Or Bar) Graphs

In column (or bar) graphs, columns (or bars) of equal width are drawn with various heights. The height of a column (or bar) represents the frequency of the corresponding variate. All columns (or bars) are drawn with equal spacing between them.

Example 2

120 students of class VIII of a certain school use different modes of travel to school as given below:

Represent the above data by a column graph.

Solution

Take the mode of travel along x-axis and the number of students (frequency) along y-axis.

Choose 1 small division = 1 student on y-axis.

Draw columns of equal width with equal spacing between them and heights corresponding to the numbers of students taking different modes of travel to school.

The column graph for the given data is shown below:

[Graph showing columns for Car (24), School Bus (48), Bicycle (18), Walking (10), and Others (20)]

Teacher's Note

Column graphs make it easy to visually compare how many students use each mode of transport, helping school administrators plan better transportation and infrastructure.

Pie Chart

The circle graphs are commonly called pie charts or pie graphs. The pie charts represent the numerical data by various sectors of a circle. As the total angle of a circle is 360-, the angle of the sector corresponding to an item is

Angle of sector = \[\frac{\text{Value of item}}{\text{Sum of values of all items}} \times 360°\]

Example 3

The following table shows the number of students in various classes in a hobby school:

Represent the above data by a pie chart.

Solution

To construct a pie chart, the angles for various sectors can be calculated as:

Total number of students = 180 + 150 + 27 + 75 + 108 = 540.

Hence, the corresponding pie chart is:

[Pie chart showing sectors for Computers (120-), Painting (100-), Pottery (18-), Paper cutting (50-), Glass Work (72-)]

Teacher's Note

Pie charts let school administrators visualize student enrollment patterns across different hobby classes, helping them allocate resources and plan new programs based on popularity.

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