ICSE Class 9 Maths Chapter 18 Statistics

Unit 5: Statistics and Graph Work

Chapter 18: Statistics

18.1 Introduction

The word Statistics came into existence towards the middle of the eighteenth century. It seems to have been derived from the Latin word 'Status' or the Italian word 'Statista' or the German word 'Statistik'; each of which means a 'Political State'.

Formerly, the study of statistics was confined only to the collection of data used for the government purposes; but nowadays, its scope is much wider.

Statistics is used in two different senses - singular and plural. When used as singular, it refers to the whole subject as a branch of knowledge which deals with statistical principles and methods used in collecting, analysing and interpreting data.

When used as plural, it refers to the numerical data, collected in a systematic manner, with some definite object in view, in any field of enquiry.

For example, Statistics of:

(i) marks of students in the ICSE examination.

(ii) number of unemployed persons in different states of India, etc.

In mathematical statistics, first of all the numerical facts are collected with some definite object in view and then these facts are organised and analysed to throw light on any sphere of enquiry.

A set of collected numerical facts is called a set of data.

For example:

Consider the marks obtained (out of 10) in a class test by 6 students of class X.

Teacher's Note

Statistics helps us understand population trends and make informed decisions; from hospital data tracking patient recovery rates to schools analyzing student performance across different demographics.

18.2 Variable

A quantity which can vary from one individual to another is called a variable.

e.g. height, weight, age, etc.

1. Continuous Variable

A variable which can take any numerical value within a certain range is called a continuous variable, e.g.

(i) wages of persons,

(ii) heights of children,

(iii) rainfall records of different cities on different days, etc.

2. Discrete (or, discontinuous) Variable

A variable which is incapable of taking all possible numerical values is called a discrete variable.

e.g. number of children in a family; (the number cannot take any value between 1 and 2, 2 and 3, 3 and 4, etc.).

Teacher's Note

Understanding continuous versus discrete variables helps us better analyze everyday measurements; temperature is continuous (98.6 degrees Fahrenheit), while the number of cars in a parking lot is discrete (you cannot have 5.5 cars).

18.3 Raw and Arrayed Data

Consider the marks obtained by 20 students in a test of 50 marks as given below:

08, 15, 23, 16, 46, 32, 42, 37, 28, 19, 45, 33, 22, 11, 20, 40, 50, 39, 34 and 49.

The data in the form, as given above, is called raw or ungrouped data.

If the above data is arranged in ascending or descending order, it is called an arrayed data.

Arrayed data in ascending order is:

08, 11, 15, 16, 19, 20, 22, 23, 28, 32, 33, 34, 37, 39, 40, 42, 45, 46, 49 and 50.

And, arrayed data in descending order is:

50, 49, 46, 45, 42, 40, 39, 37, 34, 33, 32, 28, 23, 22, 20, 19, 16, 15, 11 and 08.

Teacher's Note

Organizing data by arranging it in order makes patterns visible; when schools arrange student test scores from lowest to highest, administrators can quickly identify which students need extra support and which are excelling.

18.4 Tabulation of Data

Consider fifteen children from three families A, B and C. The family A has five children out of which three are boys and two are girls; the family B has four children out of which one is a boy and three are girls; while family C has six children out of which four are boys and two are girls.

These facts can be represented by a table as shown below:

Such a representation of data in the form of a table is called tabulation.

Teacher's Note

Tables organize information so that patterns become easy to see; a restaurant manager uses a table to track which menu items sell best during lunch versus dinner, helping them make smart decisions about inventory and staffing.

18.5 Frequency

It is a number, which tells, how many times does a particular data appear in a given set of data.

e.g. consider the following set of data:

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

In the given set of data, 1 appears 3 times; therefore, frequency of 1 is 3.

Similarly, 2 appears 4 times; therefore, frequency of 2 is 4 and so on.

18.6 Frequency Distribution

A tabular arrangement of data showing their corresponding frequencies is called a frequency distribution.

The table showing data with their corresponding frequencies is called a frequency distribution table.

1. Ungrouped Frequency Distribution

Consider the following data which gives shoe-sizes of 19 pupils:

4 3 5 6 5 6 5

6 3 6 3 3 6 4

6 6 6 4 5

In order to construct a frequency table, make a table as shown below:

In the first column, write the shoe-sizes in ascending (or, descending) order.

The shoe-sizes of different pupils are recorded in the second column (headed with Tally Marks) by marking a short vertical line called a stroke.

The marking of strokes is done as follows:

(i) The first shoe-size in the given data is 4, so make a stroke in the Tally Marks Column opposite to size 4.

(ii) The next shoe-size is 3, so make a stroke opposite to size 3 and so on.

(iii) When four strokes are made opposite to any particular shoe-size, don't make the fifth stroke in the same way but make a stroke across the first four. This gives a bundle of five strokes. The next stroke starts a new bundle.

(iv) When marking of the strokes is complete, count the strokes against each size and write in the column headed as: Frequency.

The table obtained above shows ungrouped frequency distribution.

2. Grouped Frequency Distribution

Following are the examples of grouped frequency distributions:

Given below are the marks obtained by 32 students in an examination:

29 23 30 40 11 01 15 35

40 03 12 18 30 24 25 29

31 32 25 22 27 12 13 02

07 09 19 13 32 39 25 03

Taking class intervals 1 - 10, 11 - 20, -.........., 31 - 40, make a frequency table for the above distribution.

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