ICSE Class 8 Maths Chapter 34 Collection and Tabulation of Data

Unit 6 - Statistics

Chapter 34 - Collection And Tabulation Of Data

34.1 Introduction

The word statistics is used in two different senses:

1. Singular 2. Plural

1. In the singular sense, it implies the whole subject as a branch of knowledge to be used in collecting, analysing, presenting and interpreting some numerical informations (data).

2. In the plural sense, it implies the collection of numerical data in a systematic manner with some definite object in view.

e.g. Statistics of:

(i) population (ii) taxes (iii) number of failures in your school, etc.

Teacher's Note

Statistics helps us understand patterns in daily life, from weather predictions to exam results, making data meaningful for decision-making.

34.2 Data

A set of numerical facts, collected with a definite object in view, is called data.

e.g. consider the heights (in centimetre) of 9 children in your class:

Here, the set of numbers (numerical facts representing the heights, in cm, of different children), 153, 147, 150, 152, 154 and 153 is called a set of data.

The data can be obtained by individuals (e.g. one may go from person to person to know his income, age, savings, etc.) and by government sources (e.g. the birth rate in a particular period, the rise in prices in a particular period, etc.)

Whatever be the method of collecting the data, once the data is collected, it must be arranged in a systematic form to get a fair idea of the essential points.

The arrangement of data in a systematic form, generally in a table form, is called tabulation.

e.g. consider the following tabulation:

This table represents the results of a particular examination in a certain school.

Teacher's Note

Organizing data into tables, like school examination results, helps us quickly understand trends and make informed conclusions about student performance.

34.3 Frequency

Consider the set of data: 5, 7, 3, 8, 7, 5, 5, 3, 5, 8, 7.

In this set of data, the number 5 occurs four times and we say, the frequency of number 5 is 4.

Similarly, the frequency of number 3 is 2, as it occurs two times in the given set of data, the frequency of number 7 is 3 (why?) and the frequency of number 8 is 2.

Thus frequency is a number which tells how many times does a particular data (number), appear in the given set of data.

Teacher's Note

Understanding frequency helps analyze real-world situations, such as counting how many students prefer different sports or identifying the most popular items in a survey.

34.4 Raw Data, Arrayed Data And Frequency Distribution

Let 30 students of a class score the following marks (out of 10) in a class test: 9, 8, 6, 10, 5, 6, 8, 7, 10, 5, 6, 4, 5, 7, 8, 10, 8, 9, 6, 5, 4, 5, 8, 9, 10, 6, 6, 5, 4.

The data recorded in the original form, as above, is called raw data.

If the given data is arranged in ascending or descending order of their magnitudes, it is called arrayed data or simply, an array.

Thus 4, 4, 4, 5, 5, 5, 5, 6, 6, 6, 6, 6, 7, 7, 8, 8, 8, 8, 8, 8, 8, 9, 9, 9, 10, 10, 10, 10 is an array in which, the given data is arranged in ascending order. Similarly, we can arrange it in descending order also.

The above arrayed data can also be represented in the form of a table as given below:

Such a tabular arrangement of data, showing the frequency of each number (observation), is called a frequency distribution and the table formed is called a frequency distribution table or simply, a frequency table.

Teacher's Note

Creating frequency distributions from exam scores makes it easy to see which marks were most common, helping teachers understand class performance patterns.

34.5 Constructing A Frequency Table

Example 1:

The minimum temperature (in °C) during the day at Delhi has been recorded for 30 days, as given below:

11-8 11-6 11-6 11-4 11-3 11-4

11-7 11-7 11-3 11-4 11-7 11-7

11-6 11-3 11-7 11-5 11-8 11-3

11-3 11-4 11-5 11-3 11-5 11-5

11-8 11-5 11-4 11-3 11-6 11-5

Construct a frequency table for the above data.

Solution:

Steps:

1. In the first column, write the temperatures in ascending (or descending) order.

2. The temperatures of different days are recorded in the second column (headed by Tally Marks) by making a short vertical line called a stroke. The marking of strokes is done as follows:

(i) The first temperature in the given data is 11-8, so make a stroke in the Tally marks column against temperature 11-8.

(ii) The next temperature is 11-6, so make a stroke against temperature 11-6 and so on.

(iii) When four strokes are made against any particular temperature, don't make the fifth stroke in the same way but make a stroke across the first four (like IIII). This gives a bundle of five strokes. The next stroke starts a new bundle.

(iv) When the marking of the strokes is completed, count the strokes against each and write the numbers in the column headed by Frequency.

The table obtained in Example 1 (above) represents ungrouped frequency distribution.

Teacher's Note

Tally marks provide a quick method to count data, similar to how workers might track daily tasks or inventory in a store.

34.6 Grouped Frequency Distribution

Example 2:

Given below are the marks obtained by 40 students in an examination:

40 56 24 41 51 22 59 12 26 46

51 14 23 59 58 29 41 35 36 40

42 43 36 33 38 52 23 24 13 55

18 54 20 57 30 24 43 50 26 15

Taking class intervals 10-20, 20-30, ..., 50-60; construct a frequency table for the above distribution.

Solution:

The frequency table for the given distribution is:

(Ans.)

In this frequency distribution, the mark 20 is included in the class interval 20-30 and not in 10-20. Similarly, mark 30 is included in the class interval 30-40 and not in 20-30, and so on.

Teacher's Note

Grouping data into class intervals simplifies large datasets, much like how restaurants categorize customer spending into price ranges.

34.7 Class-Intervals And Class-Limits

In example 2, given above, 10-20 is called a class-interval which is bounded by two numbers 10 and 20. These numbers (10 and 20) are called class-limits, the smaller number 10 is called the lower class limit and the larger number is called the upper class limit. Similarly, 20-30 is the second class-interval in the same example 2, 20 being the lower class limit and 30, the upper class limit and so on.

34.8 Class Mark

The class-mark of a class-interval is the value midway between its lower class limit and upper class limit.

Thus, class-mark of a class-interval = \(\frac{\text{its lower class limit} + \text{upper class limit}}{2}\)

e.g. class mark of 10-20 = \(\frac{10 + 20}{2}\) = 15;

class mark of 55-60 = \(\frac{55 + 60}{2}\) = 57.5 and so on.

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