ICSE Class 8 Maths Statistics Chapter 34 Tabulation of Data

Unit Seven - Statistics

Tabulation of Data

Diagrammatic Representation of Data

Let's Recap

1. Construct a column graph to display the sales figures recorded by a cycle manufacturer in the year 2009.

2. An advertising agency incurs the following expenditure in the month of January 2010. Construct a pie chart showing the comparison of expense heads.

Construct a pie chart showing the comparison of expense heads.

Children, please welcome our Chief Guest. She has come all the way from UK to visit our school's Annual Day function.

There are graphs and tables as well on the next page.

A website, is it? What does this beautiful pie chart show?

Madam, it shows the classification of those who visited our website according to their age groups.

After the Chief Guest leaves...

Well done children, I am proud of you.

These are tabulation of the data from which the pie chart was constructed.

Tabulation Of Data

Raw Data

Class Limits and Class Interval

Mean, Median, and Mode

Grouped Frequency Distribution

Simple Frequency Distribution

Range of Data

Grouped Data

Introduction

Statistics involves the collection, classification, presentation, and analysis of numerical facts or data.

Data needs to be collected from 'sources', by conducting a 'survey' or a search. If data on income of households in a particular colony is required, a door-to-door survey will have to be conducted. Geographical data may be obtained by searching through an atlas. Data on a larger scale, like the population of cities, can be collected from Government sources. Even for simple data like the shoe size of all your classmates you will have to do a small survey in your class. Consider the data given here.

Ankit 6, 5, Anu, Saibal, Vivek, 4, 5, 5, 6, Atula, Rohini

The above collection does not make sense unless it is presented properly.

The tabulation of data makes it easier to understand the information. In algebra the preparation of 'truth tables' helped us make graphs of equations. As the volume of data increases, new methods of tabulating the data will need to be used.

Raw Data

Consider the following data collected on the number of siblings (brothers and sisters) each student in Class VIII of a particular school has.

2 1 0 0 1 3 1 1 2 0 1 0 0 3 2 2

2 3 0 1 0 0 1 1 2 1 1 0 2 3 1 1

Each digit above represents an observation, which is the number of siblings a particular student has. This raw data does not help us get much information on the subject.

We know that the arithmetic mean, or average of n terms = \(\frac{\text{Sum of n terms}}{n}\)

The given raw data on siblings per student has 32 observations, the sum of which is 38.

Thus Arithmetic Mean of given data = \(\frac{38}{32} = 1.1875\)

Now, no student can have 1.1875 brother or sister. The mean merely indicates that most students have 'about' 2 siblings each.

Example 1: If the weights of 5 students are given as 38 kg, 41 kg, 36 kg, 39, kg, and 41 kg, find the mean of the given data.

The Arithmetic Mean of the given data

\[= \frac{38 + 41 + 36 + 39 + 41}{5} = \frac{195}{5} = 39 \text{ kg}\]

Thus, the mean weight of the 5 students is 39 kg.

Array

If the data on the number of siblings is arranged in ascending or descending order, it forms an array.

0 0 0 0 0 0 0 0 1 1 1 1 1 1 1

1 1 1 1 2 2 2 2 2 2 2 3 3 3 3

From this array we can deduce that the number of siblings each classmate has varies from 0 - 3.

In geometry we have seen how a median divides a triangle in two equal halves.

In statistics the middle term of all observations is known as the median.

Once the raw data has been written in an array, the \(\left(\frac{n+1}{2}\right)^{\text{th}}\) term is its median.

Example 2: Find the median in the following array:

7, 9, 11, 13, 15, 17, 19

There are 7 observations in the array.

The median is the \(\left(\frac{7+1}{2}\right)^{\text{th}}\) or the 4th observation.

Thus, median of the given array = 13

The given array on siblings per student has 32 observations.

The median is given as:

\[\frac{32+1}{2} = 16.5\]

The median of this array will be the average of the 16th and the 17th observations.

Thus, the median of the given array = \(\frac{1+1}{2} = 1\)

Try this! Find the mean, median, and mode of the following array: 2, 4, 6, 8, 10, 12, 14, 16

Frequency

Now, many students have 0 sibling while some have 3 siblings. The number of times an observation occurs in a set of data is known as its frequency of occurrence or simply frequency.

The value of the greatest frequency is known as the mode of the given data.

Example 3: The ages of 10 students are given as 13.5, 14.5, 14, 14, 14, 14.5, 14, 13.5, 14, and 13.5 years. Find the mode.

We observe that

3 students are 13.5 years old

5 students are 14 years old

2 students are 14.5 years old

As the maximum number of students are 14 years old, mode of the given data = 14 years.

In the given array on siblings per student, 1 is the observation that occurs the maximum number of times.

Thus, the mode of the given array = 1.

Simple Frequency Distribution

The tabular presentation of the frequency of all the observations is known as a frequency distribution. To construct a frequency distribution the observations are first listed in a column. Each occurrence of an observation is marked with a tally mark like '/'. After 4 tally marks against an observation, the fifth time an observation is repeated, it is marked with a long reverse tally mark that crosses out the first four marks. This makes it easier to read off the frequency of each observation.

is not easy to read off as 12.

is easy to read off as 12 as each crossed out bunch of marks represents a frequency of 5.

Try this! Construct a frequency table for the following data: 5, 4, 4, 3, 2, 2, 1, 5, 4, 3, 1, 5, 2, 4, 1

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