8th Class Mathematics Data Handling

DATA HANDLING

FUNDAMENTALS

The word 'statistics' is derived from the Latin word 'status' which means political state. Political states had to collect information about their citizens to facilitate governance and plan for development. Then, in course of time, statistics came to mean a branch of mathematics which deals with collection, classification, presentation and analysis of numerical data.

In this chapter, we shall learn about the classification of data, i.e., grouped data and ungrouped data, measures of central tendency, and their uses.

Data

The word 'data' means, information in the form of numerical figures or a set of given facts.

For example, the percentage of marks scored by 10 students of a class in a test are: 36, 80, 65, 75, 94, 48, 12, 64, 88 and 98.

Statistics is basically the study of numerical data. It includes methods of collection, classification, presentation, analysis of data and inferences from data. Data as can be qualitative or quantitative in nature. If one speaks of honesty- beauty, colour, etc., the data is qualitative, while height, weight, distance, marks, etc., are quantitative. Data can also be classified as: raw data, and grouped data.

Raw Data

Data obtained from direct observation is called raw data.

The marks obtained by 100 students in a monthly test are an example of raw data or ungrouped data.

In fact, little can be inferred from this data. However, arranging the marks in ascending order in the above example is a step towards making raw data more meaningful.

Grouped Data                                     

To present the data in a more meaningful way, we condense the data into convenient number of classes or groups, generally not exceeding 10 and not less than 5. This helps us in perceiving at a glance, certain salient features of data.

Some Basic Definitions

Before getting into the details of tabular representation of data, let us review some basic definitions:

Observation: Each numerical figure in a data is called an observation.

Frequency: The number of times a particular observation occurs is called its frequency.

Tabulation or Presentation of Data

A systematical arrangement of the data in a tabular form is called 'tabulation' or 'presentation' of the data. This grouping results in a table called ''frequency table' that indicates the number of scores within each group.

Many conclusion about the characteristics of the data, the behavior of variable, etc., can be drawn from this table.

The quantitative data that is to be analyzed statistically can be divided into three categories:

Individual Series: Any raw data that is collected forms an individual series.

Example:

The weights of 10 students.

36, 35,32,40,65,48,54,71, 62 and 33

Percentage of marks scored by 10 students in a test:

46, 66, 96, 99, 90, 36, 81, 73, 59, 48

Discrete Series: A discrete series is formulated from raw data. Here, the frequency of the observations is taken into consideration.

Example: Given below is the data showing the number of computers in 15 families of a locality.

1, 1, 2, 3, 2, 1, 4, 3, 2, 2, 1, 1, 1, 1, 4

Arranging the data in ascending order:

1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 3, 3, 4, 4

To count, we can use tally marks. We record tally marks in bunches of five, the fifth one crossing the other four diagonally, i.e.,

Thus, we may prepare the following frequency table.

Continuous Series: When the data contains large number of observation, we put them into different groups called 'class intervals'. Such as 1 - 10, 11 - 20, 21 - 30, 31 - 40, 41 - 50 etc.

Here, 1-10 means data whose values lies between 1 and 10 including both 1 and 10.

This form is known as 'inclusive form'. Also, 1 is called the 'lower limit⁹ and 10 is called the 'upper limit^’.

Example:

Given below are the marks (out of 50) obtained by 30 students in an examination.

Taking class intervals 1 - 10, 11 - 20, 21 - 30. 31 - 40 and 41 - 50, we prepare a frequency distribution table for the above data.

First, we write the marks in ascending order as:

Now, we can prepare the following frequency distribution table:

Now with this idea, let us review some more concepts about tabulation.

Arithmetic means of raw data (when fluencies are not given): The arithmetic mean of raw data is obtained by adding all the values of the variables and dividing the sum by the total number of values that are added.

If \[{{x}_{1}}\text{, }{{x}_{2}}........~{{x}_{n}}\] where 'n' is the total number of values, then arithmetic mean.

\[\left( \overline{x} \right)=\frac{{{x}_{1}}+{{x}_{2}},......+{{x}_{n}}}{n}=\frac{1}{n}\sum\limits_{i=1}^{n}{{{x}_{1}}}\]

Bar Graphs

• Representing data with the help of bars or rectangles of uniform width in a diagram is called a bar graph or a bar diagram.
• Each bar represents only one value of the data and hence there are as many bars as there are values in the data.
• The length of the bar indicates the value of the item. The width of the bar does not indicate anything.
• All bars should rest on the same line called the base.
• The bars may be drawn horizontally or vertically.
• A double bar graph helps us to compare two collections of data at a glance.