This chapter holds significant importance in CTET and State TETs exams. It covers various methods of data presentation and measures of central tendency. Analyzing past CTET and State TETs exams reveals that typically 1 to 2 questions are asked from this chapter each year.

Data and its Presentation

In our day-to-day life, we often encounter various types of information, such as:

• Runs made by a batsman in the last 10 test matches.
• Number of wickets taken by a bowler in the last 10 ODIs.
• Marks scored by students in a Mathematics unit test.
• Number of story books read by each of your friends.

This information, collected for study purposes, is known as data. Data is typically collected in two types based on their sources:

• Primary data: Data collected directly by the investigator for a specific purpose.
• Secondary data: Data obtained from other sources, whether published or unpublished.

Presentation of Data

Presentation of data involves organizing collected data in a simple form that is easily analyzed and interpreted. There are various methods to represent collected data:

Presentation of Data in Ascending or Descending Order

Consider the marks obtained by 10 students in a Mathematics test as given below:

Raw data: 55, 36, 95, 73, 60, 42, 25, 78, 75, 62

Now, arrange this raw data in ascending order:

Ascending order: 25, 36, 42, 55, 60, 62, 73, 75, 78, 95

By organizing data in this manner, we can easily identify the lowest and highest marks. The difference between the highest and lowest values in the data set is known as the range:

Range: 95 - 25 = 70

Presentation of Data in the Form of Ungrouped Frequency Distribution

If the number of observations in a dataset is large, arranging them in ascending or descending order can be time-consuming. In such cases, data can be presented using a frequency distribution method:

Example: Consider the marks obtained (out of 100 marks) by 25 students of Class IX:

Presentation of Data in the form of Grouped Frequency Distribution

When there are a large number of distinct values in a dataset, it's convenient to present the data in grouped frequency distribution:

(i) In Exclusive Form

(ii) In Inclusive Form

Graphical Representation of Data

Raw data can be represented in various pictorial forms to draw inferences. This process is called graphical representation of data. Some of the common types are:

1. Bar graphs
2. Histograms
3. Frequency polygon
4. Pie charts

Bar Graph

A bar graph is a pictorial representation of data where bars of uniform width are drawn with equal spacing between them. One axis (X-axis) represents the categories or variables, while the height of the bars on the other axis (Y-axis) depends on the values or frequencies of the corresponding observations.

Graphical Representation of Data - Histogram and Frequency Polygon

Histogram

A histogram is a graphical representation of a frequency distribution in exclusive form. It consists of rectangles with continuous class intervals as bases and corresponding frequencies as heights. There are no gaps between consecutive rectangles.

Example: Price Growth Percentage

Determine in which year India had the maximum growth in price:

Frequency Polygon

A frequency polygon is obtained by joining the mid-points of the upper horizontal sides of all the rectangles in a histogram. It can also be drawn independently without the histogram.

To draw a frequency polygon:

1. Calculate the class mark of each class interval.
2. Plot the class marks on the X-axis and frequencies on the Y-axis.
3. Join the points to obtain the frequency polygon.

Pie Chart (Circle Graph)

A pie chart is a pictorial representation of numerical data using non-intersecting adjacent sectors of a circle. The area of each sector is proportional to the magnitude of the data represented by the sector.

Example:

Note: In pie charts, the total quantity is distributed over a total angle of 360° (or 100%). The data is plotted with respect to only one parameter.

Measures of Central Tendency

In many frequency distributions, the average value generally lies in the central part of the distribution. These values are called measures of central tendency. Commonly used measures are mean (or arithmetic mean), median, and mode.

Arithmetic Mean [AM]

The arithmetic mean is the average of a given set of numbers. It is calculated by dividing the sum of all observations by the total number of observations.

1. Mean for Frequency Distribution:

If \( x_1, x_2, \ldots, x_n \) are the values and \( f_1, f_2, \ldots, f_n \) are their corresponding frequencies, then the mean is:

Mean, \( \bar{x} = \frac{\sum x_i \cdot f_i}{\sum f_i} \)

2. Mean for Classified Data:

If \( x_1, x_2, \ldots, x_n \) are the class marks and \( f_1, f_2, \ldots, f_n \) are their frequencies, then the arithmetic mean is:

Mean, \( \bar{x} = \frac{\sum x_i \cdot f_i}{\sum f_i} \)

9.4.2 Median

The median of a distribution is the middle value when the data are arranged in ascending or descending order.

1. Median for Ungrouped Data:

• If the number of observations \( n \) is odd, median = \( \left(\frac{n + 1}{2}\right) \)th term.
• If the number of observations \( n \) is even, median = \( \frac{n}{2} \)th term.

2. Median for Classified or Grouped Data:

Firstly, find the cumulative frequencies of all the classes and \( N/2 \), where \( N \) is the total frequency. The class whose cumulative frequency is either equal to \( N/2 \) or just greater is the median class.

Then, calculate median using the formula:

Median = \( L + \left( \frac{\left(\frac{N}{2} - C\right)}{f} \right) \times i \)

• Where L = Lower limit of the median class,
• N = Sum of frequencies,
•  f = Frequency of the median class,
• C = Cumulative frequency of the preceding class of the median class,
• i  = Class interval.

Mode

The mode of a set of observations is the value that occurs most frequently among the given observations.

Relation Between Mean, Median, and Mode

There is a mathematical relationship between the mean, median, and mode:

Mode = 3 Median - 2 Mean

Range

The range of a set of observations is the difference between the highest and lowest observations.

Range = Maximum observation - Minimum observation