Notes Data Handling - & Pedagogy Paper 2 for CTET & TET Exams - CTET & State TET

This chapter holds significant importance in CTET and State TETs exams. It covers various methods of data presentation and measures of central tendency. Analysing past papers shows that typically one or two questions are asked from this chapter each year.

Data and its Presentation

In daily life we come across many kinds of information, for example:

• 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.

Information collected for a study is called data. On the basis of how it is collected, data is classified as:

• Primary data: Data collected directly by the investigator for a specific purpose (for example, responses from a questionnaire administered by a teacher).
• Secondary data: Data obtained from existing sources (for example, data from books, reports, government publications, or websites).

Presentation of Data

Presentation of data means organising collected data in a simple, clear form so that it can be easily analysed and interpreted. Common methods of presenting data include:

• Ordering the observations (ascending or descending).
• Tabular form: ungrouped and grouped frequency distribution tables.
• Graphical form: bar graphs, histograms, frequency polygons, pie charts and ogives.

Presentation in Ascending or Descending Order

Ordering data helps to quickly identify the smallest and largest values and to compute measures like the range.

Example: Marks obtained by 10 students in a Mathematics test (raw data):

55, 36, 95, 73, 60, 42, 25, 78, 75, 62

Arrange data in ascending order:

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

Range: Maximum - Minimum = 95 - 25 = 70

Presentation in the Form of Ungrouped Frequency Distribution

If many observations repeat or the number of observations is large, present the data as a frequency distribution showing each distinct value and its frequency.

Grouped Frequency Distribution

When observations take many distinct values, it is convenient to group them into class intervals and give the frequency of each class.

Exclusive Form
In exclusive form class intervals are written so that the upper limit of one class is not included in that class when it is the lower limit of the next class (for example 0-10, 10-20, ... are treated as exclusive where 10 belongs to the second class).

Inclusive Form
In inclusive form class intervals include both the lower and upper limits as written (for example 0-10, 11-20, ... are treated as inclusive with distinct, non-overlapping ranges).

Graphical Representation of Data

Graphical methods help visualise data and reveal patterns and comparisons quickly. Common graphical forms are:

• Bar graphs
• Histograms
• Frequency polygons
• Pie charts (circle graphs)
• Ogives (cumulative frequency graphs)

Bar Graph

A bar graph uses bars of uniform width and equal spacing to represent categories on one axis (usually the x-axis) and their corresponding values or frequencies on the other axis (y-axis). Bar graphs are used for categorical data.

Histogram and Frequency Polygon

Histogram

A histogram represents a grouped frequency distribution in exclusive form. Rectangles are drawn for each class interval with bases equal to the class intervals and heights equal to the class frequencies. There are no gaps between adjacent rectangles.

Example: Price Growth Percentage

From the histogram it is clear:

• Percentage growth in year 2008 = 20%
• Percentage growth in year 2009 = 38%
• Percentage growth in year 2010 = 58%
• Percentage growth in year 2011 = 80%

Therefore, the percentage growth in price was highest in the year 2011.

Frequency Polygon

A frequency polygon is formed by joining the mid-points of the top edges of the histogram rectangles with straight lines. A frequency polygon can also be drawn without first drawing the histogram by plotting class mid-points against frequencies and joining them.

Circle Graph or Pie Chart

A circle graph or pie chart shows the relationship of parts to a whole. The whole is represented by the circle and parts are sectors whose sizes are proportional to the values they represent.

For example, if a child sleeps 8 hours in a day of 24 hours, the proportion of the circle for sleeping is:

Number of sleeping hours / Whole day = 8 / 24 = 1 / 3.

So, the sector for sleeping occupies one-third of the circle.

If the child spends 6 hours at school, then

Number of school hours / Whole day = 6 / 24 = 1 / 4.

So, the sector for school occupies one-quarter of the circle. The sizes of other sectors are found similarly.

Drawing a Pie Chart

Example: The number of students in a hostel speaking different languages is given; present this data in a pie chart.

The central angle of a component (sector) is given by:

(Value of the component / Sum of all components) × 360°

Measures of Central Tendency

Measures of central tendency summarise a large set of data by a single value that represents the centre or typical value of the data. The three common measures are Arithmetic Mean, Median and Mode.

Arithmetic Mean (AM)

The arithmetic mean is the average of the observations and is obtained by dividing the sum of all observations by the number of observations.

1. Mean for Frequency Distribution:

If x₁, x₂, ..., x_n are the values and f₁, f₂, ..., f_n are their corresponding frequencies, then the arithmetic mean is:

2. Mean for Classified (Grouped) Data:

For grouped data, use class marks (mid-points) as representative values. If x_i are class marks and f_i are frequencies, then:

Useful terms:

• Class mark (mid-point): (Upper limit + Lower limit) / 2 for each class.
• Class width (i): Difference between upper and lower limits of a class (when classes are of equal size).
• Assumed mean method: For large classes or convenient calculation, choose an assumed mean A and use deviations d_i = x_i - A to compute mean as A + (Σf_id_i / Σf_i).

Median

The median is the middle observation when the data are arranged in ascending or descending order. It divides the data into two equal parts.

1. Median for Ungrouped Data:

• If the number of observations is odd, the median is the middle value.
• If the number of observations is even, the median is the average of the two middle values.

2. Median for Grouped Data:

To find the median from grouped data:

• Compute cumulative frequencies for the classes.
• Find N/2, where N is the total frequency.
• The median class is the class whose cumulative frequency is equal to or just exceeds N/2.

Median Formula for Grouped Data

Where:

• L = Lower limit of the median class,
• N = Total frequency,
• f = Frequency of the median class,
• C = Cumulative frequency of the class preceding the median class,
• i = Class interval (class width).

Mode

The mode is the observation that occurs most frequently in a data set. A distribution may be unimodal, bimodal, or multimodal depending on the number of modes.

Example

Find the mode of the given data: 29, 25, 38, 22, 38, 25, 38, 29.

Solution: 38 occurs most frequently. Therefore, mode = 38.

Relation Between Mean, Median and Mode

For moderately skewed distributions, an empirical relation is often used:

Mode = 3 × Median - 2 × Mean

Example

If the mode of grouped data is 12 and the mean is 5, find the median.

Solution:

Mode = 12, Mean = 5.

Using Mode = 3 × Median - 2 × Mean

12 = 3 × Median - 2 × 5

12 = 3 × Median - 10

3 × Median = 12 + 10

3 × Median = 22

Median = 22 / 3 = 7.33

Range

The range of a distribution is the simplest measure of dispersion. It is the difference between the maximum and minimum observations.

Range = Maximum observation - Minimum observation

Other Important Terms and Graphs

• Cumulative frequency (cf): Running total of frequencies up to a class. Useful for finding medians, percentiles, and drawing ogives.
• Class boundaries: Actual limits of a class accounting for continuity (useful when classes are given in inclusive form).
• Frequency density: Frequency per unit class width, used when class widths are unequal. For a class, frequency density = frequency / class width.
• Ogive: A cumulative frequency graph plotted using upper class boundaries on the x-axis and cumulative frequencies on the y-axis. Useful for visually finding median and quartiles.

Pedagogical Notes for Teachers

• Begin with simple real-life examples (marks, runs, books read) to motivate the need for organisation of data.
• Teach ungrouped frequency tables first, then grouped tables, emphasising class width, class mark and cumulative frequency.
• Use physical activities (tally marks made by students) before introducing tables and graphs.
• When teaching histograms and frequency polygons, show how class mid-points are used and why histograms have no gaps between bars.
• Demonstrate how to construct a pie chart by converting values to central angles and drawing sectors using a protractor.
• Provide practice problems requiring calculation of mean, median and mode for both ungrouped and grouped data, and include problems using the assumed mean method.

Summary: This chapter introduces methods to organise and present data and explains central measures-mean, median and mode-along with practical graphical representations. Clear understanding of these concepts enables correct interpretation of data and supports teaching and assessment tasks effectively.