Chapter 5 Data Handling (Concepts)

Bar Graph & Double Bar Graph

In our everyday lives, we often come across collections of facts or figures, whether it's the number of students in each class, the favourite colours of your friends, or the marks you scored in different subjects. This collection of facts is called data. Data can be numbers, descriptions, measurements, or observations. Handling data involves collecting, organising, presenting, and interpreting this information in a meaningful way.

To make data easy to understand and compare, we often represent it visually using graphs. Two common ways to represent categorical data are using bar graphs and double bar graphs.

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Bar Graph

A bar graph is a way of representing data using rectangular bars of the same width. These bars can be drawn vertically or horizontally. The height (if vertical) or length (if horizontal) of each bar is proportional to the value or frequency it represents.

Bar graphs are useful for comparing the amounts or frequencies of different categories.

How to Draw a Bar Graph:

Follow these steps to construct a bar graph:

1. Draw Axes: Draw two perpendicular lines intersecting at a point. The horizontal line is called the x-axis, and the vertical line is called the y-axis.
2. Choose a Scale: Decide on a suitable scale for the axis that represents the values (usually the vertical axis). The scale determines the height of the bars. For example, you might decide that 1 unit length on the y-axis represents 5 students, 10 Rupees, or 100 kilograms, depending on the data. Choose a scale that allows you to represent all the data conveniently on the graph paper.
3. Mark Categories: Mark the categories (like names of subjects, months, types of vehicles, etc.) along the axis that represents the categories (usually the x-axis). Leave uniform spaces between the bars.
4. Draw Bars: For each category, draw a rectangular bar of uniform width. The height of the bar should correspond to the value of the data for that category, based on the chosen scale on the value axis.
5. Label and Title: Label both the x-axis and the y-axis clearly, indicating what they represent (e.g., "Subject", "Number of Students"). Give a suitable title to the bar graph that describes the data being presented (e.g., "Student Preferences for Subjects").

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Double Bar Graph

A double bar graph is an extension of a bar graph. It is used when you need to compare two sets of data for the same categories. For example, you might want to compare the performance of students in two different exams in the same set of subjects, or compare sales figures of two different products over several months.

In a double bar graph, for each category, a pair of bars is drawn next to each other. These bars represent the values from the two different data sets. Different colours or shading patterns are used for the two bars in each pair to distinguish them.

How to Draw a Double Bar Graph:

The steps are similar to drawing a single bar graph, with slight modifications:

1. Draw Axes: Draw two perpendicular axes (x-axis and y-axis).
2. Choose a Scale: Decide on a suitable scale for the axis representing the values (usually y-axis), covering the range of values in both data sets.
3. Mark Categories: Mark the categories along the axis representing the categories (usually x-axis). Leave space between groups of bars.
4. Draw Pairs of Bars: For each category, draw two bars side-by-side, one for each set of data. Ensure the bars within each pair have the same width, and there is some space between different pairs of bars. The height of each bar should correspond to its value on the scale.
5. Use Legend: Assign a different colour or shading pattern to the bars of each data set. Provide a legend (a small box showing the colour/pattern and the data set it represents) usually near the graph to help interpret the bars.
6. Label and Title: Label both axes and give a clear title to the double bar graph.

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Organising & Grouping Data

When we collect raw data, it's often scattered and not easy to make sense of immediately. For example, a list of marks of 50 students in a test, just as numbers written one after another, doesn't quickly tell you the highest score, the lowest score, or how many students scored within a certain range.

Organising data means arranging it in a systematic way to make it understandable and useful for analysis and interpretation. One of the fundamental ways to organise data is by using frequency distribution.

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Raw Data

The data collected in its original form, before any organisation or processing, is called raw data.

Example: The marks obtained by 20 students in a class test (out of 10 marks) might be recorded as follows:

7	5	8	6	9	7	5	8	6	7
9	5	6	8	7	9	6	5	8	7

Looking at this raw data, it's difficult to quickly answer questions like "How many students scored exactly 7 marks?" or "Which mark was scored by the highest number of students?".

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Frequency Distribution Table

To make raw data more organised and easily interpretable, we can arrange it in a frequency distribution table. This table shows each distinct value of the data and the number of times it appears. The number of times a particular value occurs in the data is called its frequency.

Steps to create a frequency distribution table for ungrouped data:

1. List all the distinct values that appear in the raw data in ascending or descending order in one column.
2. For each distinct value, go through the raw data and mark a tally mark ($|$) for every occurrence of that value. To make counting easier, make blocks of five tally marks, where the fifth mark ($\bcancel{||||}$) crosses the previous four.
3. Count the number of tally marks for each distinct value. This count is the frequency of that value. Write the frequency in a separate column.
4. Add up all the frequencies. The sum of frequencies should be equal to the total number of data points in the raw data. This helps in checking if all data points have been counted correctly.

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Grouping Data into Class Intervals

When the number of data points is large, and the values cover a wide range, listing every distinct value individually in a frequency distribution table can still be cumbersome. In such situations, it is more convenient to group the data into class intervals (or classes).

Example: Consider the following data set of marks obtained by 30 students:

10	20	36	92	95	40	50	56	60	70
92	88	80	70	72	70	36	40	36	40
92	40	50	50	56	60	70	60	60	88

Here, the marks range from 10 to 95. Creating a frequency table for each individual mark would be very long. Grouping this data makes sense.

Let's define some terms related to grouped data:

• Class Interval: A range of data values that forms a group. Examples: 0-10, 10-20, 20-30, etc.
• Lower Limit: The smallest value in a class interval (e.g., 10 is the lower limit of the interval 10-20).
• Upper Limit: The largest value in a class interval (e.g., 20 is the upper limit of the interval 10-20).
• Class Size or Width: The difference between the upper limit and the lower limit of a class interval (e.g., for 10-20, the class size is $20 - 10 = 10$).

There are two common methods for forming class intervals:

• Inclusive Method: The upper limit of a class interval is included in that interval. Example: 0-10, 11-20, 21-30. In this method, there are gaps between consecutive intervals.
• Exclusive Method (or Continuous Class Intervals): The upper limit of a class interval is NOT included in that interval, but it is included in the next interval. Example: 0-10, 10-20, 20-30. Here, a value of 10 would be included in the 10-20 interval, not 0-10. This method is preferred for graphical representation like histograms.

Steps to Create a Grouped Frequency Distribution Table:

1. Determine the range of the data (difference between the highest and lowest values).
2. Decide on a suitable number of class intervals and their class size/width. The intervals should cover the entire range of the data.
3. Form the class intervals using either the inclusive or exclusive method, ensuring no data point falls into more than one interval.
4. Go through the raw data and use tally marks to count the number of data points that fall into each class interval. In the exclusive method, remember that the upper limit is not included in the interval.
5. Count the tally marks for each class interval to get the frequency for that class.
6. Sum the frequencies to check if it matches the total number of data points.

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Histograms

We have seen how to organise raw data into a frequency distribution table and how to represent ungrouped data using bar graphs. When data is grouped into class intervals, a special type of bar graph called a histogram is used for its graphical representation. Histograms are particularly useful for visualising the distribution of continuous data.

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What is a Histogram?

A histogram is a graphical representation of a grouped frequency distribution where the class intervals are shown on the horizontal axis and the frequencies are shown on the vertical axis. It consists of a series of adjacent rectangles (bars).

Key features of a histogram:

• The base of each rectangle (bar) represents a class interval on the horizontal axis.
• The width of each bar is equal to the class size (or class width) of the interval it represents. If all class intervals have the same size, all bars will have the same width.
• The height of each bar represents the frequency of the corresponding class interval. The height is proportional to the frequency.
• Since the class intervals are continuous (using the exclusive method, like 10-20, 20-30, etc.), there are no gaps between adjacent bars in a histogram. The bars touch each other at the class boundaries.

Histograms show the shape and spread of the data distribution. They help identify where the data is concentrated.

How to Draw a Histogram:

To construct a histogram from a grouped frequency distribution table:

1. Draw Axes: Draw the horizontal and vertical axes on a graph sheet.
2. Mark Class Intervals on Horizontal Axis: Mark the class intervals along the horizontal axis (x-axis). Start with the lower limit of the first class interval at the origin or after a kink (if the first interval doesn't start at 0). Mark the upper limit of each interval. Ensure that if the class size is uniform, the points on the x-axis marking the intervals are equally spaced.
3. Choose a Scale for Vertical Axis: Choose a suitable scale for the vertical axis (y-axis) to represent the frequencies. The scale should accommodate the highest frequency. Label the y-axis as 'Frequency'.
4. Draw Rectangular Bars: For each class interval, draw a rectangular bar. The base of the bar lies on the horizontal axis, covering the range of the class interval. The height of the bar is drawn up to the frequency of that class interval on the vertical axis, according to the chosen scale.
5. Ensure No Gaps: Draw the bars adjacent to each other. The right edge of one bar coincides with the left edge of the next bar.
6. Use Kink/Break (if needed): If the first class interval starts at a value significantly greater than 0, use a zig-zag line (called a kink or break) on the horizontal axis near the origin to indicate that the scale between 0 and the start of the first interval is not shown proportionally.
7. Label and Title: Label the horizontal axis appropriately (e.g., 'Height in cm', 'Marks', 'Ages') and give a suitable title to the histogram (e.g., "Histogram showing Distribution of Heights").

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Difference Between a Bar Graph and a Histogram

Although a histogram looks similar to a bar graph, there are fundamental differences between them. The choice between a bar graph and a histogram depends on the type of data you are representing.

Basis of Difference	Bar Graph	Histogram
Data Type	It is used to represent categorical or discrete data. Each bar represents a distinct category.	It is used to represent continuous data that is grouped into class intervals.
Gaps between Bars	There are gaps between the bars to indicate that the categories are separate and distinct.	There are no gaps between the bars, as they represent a continuous range of data. The bars are adjacent.
Horizontal Axis (X-axis)	The x-axis represents distinct categories (e.g., fruits, months, countries).	The x-axis represents a continuous numerical scale divided into class intervals.
Bar Width	The width of the bars has no mathematical significance and is chosen for visual clarity. All bars are typically of the same width.	The width of the bars is significant as it represents the class interval. The width can be uniform or variable.
Ordering of Bars	The bars can be reordered in any way (e.g., alphabetically, by size) without changing the meaning of the graph.	The bars cannot be reordered, as they are fixed by the numerical sequence of the class intervals on the x-axis.

Pie Chart

Visual representations of data help us understand information quickly. While bar graphs are excellent for comparing discrete categories or comparing quantities over time, a pie chart is ideal for showing how different parts make up a whole. It's like slicing a pie, where each slice represents a portion of the entire pie.

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What is a Pie Chart?

A pie chart, also known as a circle graph, is a circular statistical graph that is divided into sectors (like slices of a pie). Each sector represents a category or component of the data. The entire circle represents the total or the sum of all the components (which corresponds to $100\%$ of the data or $360^\circ$ of the central angle).

The size of each sector in a pie chart is proportional to the value or frequency of the data it represents. This proportionality is determined by the angle of the sector at the center of the circle. A larger share of the total data corresponds to a larger sector with a larger central angle.

Key Idea: Proportionality

The value of a component is proportional to the angle of its sector at the center of the circle.

From this proportionality, we can derive the formula for calculating the central angle for each component:

The sum of all the central angles for all the components must always add up to $360^\circ$ (a full circle).

How to Draw a Pie Chart:

To construct a pie chart, you need the data for different categories and their corresponding values. Follow these steps:

1. Calculate Total Value: Sum up the values of all the individual components or categories to find the total value of the data.
2. Calculate Central Angles: For each component, calculate the measure of the central angle it should represent using the formula: Central Angle $= (\frac{\text{Value of the component}}{\text{Total value}}) \times 360^\circ$. Ensure that the sum of all calculated central angles is $360^\circ$.
3. Draw Circle: Draw a circle of a suitable radius using a compass.
4. Draw First Radius: Draw any radius of the circle. This will be the starting line for drawing the first sector.
5. Draw Sectors: Starting from the first radius, use a protractor to measure and draw the central angle for the first component. Draw the second radius to complete the first sector. Now, use this second radius as the starting line for the next sector and draw its central angle. Continue this process for all components. The last sector's boundary should align with the first radius drawn.
6. Label Sectors: Label each sector clearly with the name of the category it represents. You can also write the value of the component or its percentage of the total within or near the sector.
7. Add Title: Give a suitable title to the pie chart that describes the data it represents.

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Probability and Related Terms

In our daily lives, we often encounter situations where the outcome is uncertain. For example, when it is cloudy, it might rain or it might not. When you participate in a competition, you might win or lose. Probability is a branch of mathematics that deals with measuring the likelihood of such events occurring. It helps us quantify uncertainty.

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Introduction to Probability

Probability provides a numerical measure for the chance of a specific outcome happening in an experiment. The value of probability is always between 0 and 1. A probability of 0 means the event is impossible, while a probability of 1 means the event is certain to happen.

Related Terms in Probability:

To understand probability better, let's define some key terms:

1. Experiment: An operation or activity that produces a set of possible results. The results of an experiment are called outcomes.

Examples of experiments:

• Tossing a coin.
• Rolling a standard six-sided die.
• Drawing a card from a deck of cards.
• Spinning a spinner.

2. Random Experiment: An experiment in which all possible outcomes are known, but the specific outcome of any single trial cannot be predicted in advance. All possible outcomes must have an equal chance of occurring (this is assumed in basic probability).

Examples: Tossing a fair coin, rolling a fair die.

3. Outcome: A single possible result of a random experiment. It is one of the results that can happen when you perform the experiment.

Examples:

• When tossing a coin, 'Head' is an outcome, and 'Tail' is another outcome.
• When rolling a die, '1', '2', '3', '4', '5', '6' are individual outcomes.

4. Sample Space: The set of all possible outcomes of a random experiment. It is usually denoted by the letter 'S'. Listing the sample space helps us identify all the potential results.

Examples:

• Experiment: Tossing a coin. Sample space S = {Head, Tail} or {H, T}. The total number of possible outcomes is 2.
• Experiment: Rolling a standard die. Sample space S = {1, 2, 3, 4, 5, 6}. The total number of possible outcomes is 6.
• Experiment: Tossing two coins. Sample space S = {HH, HT, TH, TT}. The total number of possible outcomes is 4.

5. Event: An event is a collection of one or more outcomes from the sample space of an experiment. It is a subset of the sample space. Events are often denoted by capital letters like E, A, B, etc.

Examples:

• Experiment: Rolling a die (S = {1, 2, 3, 4, 5, 6}).
  • Event E1: Getting an even number. The outcomes favorable to this event are {2, 4, 6}. So, E1 = {2, 4, 6}. The number of favourable outcomes for E1 is 3.
  • Event E2: Getting a number less than 3. The outcomes favorable to this event are {1, 2}. So, E2 = {1, 2}. The number of favourable outcomes for E2 is 2.
  • Event E3: Getting a 5. The outcome favorable to this event is {5}. So, E3 = {5}. The number of favourable outcomes for E3 is 1.
• Experiment: Tossing a coin (S = {H, T}).
  • Event E: Getting a head. The outcome favorable to this event is {H}. So, E = {H}. The number of favourable outcomes for E is 1.

Favourable Outcomes: The outcomes that satisfy the condition of a particular event are called the favourable outcomes for that event.

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Probability of an Event

The probability of an event E, denoted by P(E), is a numerical measure of how likely the event is to occur. For a random experiment where all outcomes are equally likely, the probability of an event E is defined as the ratio of the number of outcomes favourable to the event to the total number of possible outcomes in the sample space.

Formula for Probability:

Using symbols, if $n(\text{E})$ is the number of favourable outcomes for event E, and $n(\text{S})$ is the total number of possible outcomes in the sample space S, then:

Important Points about Probability:

• Equally Likely Outcomes: The outcomes of an experiment are said to be equally likely if there is no reason to expect one outcome in preference to another. For example, in the toss of a fair coin, 'Head' and 'Tail' are equally likely outcomes. In the roll of a fair die, each of the numbers 1 to 6 is an equally likely outcome.
• The probability of any event E is always a number between 0 and 1, inclusive. $0 \leq \text{P(E)} \leq 1$.
• Impossible Event: An event that can never occur is called an impossible event. The probability of an impossible event is 0. Example: Getting a 7 when rolling a standard die.
• Sure Event (or Certain Event): An event that is certain to occur is called a sure event. The probability of a sure event is 1. Example: Getting a number less than 7 when rolling a standard die.

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Understanding a Standard Deck of Cards

Many probability problems involve a standard deck of 52 playing cards. Understanding its composition is crucial.

A standard deck has 52 cards in total. These cards are divided into four suits:

• Spades (♠) and Clubs (♣), which are Black in colour.
• Hearts (♥) and Diamonds (♦), which are Red in colour.

Summary of Suits and Colours:

• Total Cards = 52
• Red Cards = 26 (13 Hearts + 13 Diamonds)
• Black Cards = 26 (13 Spades + 13 Clubs)

Each of the four suits has 13 cards, ranked as follows:

Ace (A), 2, 3, 4, 5, 6, 7, 8, 9, 10, Jack (J), Queen (Q), King (K)

The cards can be further categorised:

• Face Cards (or Picture Cards): These are the cards with faces on them. In each suit, there are 3 face cards: Jack, Queen, and King.
  • Total Face Cards = 3 cards/suit × 4 suits = 12 Face Cards.
  • (6 Red Face Cards and 6 Black Face Cards)

• Number Cards: These are the cards from 2 to 10. There are 9 number cards in each suit.
  • Total Number Cards = 9 cards/suit × 4 suits = 36 Number Cards.
• Aces: There is one Ace in each suit, making a total of 4 Aces.

Property	Spades (♠)	Clubs (♣)	Hearts (♥)	Diamonds (♦)	Total
Colour	Black	Black	Red	Red	26 Black, 26 Red
Total Cards per Suit	13	13	13	13	52
Aces	1	1	1	1	4
Number Cards (2-10)	9	9	9	9	36
Face Cards (J, Q, K)	3	3	3	3	12

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