Statistics

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CBSE Learning Objectives – Key Concepts & Skills You Must Know

Understand the importance of graphical representation of data.
Identify different types of graphical representations:
- Bar graphs
- Histograms
- Frequency polygons
Construct bar graphs using given data.
Analyze and interpret histograms and frequency polygons.
Compare data sets using frequency polygons.

CBSE Revision Notes & Quick Summary for Last-Minute Study

Chapter 12: Statistics

12.1 Graphical Representation of Data

Graphical representation makes data easier to understand compared to tables.
Types of graphical representations:
- Bar graphs
- Histograms (uniform and varying widths)
- Frequency polygons

Bar Graphs

A pictorial representation of data using bars of uniform width.
Example: Birth months of students.

Histograms

Used for continuous data.
Example: Weights of students.

Example of Histogram

Weights (in kg)	Number of students
30.5 - 35.5	9
35.5 - 40.5	6
40.5 - 45.5	15
45.5 - 50.5	3
50.5 - 55.5	1
55.5 - 60.5	2
Total	36

Frequency Polygons

Used for continuous data and comparing different datasets.
Example: Performance of two sections in a test.

Example of Frequency Polygon

Classes	Class-marks	Frequency
140 - 150	145	5
150 - 160	155	10
160 - 170	165	20
170 - 180	175	9
180 - 190	185	6
190 - 200	195	2
Total		52

Important Diagrams

Histogram: Displays frequency distribution of marks among students.
Frequency Polygon: Plotted points connected by lines to show frequency distribution.

Example of Marks Distribution

Marks	Number of students
0 - 10	5
10 - 20	10
20 - 30	4
30 - 40	6
40 - 50	7
50 - 60	3
60 - 70	2
70 - 80	2
80 - 90	3
90 - 100	9
Total	51

Common Mistakes

Misleading histograms due to varying widths of bars.
Ensure areas of rectangles in histograms are proportional to frequencies.

Tips

Always check the scale and intervals used in graphs.
Use frequency polygons for better comparison between datasets.

CBSE Exam Tips, Important Questions & Common Mistakes to Avoid

Common Mistakes and Exam Tips

Common Pitfalls

Misleading Histograms: Ensure that the areas of the rectangles in a histogram are proportional to the frequencies. If the widths of the rectangles vary, this can lead to a misleading representation of the data.
Frequency Polygon Completion: When drawing a frequency polygon, remember to include points for classes with zero frequency to complete the polygon correctly.
Class Interval Representation: Be cautious when representing class intervals with varying widths; the lengths of the rectangles must be adjusted accordingly to maintain proportionality.

Tips for Avoiding Mistakes

Check Class Sizes: Always verify that the class sizes are consistent when interpreting histograms. If they vary, adjust the lengths of the rectangles to reflect the correct proportions.
Use Continuous Class Intervals: When creating frequency distributions, ensure that class intervals are continuous to avoid gaps that can misrepresent the data.
Draw Accurate Graphs: When drawing graphs, ensure that all axes are labeled correctly, and the scales used are appropriate for the data being represented.

CBSE Quiz & Practice Test – MCQs, True/False Questions with Solutions

Multiple Choice Questions

Histogram

Frequency polygon

Bar graph

Line graph

Correct Answer: C

Solution:

A bar graph is suitable for comparing discrete data such as the number of girls per thousand boys in different sections.

Correct Answer: B

Solution:

The class mark for the interval 160-170 is 165. The frequency for this interval is 20, so the y-coordinate of the point is 20.

Histogram

Frequency polygon

Pie chart

Bar graph

Correct Answer: D

Solution:

A bar graph is suitable for showing the distribution of expenditures across different categories.

Correct Answer: B

Solution:

The class mark for the interval 70-80 is 75. The frequency for this interval is 2, so the y-coordinate of the point is 2.

Bar graph

Histogram

Pie chart

Line graph

Correct Answer: B

Solution:

A histogram is used for displaying a grouped frequency distribution with continuous classes.

Bar graph

Histogram

Frequency polygon

Pie chart

Correct Answer: C

Solution:

Frequency polygons are useful for comparing two different sets of data of the same nature, such as the performance of two different sections of the same class.

The height of the bars

The width of the bars

The color of the bars

The spacing between the bars

Correct Answer: A

Solution:

When class intervals have varying widths, the height of the bars must be adjusted so that the area of each bar is proportional to the frequency.

To make the polygon symmetrical

To ensure the polygon starts and ends at the horizontal axis

To increase the number of data points

To make the graph more colorful

Correct Answer: B

Solution:

Adding class intervals with zero frequency ensures that the frequency polygon starts and ends at the horizontal axis, making the area of the polygon the same as the area of the histogram.

0.4

0.2

Correct Answer: B

Solution:

The height of the rectangle in a histogram is equal to the frequency when the width is constant. Therefore, the height is 2.

The total number of data points

The frequency of data within a specific interval

The average value of the data

The median of the data set

Correct Answer: B

Solution:

The height of each bar in a histogram represents the frequency of data points within a specific interval.

Correct Answer: A

Solution:

In a histogram, the height of each rectangle is proportional to the frequency of the class interval. For the interval 40-50, the frequency is 7, so the height of the rectangle is 7.

Education

Groceries

Entertainment

All are equal

Correct Answer: A

Solution:

The expenditure on education is ₹5000, which is higher than groceries (₹4000) and entertainment (₹1000). Therefore, the bar for education will be the highest.

The height of the bars.

The width of the bars.

The scale of the axes.

The area of the bars.

Correct Answer: D

Solution:

When the widths of bars vary, the area of each bar must be proportional to the frequency to accurately represent the data.

Bars of varying width

Continuous data representation

Bars of uniform width with equal spacing

Representation of cumulative frequency

Correct Answer: C

Solution:

A bar graph features bars of uniform width with equal spacing between them.

The heights of the bars

The widths of the bars

The spacing between the bars

The color of the bars

Correct Answer: A

Solution:

When the widths of the rectangles are varying, the lengths (heights) of the rectangles must be adjusted to ensure the areas are proportional to the frequencies.

Rent

Groceries

Entertainment

All are equal

Correct Answer: A

Solution:

In a bar graph, the height of each bar represents the value of the category. Since the expenditure on rent is ₹5000, which is higher than groceries (₹4000) and entertainment (₹1000), the bar for rent will be the highest.

The height of each bar is proportional to the frequency.

The area of each bar is proportional to the frequency.

The width of each bar is proportional to the frequency.

The bars are separated by gaps.

Correct Answer: B

Solution:

In histograms with varying widths, the area of each bar is proportional to the frequency, not just the height.

Bar graph

Histogram

Pie chart

Line graph

Correct Answer: B

Solution:

A histogram is used for representing grouped frequency distributions with continuous classes.

Bar Graph

Histogram

Frequency Polygon

Pie Chart

Correct Answer: C

Solution:

Frequency polygons are particularly useful for comparing two different sets of data of the same nature, such as the performance of two sections of a class.

Bar graph

Frequency polygon

Pie chart

Scatter plot

Correct Answer: B

Solution:

Frequency polygons are useful for comparing two different sets of data of the same nature.

The heights should remain unchanged.

The heights should be proportional to the class width.

The heights should be inversely proportional to the class width.

The heights should be adjusted based on the total number of classes.

Correct Answer: C

Solution:

In a histogram with varying class widths, the heights of the bars should be inversely proportional to the class width to ensure the areas of the bars are proportional to the frequencies.

To ensure all data points are included.

To make the polygon symmetrical.

To make the area of the frequency polygon equal to the area of the histogram.

To simplify the calculation of the mean.

Correct Answer: C

Solution:

Adding class intervals with zero frequency before the lowest class and after the highest class ensures that the area of the frequency polygon matches the area of the histogram.

It shows trends over time

It provides a visual comparison of individual items

It displays cumulative data

It is used for continuous data

Correct Answer: B

Solution:

Bar graphs provide a visual comparison of individual items, making the data easier to understand.

Correct Answer: A

Solution:

The area of a rectangle in a histogram is calculated as the product of its height (frequency) and width. For the interval 40.5-45.5, the frequency is 16 and the width is 5 kg, so the area is 16 * 5 = 80.

Representing categorical data.

Comparing two or more distributions.

Displaying data with large class intervals.

Showing data trends over time.

Correct Answer: B

Solution:

Frequency polygons are useful for comparing two different sets of data of the same nature.

The total number of observations.

The frequency of the class interval.

The proportion of observations in the class interval.

The cumulative frequency up to that class interval.

Correct Answer: B

Solution:

In a histogram, the area of each rectangle is proportional to the frequency of the class interval.

Bar graph

Histogram

Line graph

Pie chart

Correct Answer: B

Solution:

A histogram is suitable for representing grouped frequency distribution with continuous classes, such as test scores grouped into intervals.

Correct Answer: B

Solution:

The y-coordinate of the point representing the class mark of the interval 40-50 is equal to the frequency of that interval, which is 20.

Bar graph

Histogram

Line graph

Scatter plot

Correct Answer: B

Solution:

A histogram is used for representing a grouped frequency distribution with continuous classes.

Line graph

Histogram

Bar graph

Frequency polygon

Correct Answer: C

Solution:

A bar graph is suitable for representing categorical data such as expenditures under different heads.

₹4000

₹5000

₹6000

₹7000

Correct Answer: A

Solution:

Since the scale is 1 unit = ₹1000, a bar of height 4 units represents an expenditure of 4 x ₹1000 = ₹4000 on groceries.

2 units

4 units

10 units

20 units

Correct Answer: B

Solution:

Since the frequency for the interval 50-60 is 10 and the scale is 1 unit = 5 students, the height of the bar should be 10 / 5 = 2 units.

2.67

3.5

Correct Answer: A

Solution:

To maintain proportionality, the length of the rectangle is adjusted by the formula: (Frequency × Minimum class size) / Width of the interval. Here, (8 × 10) / 30 = 2.67.

A graph that uses bars to represent data

A line graph that connects the midpoints of the tops of the bars in a histogram

A circular chart divided into sectors

A graph that shows data using dots

Correct Answer: B

Solution:

A frequency polygon is a line graph that connects the midpoints of the tops of the bars in a histogram.

Ensuring equal spacing between bars.

Using bars of uniform width.

Making the height of each bar proportional to the frequency of the data.

Connecting the tops of the bars with a line.

Correct Answer: D

Solution:

In a bar graph, it is not necessary to connect the tops of the bars with a line. This is a feature of a frequency polygon, not a bar graph.

They represent the midpoints of the class intervals.

They represent the endpoints of the class intervals.

They represent the total frequency of the data.

They represent the cumulative frequency.

Correct Answer: A

Solution:

In a frequency polygon, the points plotted are the midpoints of the class intervals, which are then joined by line segments.

1.5

150

Correct Answer: A

Solution:

The height of the rectangle is calculated as the frequency divided by the interval width. For the interval 60-70, the frequency is 15 and the width is 10, so the height is 15/10 = 1.5.

3.2

Correct Answer: A

Solution:

In a histogram, the height of the rectangle is equal to the frequency when the width is uniform. Hence, the height of the rectangle for the interval 40.5-45.5 is 16.

320

Correct Answer: A

Solution:

The area of the rectangle in a histogram is given by the product of the frequency and the width of the class interval. Here, the area is

16 \times 5 = 80

Correct Answer: B

Solution:

The y-coordinate of the point on the frequency polygon corresponding to the class mark of an interval is equal to the frequency of that interval, which is 12.

Section A

Section B

Both performed equally

Cannot be determined

Correct Answer: A

Solution:

In the 30-40 interval, Section A has a frequency of 12, whereas Section B has a frequency of 10. Therefore, Section A performed better in this interval.

Class boundaries

Class midpoints

Frequency

Cumulative frequency

Correct Answer: B

Solution:

In a frequency polygon, class midpoints are plotted on the horizontal axis.

Bars are of varying widths

Bars have equal spacing

Bars are of uniform width

Bars are circular

Correct Answer: A

Solution:

In a histogram, the bars can have varying widths, and the area of the bars is proportional to the frequency.

Line graph

Bar graph

Pie chart

Histogram

Correct Answer: B

Solution:

A bar graph is effective for comparing different categories, such as causes of fatality.

Bars of uniform width

Equal spacing between bars

Continuous classes

Depicts variables on one axis

Correct Answer: C

Solution:

Bar graphs typically have bars of uniform width with equal spacing between them, but they do not represent continuous classes.

To display categorical data

To represent data in a circular format

To compare different sets of continuous data

To show the relationship between two variables

Correct Answer: C

Solution:

Frequency polygons are used to compare different sets of continuous data.

Frequency

Variable

Percentage

Time

Correct Answer: B

Solution:

In a bar graph, the variable is typically represented on the x-axis.

The area of each bar is proportional to the frequency of the class interval.

The area of each bar is proportional to the width of the class interval.

The area of each bar is proportional to the square of the frequency.

The area of each bar is equal for all intervals.

Correct Answer: A

Solution:

In a histogram with varying class widths, the area of each bar is proportional to the frequency of the class interval.

Histogram

Frequency polygon

Bar graph

Scatter plot

Correct Answer: C

Solution:

A bar graph is ideal for comparing different categories of expenditures, making it easier to visualize spending.

It uses bars to represent data

It connects midpoints of histogram bars with line segments

It displays data in circular form

It is used for categorical data

Correct Answer: B

Solution:

A frequency polygon connects the midpoints of the tops of the bars of a histogram with line segments.

Bars are of uniform width

Bars are drawn with no gaps between them

It represents categorical data

The area of the bars is proportional to the frequency

Correct Answer: C

Solution:

A histogram represents continuous data, not categorical data.

The data is not continuous.

The class intervals are of varying sizes.

The frequencies are not proportional.

The data is categorical.

Correct Answer: B

Solution:

In a histogram, if the width of the bars is not uniform, it indicates that the class intervals are of varying sizes.

A bar graph represents continuous data

A histogram has gaps between bars

A bar graph is used for categorical data with spaces between bars

A histogram uses pie slices

Correct Answer: C

Solution:

A bar graph is used for categorical data and has spaces between the bars, whereas a histogram is used for continuous data with no spaces between bars.

To display the distribution of a continuous data set.

To compare different categories of data.

To show the relationship between two variables.

To represent cumulative frequency.

Correct Answer: B

Solution:

Bar graphs are used to compare different categories of data, making it easier to visualize differences.

Draw a pie chart

Plot the class marks on the horizontal axis

Calculate the mean of the data

List all data points individually

Correct Answer: B

Solution:

The first step in constructing a frequency polygon is to plot the class marks on the horizontal axis and frequencies on the vertical axis.

Bar graphs have gaps between bars, while histograms do not.

Histograms are used for categorical data, while bar graphs are for numerical data.

Bar graphs show frequency distribution, while histograms show proportions.

Histograms have equal bar widths, while bar graphs do not.

Correct Answer: A

Solution:

Bar graphs typically have gaps between bars to represent discrete data, whereas histograms have no gaps to represent continuous data.

Bar graph

Histogram

Pie chart

Line graph

Correct Answer: B

Solution:

A histogram is used to represent a grouped frequency distribution with continuous classes.

To represent data in a tabular form

To show the frequency distribution of continuous data

To compare individual data points

To display data using pie charts

Correct Answer: B

Solution:

A histogram is used to represent the frequency distribution of continuous data by showing the number of data points that fall within specified ranges.

True or False

Correct Answer: True

Solution:

Frequency polygons are useful for comparing two different sets of data of the same nature, as they provide a visual comparison of frequencies across different intervals.

Correct Answer: True

Solution:

In a histogram, the areas of the rectangles are proportional to the frequencies, which is why the width of the bars is significant.

Correct Answer: False

Solution:

In a bar graph, the width of the bars is not important; it is the height of the bars that represents the value of the variable.

Correct Answer: False

Solution:

In a bar graph, the width of the bars is not important as long as they are of uniform width and have equal spacing between them.

Correct Answer: True

Solution:

A frequency polygon is created by plotting points at the midpoints of class intervals and connecting them with line segments.

Correct Answer: False

Solution:

In a histogram, the width of the bars is important because the area of each bar (width times height) should be proportional to the frequency.

Correct Answer: False

Solution:

Histograms are used to represent continuous data, while bar graphs are typically used for categorical data. The key difference is that histograms have no gaps between bars, indicating continuity.

Correct Answer: False

Solution:

In a bar graph, the width of the bars is not important as long as they are uniform. The focus is on the height of the bars, which represents the data values.

Correct Answer: True

Solution:

In a histogram, the area of each rectangle is proportional to the frequency of the data it represents, especially when the class widths are not uniform.

Correct Answer: False

Solution:

Frequency polygons are used when the data is continuous and very large, making them suitable for comparing different data sets.

Correct Answer: True

Solution:

A bar graph uses bars of uniform width with equal spacing to represent data, making it easier to compare different categories.

Correct Answer: True

Solution:

According to the bar graph data, the maximum number of students were born in August.

Correct Answer: True

Solution:

When class widths vary, the heights of the rectangles in a histogram must be adjusted to ensure that the areas are proportional to the frequencies, otherwise it can give a misleading picture.

Correct Answer: True

Solution:

A frequency polygon is formed by joining the mid-points of the tops of the rectangles in a histogram with line segments.

Correct Answer: False

Solution:

A frequency polygon can be drawn even if the class intervals are of varying widths. The key is to plot the class marks and join them with line segments.

Correct Answer: True

Solution:

If the widths of the rectangles in a histogram are not uniform, it can give a misleading representation of the data, as the areas of the rectangles are not directly proportional to the frequencies.

Correct Answer: True

Solution:

In a bar graph, the width of the bars is not important as long as they are of uniform width and equally spaced.

Correct Answer: True

Solution:

In a frequency polygon, the points are plotted at the midpoints of the class intervals, and these points are connected by straight lines.

Correct Answer: True

Solution:

When the widths of bars in a histogram vary, adjustments must be made to ensure the areas of the bars are proportional to the frequencies.

Correct Answer: True

Solution:

Frequency polygons are particularly useful for comparing two different sets of data because they allow for easy visual comparison of trends and distributions.

Correct Answer: True

Solution:

In a histogram with varying class sizes, the lengths of the rectangles are adjusted to ensure the areas are proportional to the frequencies.

Correct Answer: False

Solution:

Frequency polygons are used when the data is continuous and very large.

Correct Answer: True

Solution:

Frequency polygons are particularly useful for comparing two datasets, as they provide a clear visual comparison.

Correct Answer: True

Solution:

When class widths vary, the lengths of the rectangles in a histogram must be adjusted so that their areas remain proportional to the frequencies.

Correct Answer: True

Solution:

In a bar graph, the width of the bars is not significant; what matters is that the bars are of uniform width with equal spacing between them.

Statistics

AI Learning Assistant

CBSE Learning Objectives – Key Concepts & Skills You Must Know

CBSE Revision Notes & Quick Summary for Last-Minute Study

Chapter 12: Statistics

12.1 Graphical Representation of Data

Bar Graphs

Histograms

Example of Histogram

Frequency Polygons

Example of Frequency Polygon

Important Diagrams

Example of Marks Distribution

Common Mistakes

Tips

CBSE Exam Tips, Important Questions & Common Mistakes to Avoid

Common Mistakes and Exam Tips

Common Pitfalls

Tips for Avoiding Mistakes

Multiple Choice Questions

Q1.A survey shows the number of girls per thousand boys in different sections of society. Which graphical representation is most suitable to display this data?

Correct Answer: C

Q2.In a frequency polygon, the class marks are plotted against the frequencies. If the class intervals are 140-150, 150-160, ..., 190-200 with frequencies 5, 10, 20, 9, 6, and 2 respectively, what is the y-coordinate of the point corresponding to the class mark of the interval 160-170?

Correct Answer: B

Q3.A family with a monthly income of ₹20,000 plans their expenditures under various heads. Which graphical representation would best help visualize the proportion of income spent on each category?

Correct Answer: D

Q4.A frequency polygon is drawn using the class marks of intervals 0-10, 10-20, ..., 90-100. If the frequency for the interval 70-80 is 2, what should be the y-coordinate of the point for this interval?

Correct Answer: B

Q5.Which graphical representation is used to display a grouped frequency distribution with continuous classes?

Correct Answer: B

Q6.Which graphical representation is most suitable for comparing the performance of two different sections of students in a class?

Correct Answer: C

Q7.In a histogram, if the class intervals have varying widths, what must be adjusted to accurately represent the data?

Correct Answer: A

Q8.What is the primary purpose of adding class intervals with zero frequency when constructing a frequency polygon?

Correct Answer: B

Q9.In a histogram representing the distribution of students' weights, the interval 50.5-55.5 kg has a frequency of 2. If the width of this interval is 5 kg, what is the height of the rectangle for this interval?

Correct Answer: B

Q10.In a histogram, what does the height of each bar represent?

Correct Answer: B

Correct Answer: A

Q12.A bar graph represents the monthly expenditures of a family with expenditures on education, groceries, and entertainment being ₹5000, ₹4000, and ₹1000 respectively. Which category has the highest bar?

Correct Answer: A

Q13.If a histogram has varying widths of bars, what must be adjusted to accurately represent the data?

Correct Answer: D

Q14.Which of the following is a key feature of a bar graph?

Correct Answer: C

Q15.If a histogram has bars of varying widths, what must be adjusted to ensure the areas are proportional to the frequencies?

Correct Answer: A

Q16.In a bar graph representing monthly expenditures, if the expenditures on rent, groceries, and entertainment are ₹5000, ₹4000, and ₹1000 respectively, which category has the highest bar?

Correct Answer: A

Q17.Which of the following is true about histograms with varying widths of bars?

Correct Answer: B

Q18.Which graphical representation is most suitable for displaying the distribution of students' weights in a class where the data is grouped into continuous classes?

Correct Answer: B

Q19.A teacher wants to compare the performance of students in two different sections of a class using a graphical method. Which of the following graphical representations would be most suitable for this purpose?

Correct Answer: C

Q20.A teacher wants to compare the performance of two sections of students in a mathematics test. Which graphical representation should she use?

Correct Answer: B

Q21.When constructing a histogram for a dataset with varying class widths, how should the heights of the bars be adjusted?

Correct Answer: C

Q22.In a frequency polygon, why is it necessary to add class intervals with zero frequency before the lowest class and after the highest class?

Correct Answer: C

Q23.What is the main advantage of using a bar graph?

Correct Answer: B

Q24.A histogram is drawn for the weights of students in a class with intervals 30.5-35.5, 35.5-40.5, ..., 60.5-65.5. If the number of students in the interval 40.5-45.5 is 16, what should be the area of the rectangle for this interval if the width is 5 kg?

Correct Answer: A

Q25.A frequency polygon is particularly useful for:

Correct Answer: B

Q26.In a histogram, what does the area of each rectangle represent?

Correct Answer: B

Q27.A teacher wants to analyze the performance of students in a test. She decides to group the scores into intervals of 10 marks each. Which of the following graphical representations would be most suitable for this data?

Correct Answer: B

Q28.In a frequency polygon, if the class interval 30-40 has a frequency of 15 and the class interval 40-50 has a frequency of 20, what is the y-coordinate of the point representing the class mark of the interval 40-50?

Correct Answer: B

Q29.A teacher wants to represent the distribution of student weights in her class. Which graphical representation should she use if the data is grouped into continuous classes?

Correct Answer: B

Q30.A family with a monthly income of ₹20,000 plans their expenditures under various heads. Which type of graph would best represent this data?

Correct Answer: C

Q31.A bar graph is used to represent the monthly expenditure of a family. If the expenditure on groceries is represented by a bar of height 4 units, and the scale is 1 unit = ₹1000, what is the expenditure on groceries?