Class 9 Maths Chapter 12: Statistics

Statistics is Chapter 12 of NCERT Class 9 Maths. In the current syllabus it focuses on presenting data visually through bar graphs, histograms (including those with unequal class widths) and frequency polygons. Watch the video lesson below, then use these notes and solved examples to practise drawing and reading these graphs.

What you will learn

Basic terms: frequency, class interval, class size and class mark
How to draw and read a bar graph
How to draw a histogram for equal and unequal class widths
How to make a frequency polygon, with or without a histogram
Choosing a sensible scale and the right graph for a given set of data

Video Lessons

Statistics - Part 1

Key Concepts and Formulas

Data Terms You Need

Data is a collection of facts or numbers gathered for a purpose. Large data is usually grouped into a frequency distribution table before it is drawn as a graph.

Class size = upper limit − lower limit
Class mark = (lower limit + upper limit) ÷ 2
Inclusive to continuous (gap of 1):
lower limit − 0.5, upper limit + 0.5

Frequency: the number of times an observation occurs, or the number of observations in a class.
Class interval: a group such as 20-30. Here 20 is the lower limit and 30 is the upper limit. In such continuous classes, the upper limit is not included in that class.
Classes like 1-10, 11-20 are not continuous and must be adjusted before drawing a histogram.

Bar Graphs

A bar graph shows data using bars of equal width with equal gaps between them. It suits categories such as favourite sports or monthly sales.

Bar height = value ÷ scale
e.g. scale 1 cm = 50 students

One axis shows the categories and the other shows the values.
The height of each bar is proportional to the value it represents.
Choose a scale so that the tallest bar fits on the graph paper.

Histograms with Equal Class Widths

A histogram is used for continuous grouped data. It looks like a bar graph but the rectangles are drawn side by side with no gaps.

Width of rectangle = class size
Height of rectangle = frequency

Class intervals go on the x-axis and frequencies on the y-axis.
If the x-axis does not start at 0, draw a small kink (zig-zag) near the origin to show the break in scale.

Histograms with Unequal Class Widths

In a histogram, it is the area of each rectangle that must be proportional to the frequency. When the widths are different, using raw frequencies as heights gives a wrong picture.

Adjusted frequency =
(minimum class size ÷ class size) × frequency
Height of rectangle = adjusted frequency

Histogram with unequal class widths using adjusted frequencies 6, 9, 7 and 4 as bar heights, Class 9 Maths Statistics — Wider classes are drawn with adjusted heights

Take the minimum class size as the standard width.
Label the y-axis to show that heights are adjusted frequencies.

Frequency Polygons

A frequency polygon is a closed figure that shows the shape of a distribution. It is very useful for comparing two sets of data on the same axes.

Plot points: (class mark, frequency)
Add (first mark − class size, 0)
Add (last mark + class size, 0)

Histogram of student heights with a frequency polygon joining class marks and closing at zero frequency, Class 9 Statistics — Frequency polygon drawn over a histogram

From a histogram: join the mid-points of the tops of the rectangles with straight lines.
The two zero-frequency end points come from imaginary classes before the first class and after the last class, so the polygon closes on the x-axis.
The area under the frequency polygon equals the area of the histogram.

Solved Examples

Example 1: Find the class mark and class size of the class 35-45.

Class mark = (35 + 45) ÷ 2 = 80 ÷ 2 = 40
Class size = 45 − 35 = 10
∴ Class mark = 40, class size = 10

Example 2: The class marks of a distribution are 12, 20, 28 and 36. Find the classes.

Class size = 20 − 12 = 8
Half of class size = 4
First class = 12 − 4 to 12 + 4 = 8-16
∴ Classes: 8-16, 16-24, 24-32, 32-40

Example 3: Make the classes 1-10, 11-20 and 21-30 continuous.

Gap = 11 − 10 = 1, half gap = 0.5
Subtract 0.5 from lower, add 0.5 to upper
∴ 0.5-10.5, 10.5-20.5, 20.5-30.5

Example 4: Classes 0-10, 10-20, 20-40 and 40-70 have frequencies 6, 9, 14 and 12. Find the heights of the rectangles for a histogram.

Minimum class size = 10
0-10: (10 ÷ 10) × 6 = 6
10-20: (10 ÷ 10) × 9 = 9
20-40: (10 ÷ 20) × 14 = 7
40-70: (10 ÷ 30) × 12 = 4
∴ Heights: 6, 9, 7, 4

Example 5: Data has classes 20-30, 30-40, 40-50, 50-60 and 60-70. Where does its frequency polygon start and end?

Imaginary classes: 10-20 and 70-80
Class marks: (10 + 20) ÷ 2 = 15
(70 + 80) ÷ 2 = 75
∴ Starts at (15, 0), ends at (75, 0)

Example 6: In a bar graph with scale 1 cm = 50 students, how tall are the bars for 450 and 275 students?

450 ÷ 50 = 9 cm
275 ÷ 50 = 5.5 cm
∴ Bars are 9 cm and 5.5 cm tall

Example 7: A histogram shows heights of students: 140-145 cm (5), 145-150 cm (12), 150-155 cm (18), 155-160 cm (9). How many students are shorter than 150 cm, and which class has the most students?

Shorter than 150 cm = 5 + 12 = 17
Largest frequency = 18 (class 150-155)
∴ 17 students; class 150-155 cm

Common Mistakes to Avoid

Leaving gaps between the rectangles of a histogram, or drawing a bar graph with no gaps. Histograms have no gaps; bar graphs do.
Plotting raw frequencies when class widths are unequal. Always compute adjusted frequencies first.
Joining the class limits instead of the class marks in a frequency polygon, or leaving the polygon open at the ends.
Forgetting to convert inclusive classes such as 1-10, 11-20 into continuous classes before drawing a histogram.

Quick Revision

Class mark = (lower + upper) ÷ 2
Class size = upper − lower
Adjusted frequency =
(min class size ÷ class size) × frequency

Bar graph: equal widths, equal gaps; histogram: no gaps, continuous classes
Frequency polygon: join class marks and close it with zero-frequency classes at both ends
Draw a kink on the axis when the scale does not start from 0