This document provides a summary of a lecture on frequency graphs. It discusses histograms, frequency polygons, smoothed frequency curves, and cumulative frequency distributions. Histograms represent class frequencies as vertical rectangles, with the total area equal to the total frequency. Frequency polygons connect the midpoints of class intervals by straight lines. Smoothed frequency curves approximate the histogram as class intervals get smaller. Cumulative frequency distributions cumulate the frequencies from lowest to highest class to create an ogive curve for determining positional measures. The lecture aims to help trainees understand and interpret different types of frequency graphs.

1. Lecture Series on
Statistics
No. BS_4 Contd.
Date – 10.08.2008
“ Frequency Graphs "
By
Dr. Bijaya Bhusan Nanda, Ph. D. (Stat.)

2. CONTENT
 Histogram
 Frequency polygon
 Smoothed frequency
curve
 Cumulative frequency
curve or ogives

3. Learning Objective
 The Trainees will be able to construct
and interpret frequency graphs.

4. What is Histogram ?
(Definition)
 The histogram is a special type of
bar graph that represents frequency
or relative frequency of continuous
distribution.
 Histograms are appropriate for
continuous, quantitative variables.
 A normal curve can be superimposed
onto the histogram

5.  It represents the class frequencies in the
form of vertical rectangles erected over
respective class intervals.
 Total area of the rectangles is equivalent
to total frequency.

6. How to Construct a Histogram?
 Class intervals are decided and the
class frequencies are obtained.
 The class intervals are marked along
the X – axis.
 A set of adjacent rectangles are
erected over the C.Is with area of
each rectangle being proportional to
the corresponding CI.

7. Significance of Histogram?
 It helps in the understanding of the
frequency distribution.
• Skew ness of the distribution or its
deviation from the symmetry
• Peakedness of the distribution
• Comparison of two frequency distribution
LOOK at the Following data

8. Histogram of Age of Respondents
200
180
160
140
120
100
80
60
Frequency
40 Std. Dev = 17.29
20 Mean = 44.5
0 N = 600.00
22.5 32.5 42.5 52.5 62.5 72.5 82.5 92.5
Age of Respondent
Source of Data: 1991 General Social Survey

9. Highest Year of School: Mother Highest Year of School : Respondent
300
300
200 200
100 100
Frequency
Frequency
Std. Dev = 3.47 Std. Dev = 2.82
Mean = 11.2 Mean = 13.3
0 N = 500.00 0 N = 598.00
0.0 2.5 5.0 7.5 10.0 12.5 15.0 17.5 20.0 4.0 6.0 8.0 10.0 12.0 14.0 16.0 18.0 20.0
Highest Year School Completed, Mother Highest Year of School Completed

10. LINE CHART
No. of Couples vrs. No. of Families

11. FREQUENCY POLYGON
Frequency polygon is a special type
of line graph representing frequency
distribution
 Frequency polygon can be drawn both for
continuous and discrete data.
 Comparison of two frequency
distributions is easier through the
superimposition of two histograms or
their resulting frequency polygons

12. How to draw Frequency
Polygon?
 CIs decided & the CFs obtained.
 The CI are marked along the X – axis.
 Dot is put above the midpoint of each
CI represented on the horizontal axis
corresponding to the frequency of the
relevant CI.
 Connect the dots by straight lines

13.  The frequency polygon can also be
drawn by joining the mid-points of
the tops of the rectangles through
straight lines in the histogram.
 The mid-points of the tops of the
first and last rectangles are extended
to the mid-points of the classes at
the extreme having zero frequencies.

14. Significance of Frequency polygon
 It helps in the understanding the
frequency distribution of data
• Skew ness of the distribution or its
deviation from the symmetry
• Peakedness of the distribution
• Comparison of two frequency distribution
• This is useful for Continuous and discrete
distribution

15. Frequency Polygon of Age
Distribution
200
150
Frequency
100
50
0
22.5 32.5 42.5 52.5 62.5 72.5 82.5 92.5
Midpoint of the Age Interval

16. SMOOTHED FREQUENCY CURVE
For any continuous frequency distribution, if
the class intervals become smaller and
smaller, resulting in the increase of the number
of class intervals, the frequency polygon tends
to a smooth curve called frequency curve.

17. The area under the frequency curve represents the
total frequency and approximates the area bounded
by rectangles in the histogram
Advantage in statistical analysis:
Does not put any restriction on the choice of
CI.
Frequency function can be expressed by a
mathematical function.
It is easy to compare two frequency distributions
through their frequency curves.
Histograms and frequency curves may also be
drawn using relative frequency distributions.

18.  Depending upon the nature of the
frequency distributions, the
frequency curves may be of different
shapes.
• symmetrical frequency curve, and
• Skewed or asymmetrical
frequency curve

19. Symmetrical Frequency Curve
• The symmetrical frequency curves
look like bell-shaped curves where
the highest frequency occurs in the
central class and other frequencies
gradually decrease symmetrically
on both sides

20. Skewed or asymmetrical
frequency curve
 Moderately skewed with the highest frequency
learning either towards left or right (long right tail or
long left tail)
 Extreme asymmetrical form like J-shaped or U-
shaped.
 A frequency curve with long right tail indicates that
lower values occur more often than the extreme
higher values, e.g., distribution of income of families
in a locality.
 Similarly, in a long left tailed frequency distribution,
the extreme higher values occur more often than the
extreme lower values, e.g., educated members
among income groups.

21.  U Shaped frequency curve
• Mortality according to age group
 J Shaped frequency curve
• Distribution of death rate by age
group in a population with low IMR

22. CUMULATIVE FREQUENCY DISTRUBUTION
 A cumulative frequency distribution
may construct from the original
frequency distribution by cumulating or
adding together frequencies
successively:
 The cumulative frequency distributions
so constructed are called upward
cumulative frequency distributions
because frequencies are cumulated
from the lowest class interval to the
highest class interval

23. OGIVE
 OGIVE; The graphical representation
of cumulative frequency distrn.
 Helpful to find out number or
percentage of observations above or
below a particular value.
 Offer a graphical technique for
determining positional measures
such as median, quartiles, deciles,
percentiles, etc

24. Cumulative frequency
120
100
80
60
40
20
0
0
8
4
0
92
6
2
68
76
84
10
12
10
11
13
14
Quantity of gluc oza (mg%)
Figure 2.8 Cumulative frequency distribution for quantity of glucose
(for data in Table 2.1)

25. Next Session
 Descriptive Statistics – Measures of
Central Tendency

26. THANK YOU