“ Frequency Graphs "

By

Dr. Bijaya Bhusan Nanda, Ph. D. (Stat.)

Lecture Series on

Statistics -HSTC

Histogram

Frequency polygon

Smoothed frequency curve

Cumulative frequency curve or ogives

CONTENT

Learning Objective

The Trainees will be able to construct and interpret frequency graphs.

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

What is Histogram?

(Definition)

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.

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.

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

Age of Respondent

92.582.572.562.552.542.532.522.5

Histogram of Age of Respondents

Source of Data: 1991 General Social Survey

Fre

qu

en

cy

200

180

160

140

120

100

80

60

40

20

0

Std. Dev = 17.29

Mean = 44.5

N = 600.00

Highest Year School Completed, Mother

20.017.515.012.510.07.55.02.50.0

Highest Year of School: Mother

Fre

qu

en

cy

300

200

100

0

Std. Dev = 3.47

Mean = 11.2

N = 500.00

Highest Year of School Completed

20.018.016.014.012.010.08.06.04.0

Highest Year of School : Respondent

Fre

qu

en

cy

300

200

100

0

Std. Dev = 2.82

Mean = 13.3

N = 598.00

LINE CHART

No. of Couples vrs. No. of Families

Frequency polygon is a special type of

line graph representing frequency

distribution

FREQUENCY POLYGON

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

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

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.

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

Frequency Polygon of Age

Distribution

0

50

100

150

200

22.5 32.5 42.5 52.5 62.5 72.5 82.5 92.5

Midpoint of the Age Interval

Fre

qu

en

cy

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.

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.

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

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

Skewed or asymmetrical

frequency curve Moderately skewed with the highest frequency

leaning 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.

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

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

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

OGIVE

0

20

40

60

80

100

120

68 76 84 92 100

108

116

124

132

140

Quantity of glucoza (mg%)

Cu

mu

lati

ve f

req

uen

cy

Figure 2.8 Cumulative frequency distribution for quantity of glucose

(for data in Table 2.1)

Next Session

Descriptive Statistics – Measures of Central Tendency

THANK YOU