Lecture Series on Statistics -HSTC. Frequency Graphs " Dr. Bijaya Bhusan Nanda, Ph. D. (Stat.)

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1 Lecture Series on Statistics -HSTC 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.

Histograms are appropriate for continuous,

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 Frequency 200 Histogram of Age of Respondents Std. Dev = Mean = 44.5 N = Age of Respondent Source of Data: 1991 General Social Survey

9 Frequency Frequency 300 Highest Year of School: Mother 300 Highest Year of School : Respondent Std. Dev = 3.47 Mean = 11.2 N = Std. Dev = 2.82 Mean = 13.3 N = 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

The frequency polygon can also be drawn by

The mid-points of the tops of the first and

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 Frequency Polygon of Age Distribution 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 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.

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 Quantity of glucoza (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

UNIT 15 GRAPHICAL PRESENTATION OF DATA-I

UNIT 15 GRAPHICAL PRESENTATION OF DATA-I Graphical Presentation of Data-I Structure 15.1 Introduction Objectives 15.2 Graphical Presentation 15.3 Types of Graphs Histogram Frequency Polygon Frequency Curve