Lecture Series on Statistics No. BS_4 Contd. Date – 10.08.2008 “ Frequency Graphs " By Dr. Bijaya Bhusan Nanda, Ph. D. (Stat.)
CONTENT Histogram Frequency polygon Smoothed frequency curve Cumulative frequency curve or ogives
Learning Objective The Trainees will be able to construct and interpret frequency graphs.
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
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
Histogram of Age of Respondents 200 180 160 140 120 100 80 60Frequency 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
Highest Year of School: Mother Highest Year of School : Respondent 300 300 200 200 100 100 FrequencyFrequency 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
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
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 200 150Frequency 100 50 0 22.5 32.5 42.5 52.5 62.5 72.5 82.5 92.5 Midpoint of the Age Interval
SMOOTHED FREQUENCY CURVEFor any continuous frequency distribution, ifthe class intervals become smaller andsmaller, resulting in the increase of the numberof class intervals, the frequency polygon tendsto a smooth curve called frequency curve.
The area under the frequency curve represents thetotal frequency and approximates the area boundedby rectangles in the histogramAdvantage 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 distributionsthrough their frequency curves.Histograms and frequency curves may also bedrawn 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 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.
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 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
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)
Next Session Descriptive Statistics – Measures of Central Tendency