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Chapter 11 ,Measures of Dispersion(statistics)

Ananya Sharma
Ananya Sharma

This document discusses various measures of dispersion, which describe how spread out or varied the values in a data set are. It describes absolute measures like range and relative measures like coefficient of range. It also discusses interquartile range, mean deviation, standard deviation, and the Lorenz curve. Quartile deviation and mean deviation are simple measures but are less accurate and reliable. Standard deviation is a more certain measure but gives more importance to extreme values. The Lorenz curve measures deviation from equal distribution for variables like income, wealth, wages and more.

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Chapter 11 Measures of Dispersion
Chapter 11 ,Measures of Dispersion(statistics)
ABSOLUTE AND RELATIVE MEASURES OF DISPERSION When dispersion of the series is expressed in terms of the original unit of the series, it is called absolute measure of dispersion. The relative measure of dispersion expresses the variability of data in terms of some relative value or percentage.
Range R = Upper Limit of the Last Class Interval Lower Limit of the First Class Interval Or R = H L Where R=Range; H=Highest value in the series; L=Lowest value in the series Coefficient of range CR = H L H + L Where CR=Coefficient of range; H=Highest value in the series; L=Lowest value in the series
Inter Quartile Range = Q3 Q1 Quartile Deviation = Q3 Q1 2 Also called Semi-inter Quartile Range Coefficient of Quartile Deviation Coefficient of QD= Q3 Q1歎 Q3 + Q1 = Q3 Q1 2 2 Q3 + Q1
QUARTILE DEVIATION Merits Simple Less effect of Extreme Values Demerits Not based on all Values Formation of Series not known Instability
Mean deviation is the arithmetic average of deviations of all the values consideration taken from a statistical average (mean, median or mode) of series. In taking deviation of values, algebraic signs + and are not taken into consideration, that is negative deviations are also treated as positive deviations
If deviations are taken from median MDm = | X M | or | dm | N N If deviations are taken from arithmetic average MDx = | X X | or | d x | N N Where, MD = Mean deviation ; X M = Deviation from the median ; X - X = deviation from the arithmetic average ; N = Number of items
Coefficient of MD from Mean = MDx X = Mean Deviation Arithmetic Mean Coefficient of MD from Median = MDm M = Mean Deviation Median Coefficient of MD from Mode = MDz Z = Mean Deviation Mode
MEAN DEVIATION Merits Simple Based on all Values Less Effect of Extreme Values Demerits Inaccuracy Not Capable of Algebraic Treatment Unreliable
Standard Deviation is the Square root of the Arithmetic Mean of the squares of deviations of the items from their mean value. This is generally denoted by (sigma) of the Greek language. COEFFICIENT OF STANDARD DEVIATION = X
Merits Based on all Values Certain Measure Little Effect of a Change in Sample Algebraic Treatment Demerits Difficult More Importance to Extreme Value
Lorenz Curve is a measure of deviation of actual distribution from the line of equal distribution. Lorenz curve as a measure of dispersion is presently applied to the following parameters, viz., 1. Distribution of income 2. Distribution of wealth 3. Distribution of wages 4. Distribution of profits 5. Distribution of production 6. Distribution of population
By Less than orMore than ogives methoda frequency distribution series is first converted into a less than ormorethan cumulative series as in the cas e of ogives, data arepresented graphically to make a less than or more than ogive, N/2 item of the series is determined and from this point (on the y-axis of the graph)a perpendicular is drawn to the right to cut the cumulative frequency curve.The medianvalue is the o newhere cumulative frequency curvecuts corresponding to x-axis. Less than and more than ogive curve method present the data graphical ly in the form of less than and more than ogives simultaneously. The two ogives aresuperimposed upon eachother t o determine the median value. Mark the point where the ogive curvecut each other, draw a perpendicular from that point on x- axis, the corresponding value on the x-axis would be the median value.
Prepare a histogram from the data. Find out the rectangle whose height is the highest. This will be the modal class. Draw two lines - one joining the top right point of the rectangle preceding the modal class with top right point of the modal class. The other joining the top left point of the modal classwith the top left point of the post modal class. From the point of i ntersection of these two diagonal lines, drawa perpendicular on horizo ntal axisi.e. xaxisthe point where this perpendicular linemeets x- axis, gives us the value of mode.
Chapter 11 ,Measures of Dispersion(statistics)

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Chapter 11 ,Measures of Dispersion(statistics)

  • 1. Chapter 11 Measures of Dispersion
  • 3. ABSOLUTE AND RELATIVE MEASURES OF DISPERSION When dispersion of the series is expressed in terms of the original unit of the series, it is called absolute measure of dispersion. The relative measure of dispersion expresses the variability of data in terms of some relative value or percentage.
  • 4. Range R = Upper Limit of the Last Class Interval Lower Limit of the First Class Interval Or R = H L Where R=Range; H=Highest value in the series; L=Lowest value in the series Coefficient of range CR = H L H + L Where CR=Coefficient of range; H=Highest value in the series; L=Lowest value in the series
  • 5. Inter Quartile Range = Q3 Q1 Quartile Deviation = Q3 Q1 2 Also called Semi-inter Quartile Range Coefficient of Quartile Deviation Coefficient of QD= Q3 Q1歎 Q3 + Q1 = Q3 Q1 2 2 Q3 + Q1
  • 6. QUARTILE DEVIATION Merits Simple Less effect of Extreme Values Demerits Not based on all Values Formation of Series not known Instability
  • 7. Mean deviation is the arithmetic average of deviations of all the values consideration taken from a statistical average (mean, median or mode) of series. In taking deviation of values, algebraic signs + and are not taken into consideration, that is negative deviations are also treated as positive deviations
  • 8. If deviations are taken from median MDm = | X M | or | dm | N N If deviations are taken from arithmetic average MDx = | X X | or | d x | N N Where, MD = Mean deviation ; X M = Deviation from the median ; X - X = deviation from the arithmetic average ; N = Number of items
  • 9. Coefficient of MD from Mean = MDx X = Mean Deviation Arithmetic Mean Coefficient of MD from Median = MDm M = Mean Deviation Median Coefficient of MD from Mode = MDz Z = Mean Deviation Mode
  • 10. MEAN DEVIATION Merits Simple Based on all Values Less Effect of Extreme Values Demerits Inaccuracy Not Capable of Algebraic Treatment Unreliable
  • 11. Standard Deviation is the Square root of the Arithmetic Mean of the squares of deviations of the items from their mean value. This is generally denoted by (sigma) of the Greek language. COEFFICIENT OF STANDARD DEVIATION = X
  • 12. Merits Based on all Values Certain Measure Little Effect of a Change in Sample Algebraic Treatment Demerits Difficult More Importance to Extreme Value
  • 13. Lorenz Curve is a measure of deviation of actual distribution from the line of equal distribution. Lorenz curve as a measure of dispersion is presently applied to the following parameters, viz., 1. Distribution of income 2. Distribution of wealth 3. Distribution of wages 4. Distribution of profits 5. Distribution of production 6. Distribution of population
  • 14. By Less than orMore than ogives methoda frequency distribution series is first converted into a less than ormorethan cumulative series as in the cas e of ogives, data arepresented graphically to make a less than or more than ogive, N/2 item of the series is determined and from this point (on the y-axis of the graph)a perpendicular is drawn to the right to cut the cumulative frequency curve.The medianvalue is the o newhere cumulative frequency curvecuts corresponding to x-axis. Less than and more than ogive curve method present the data graphical ly in the form of less than and more than ogives simultaneously. The two ogives aresuperimposed upon eachother t o determine the median value. Mark the point where the ogive curvecut each other, draw a perpendicular from that point on x- axis, the corresponding value on the x-axis would be the median value.
  • 15. Prepare a histogram from the data. Find out the rectangle whose height is the highest. This will be the modal class. Draw two lines - one joining the top right point of the rectangle preceding the modal class with top right point of the modal class. The other joining the top left point of the modal classwith the top left point of the post modal class. From the point of i ntersection of these two diagonal lines, drawa perpendicular on horizo ntal axisi.e. xaxisthe point where this perpendicular linemeets x- axis, gives us the value of mode.