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Summary

 

  • The tendency of a given set of observations to cluster around a single, central or a middle value is known as central tendency. This value which represents the given set of observations is described as the average value or the measure of central tendency.
  • Following are the different measures of central tendency
    • Arithmetic Mean (A.M.)
    • Median (Me)
    • Mode (Mo or Z)
    • Geometric Mean (G.M.)
    • Harmonic Mean (H.M.)
  • Arithmetic Mean of the given observations can be found using the relation
     
    Description: 98817.png 
  • When there is a simple frequency distribution relating to an attribute, we have
     
    Description: 108156.png 
  • To simplify calculations, we can calculate A.M. using
     
    Description: 108152.png 
     
    where,
     
    Description: 108148.png
     
    xi  Mid value of the ith class interval
     
    A  Assumed mean
     
    C  Class length
     
    Description: 108144.png  
  • The sum of deviations taken from the arithmetic mean is zero.
  • Arithmetic mean is affected due to the change of origin and scale. 
  • Combined arithmetic mean of 3 sets of observations is given by
     
     
    Description: 108224.png 
  • Median is the middle most value when the observations are arranged either in an ascending order or in an descending order of their magnitudes. It can be calculated as
     
    Description: 108220.png
     
    Description: 108216.png  
  • In case of a simple frequency distribution, median is calculated using
     
    Description: 108212.png 
  • When a continuous or grouped frequency distribution is given, we can find the median by the following relation:
     
    Description: 108276.png 
    where,
     
    N  total frequency
     
    l  lower class boundary value of the median class
     
    f  frequency of the median class
     
    Cf  less than cumulative frequency upto l
     
    C  class length of the median class
  • Median is affected by change in origin as well as change in scale.
  • For a set of observations, the sum of absolute values of deviations is minimum when the deviations are taken from the median.
  • Quartiles are partition values which divide the given set of observations into four equal parts when the observations are arranged in ascending or descending order. For discrete data, quartile values can be found using the relation
     
    Description: 108270.pngvalue in the arranged series
     
    where, i = 1, 2, 3 
  • For a continuous frequency distribution
     
    Description: 108380.png
     
    where,
     
    i = 1, 2, 3
     
    N  total frequency
     
    l  lower class boundary value of the ith quartile class
     
    f  frequency of the ith quartile class
     
    Cf  cumulative frequency upto l
     
    C  class length of the ith quartile class
  • Deciles are the partition values that divide a given set of observations into 10 equal parts. For discrete data, decile values can be found using the relation
     
    Description: 108373.png where, i = 1, 2, 3 ... 9
  • For a continuous frequency distribution
     
    Description: 108367.png 
     
    where,
     
    i = 1, 2, 3 ... 9
     
    N  total frequency
     
    l  lower class boundary value of the ith decile class
     
    f  frequency of the ith decile class
     
    Cf  cumulative frequency upto l
     
    C  class length of the ith decile class
  • Percentiles are partition values that divide the given set of observations into hundred equal parts. For discrete data, percentile values can be found using the relation
     
    Description: 108361.png where, i = 1, 2, 3 ... 99.
  • For a continuous frequency distribution
     
    Description: 108355.png 
     
    where,
     
    i = 1, 2, 3 ... 99.
     
    N  total frequncy
     
    l  lower class boundary value of the ith percentile class
     
    f  frequency of the ith percentile class
     
    Cf  cumulative frequency upto l
     
    C  class length of the ith percentile class
  • Mode (Z) is that value which has the maximum concentration of observations around it. This can also be described as the most common value among the given observations.
     
    In discrete frequency distribution, the observation having the highest frequency is called Mode.
  • If in an ungrouped data, any observation is not repeating, then Mode (Z) can be found using the relation
     
    Mean – Mode = 3(Mean – Median)
  • In continuous frequency distribution, following formula is applicable:
    Description: 108479.png 
     
    where,
     
    l  LCB of the modal class
     
    f0  frequency of the modal class
     
    f1  frequency of the pre-modal class
     
    f1  frequency of the post modal class
     
    C  class length of the modal class
  • Mode is affected by change in origin as well as change in scale.
  • Geometric mean is defined as the nth root of the product of n observations.
     
     
    For a frequency distribution
     
    Description: 108473.png 
  • If z = xy, then G.M. of z = (G.M. of x) × (G.M. of y)
     
    Description: 108467.png 
  • Harmonic mean is defined as the reciprocal of the A.M. of reciprocals of observations.
     
    Description: 108461.png 
  • When the given data is a grouped frequency distribution, the H.M. can be found using the relation
     
    Description: 108455.png 
  • For any set of positive observations, A.M. ≥ G.M. ≥ H.M.
  • For any two positive observations A.M. × H.M. = G.M2
  • The degree to which numerical data tend to spread about an average value is called the variation or dispersion of data.
  • Measures of dispersion may be classified into:
    1. Absolute measures of dispersion
    • range
    • mean deviation
    • standard deviation
    • quartile deviation
    1. Relative measures of deviation
    • coefficient of range
    • coefficient of mean deviation
    • coefficient of variation
    • coefficient of quartile deviation
  • For a given set of observations, the difference between the highest and the lowest observation is called range.
  • If H and L denote the highest and lowest observations, then
     
    Range = H – L
     
    Corresponding relative measure of dispersion is
     
    Description: 108449.png 
  • Range remains unaffected due to a change of origin but is affected in the same ratio due to a change in scale.
  • The mean deviation about a measure of central tendency A, when the observations are a set of discrete data, is expressed as
     
    Description: 108443.png 
     
    For a grouped frequency distribution, mean deviation about A is given by
     
    Description: 108437.png
     
    Coefficient of mean deviation about Description: 108431.png  
  • Mean deviation takes its minimum value when the deviations are taken from the median.
  • Mean deviation remains unchanged due to a change of origin but changes in the same ratio due to a change in scale.
  • The standard deviation of a set of values is the root mean square deviation, when deviations are taken from the A.M. of observations. For discrete data,
     
    Description: 107441.png 
     
    For a frequency distribution,
     
    Description: 107434.png 
     
    Coefficient of variationDescription: 107428.png 
  • The square of standard deviation known as variance is also sometimes regarded as a measure of dispersion.
  • If all the observations assumed by a variable are constant, i.e., equal, then S = 0.
  • S.D. remains unaffected due to a change of origin, but is affected in the same ratio due to a change of scale.
  • If there are two groups containing n1 and n2 observations, x1 and x2 are the respective A.M.s and S1 and S2 are the respective S.D.s, then the combined S.D. is given by
     
    Description: 99919.png 
  • Quartile deviation is given by Description: 99913.png
  • The corresponding relative measure: Description: 99907.png




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