What is the relationship between Arithmetic Mean, Geometric Mean, and Harmonic Mean? Why is the Arithmetic Mean higher than the Geometric Mean and the Geometric Mean higher than the Harmonic Mean?

Measures of Central Tendency

Q.1 What do you mean by Central Tendency Measures? Define the terms arithmetic mean, medium, and mode.

    The central tendency of a variable refers to a typical value around which other values tend to cluster; hence, the value reflecting the central tendency of the series is referred to as measurements of central tendency or average.

   Clark says, “Average is an attempt to find one single figure to describe whole of figures.”

Arithmetic Mean (X): The arithmetic mean is the most common and extensively used metric for expressing the complete data set by a single number. Its value is calculated by adding all the items and dividing the sum by the number of items.

           There are two kinds of arithmetic means:

•      Simple Arithmetic mean

•      Weighted arithmetic mean

Calculation of Arithmetic Mean:

      

 

Step-Deviation Method in Individual Series: Not applicable            

Step-Deviation Method in Discrete Series:

Median (M):

       The median is the variable value that divides the group into two equal halves, one with all values larger than the median and the other with all values less than the median.

           Calculation of Median:

           a) Individual Series: Sort the variables into ascending or descending order.

b) Discrete Series:

•  Sort the variables in ascending or descending order, and then compute the cumulative frequencies.

•   The median will be the value (X) corresponding to this in the cumulative frequency.

c) Continuous Series:

•      Arrange the variables ascending or descending.

•      Cumulative frequencies should be calculated.

•      (N/2) is used to get the median class.

Mode (Z):

The value that appears the most frequently in a series is known as the mode, and it is the value of the item around which frequencies are most densely packed.

Calculation of Mode:

a) Individual Series:

        I.            By inspection - value is the most frequently repeated.

     II.            Individual series are converted into distinct series.

           b) Discrete Series: 

        I.            By inspection – the value with the highest frequency.

     II.            By categorizing.

   III.            According to the empirical connection.

a)     Continuous Series:

                                                        I.            First, determine the model class by inspection or grouping.

                                                     II.            Next, use the following formula -

Q.2 What are the fundamental components of an Ideal Average?

                      I. It should be simple to grasp.

                     II. Well-defined and rigorously defined.

                    III. Based on all observations.

                   IV. Easy to compute.

                    V. Less susceptible to fluctuations.

                   VI. Algebraic treatment is possible.

          VII. Stability of sampling.

Q3. What exactly is Geometric Mean? Give the Algebraic Characteristics of Geometric Mean and explain why it is helpful.

The nth root of the product of N items or values is the geometric mean.

Geometric Mean Algebraic Characteristics:

   I. If each component is replaced with a geometric mean, the result remains unaltered.

  II.  Geometric mean cannot be calculated if some item in the series has a negative or zero value.

 III. The product of the geometric mean's related ratios is always equal.

 IV. Changing the order of the items has no effect.

It is permissible or beneficial to employ the geometric mean: -

     ·  When looking for ratios or percentages.

     ·  When determining rate of rise or rate of reduction.

     ·  When the various values are vastly different.

Q.4 What Does the Word Harmonic Mean? In what situations is Harmonic Mean used?

A series' harmonic mean is the reciprocal of the arithmetic mean of the reciprocal of its item values.

Calculation of Harmonic Mean (H.M.):

Individual Series Discrete & Continuous Series

H.M. = Reciprocal 

The Harmonic Mean is utilized in the following situations:

·        To calculate average speed or velocity.

·        To determine the average price.

·   If the variable item in the question is to be kept constant in the answer, or vice versa, harmonic mean will be computed.

Q.5 What exactly is Partition Value? Can you provide a method for computing different Partition Values?

Partition values are the values of the elements that split the series into multiple sections. Quartiles, Quintiles, Octiles, Deciles, and Percentiles are four, five, eight, ten, and hundred equal portions of a variable. The preceding partition values provide an understanding of the construction of the series that is utilized in the computation of dispersion and skewness.

Partition Values are calculated using the following formula:

Q.6 What is the relationship between Arithmetic Mean, Geometric Mean, and Harmonic Mean? Why is the Arithmetic Mean higher than the Geometric Mean and the Geometric Mean higher than the Harmonic Mean?

The relationship between the arithmetic mean, geometric mean, and harmonic mean may be expressed as —

When computing arithmetic mean, larger values are given more weight than tiny ones, however in harmonic mean, smaller values are given considerably more weight than larger values. As a result, the arithmetic mean is bigger than the geometric mean, and the geometric mean is greater than the harmonic mean.