Mean(s)

The most important objective of a statistical analysis is to calculate a single value that represents the characteristics of the entire available raw data.  This single value representing the entire data is called the 'central value' or an 'average'.  This value is the point around which all the other values of the data cluster. 

 

Types of averages

 

 

Arithmetic Mean

 

The arithmetic mean or just 'mean' is the most simple and frequently used average Arithmetic Mean is represented by notation  (read as x-bar).

 

The mean of a collection of observations x₁, x₂, …………. , x_n is given by:

 

     = (1/n) (x₁+x₂+………x_n)

     = ∑x/n

                       _n

     = (1/n) ∑x_i

                      ⁱ⁼¹

 

where –

 

is the sample mean

i is the set of natural numbers

  _n

∑x_i is the sum of the values of all observations

 ⁱ⁼¹

n is the number of elements

∑ indicates that all the values of x are summed together.

 

Example (ungrouped data)

Absentee list of drivers of the transport department over a span of 90 days

 

The mean of the ungrouped data can be calculated as follows:

 

     = ∑x/n

     = 8+6+6+7+4+5+6+2+4+7

             ----------------------------

                             10

          = 55/10

          = 5.5 days leave per driver out of 90 days.

 

 

Example (grouped data)

A frequency distribution consists of data that are grouped into classes.  Every observation (value) is placed in one of the classes. 

 

 

To compute the arithmetic mean of grouped data, we have to calculate the midpoint of each class and multiply each mid point (class mark) by frequency of observations in the corresponding class.  Then, we have to add all these results and divide the sum by the total number of observations.

 

Mid point (class mark)         = x = (lower limit + upper limit)/2

The formula for computing Arithmetic mean for grouped data is:

     = ∑(f x x)/n

where,

          ∑ = Notation for 'Sum'

          f = number of observations in each class

          x = class mark (mid point of each class)

          n = number of observations in the sample

 

 

     = 4456000/600 = Rs. 7426.66.

 

 

Geometric Mean

 

One often comes across quantities that change over a period of time, and may need to know the average rate of change over a period of time.  Arithmetic mean is inaccurate in tracing such a change.  Hence a new measure of central tendency is required called the 'Geometric Mean'.

 

Geometric Mean (GM) is defined as nth root of the product of n observations.

 

G.M.    =        _n√product of all the values

          =        _n√x₁ x x₂ x ……………x_n

          =        (x₁ x x₂ x ………….x_n)^1/n

 

where 'n' is the number of values

 

Example

Growth Rate of Textile units

 

 

GM      = ₅√1.07 x 1.08 x 1.10 x 1.12 x 1.18

          = 1/1.1093

 

1.1093 is the average growth factor. The growth rate is calculated as 1.1093-1 = 0.1093 = 10.93% p.a.

 

Harmonic Mean

 

Harmonic mean is based on the reciprocals of numbers averaged.  It is defined as the reciprocal of the arithmetic mean of the reciprocal of the given individual observations.  Thus by definition:

 

 

Example

Gopi walks from his house to the bank at a speed of 2 kmph, while returning from the bank to the house, his speed was 3 kmph.  Calculate the average speed for the whole walk.

 

Average speed         =        N / {(1/X1) = (1/X2)}

                             =        2 / (1/2+1/3)

                             =        12/5

                             =        2.4 kmph

 

 

 

Merits of Geometric Mean

1. It is rigidly defined.
2. It is based on all observations and is sensitive to changes therein.
3. It gives less weight to large values and more to small values.
4. It is capable of further mathematical treatment.

 

Limitations of Geometric Mean

1. It is relatively difficult to understand and calculate as compared to the Arithmetic Mean.
2. It may not be actually present in the data.
3. The loss of a single observation makes it impossible to calculate the mean correctly.
4. A Geometric Mean with zero value cannot be compounded with similar other non-zero values or values with negative sign.
5. It is unduly affected by sampling fluctuations or extreme or negative values.