Mean

Before learning about mean, let us first know what is a central tendency? Professor Bowley tells that Averages are statistical fixed quantities that enable us to get a picture of the importance of the whole in a single effort. In other words, an average is the value of the variable which represents the entire distribution. The 5 measures of central tendency that are in general use are as follows:

• Arithmetic Mean/Mean
• Median
• Mode
• Geometric Mean
• Harmonic Mean

What is an Arithmetic mean?

Arithmetic mean of a given set of observations is defined as the sum divided by the number of observations. It is usually represented by or x bar. Symbolically, we can write it as,

= 1 / n (x₁+x₂+……+x_n ) = 1 / n ∑ x_i where i = 1 to n

In case of the frequency distribution, i.e., x and f are provided then, it can be symbolically written as,

= 1 / N ( x₁f₁+x₂f₂+……+x_nf_n ) = 1 / N ∑ x_if_i where ∑ fi = N; where i = 1 to n

In the case of a grouped/continuous frequency distribution, x is the mid-value of the corresponding class. If the values of x and f are large, then the mean calculation by the above formula would be very tedious and time-consuming. In such cases, the arithmetic values are reduced to a great extent by taking the deviations of the given values from any arbitrary point A as shown below:

= A + 1 / N ∑ f_id_i where i = 1 to n and d = x_i - A

Properties of Arithmetic Mean:

First Property:

Algebraic sum of the deviations(d) of a set of values from their arithmetic mean is zero.

Second Property:

The sum of the squares of the deviations of a set of values is minimum when taken about mean.

Third Property:

This is also referred as Mean of the composite series.

If xi bar (xi bar) where i = 1 to k are the means of the k component series of sizes ni, (I = 1 to k) respectively, then, the mean of the x bar of the composite series is obtained on combining the component series is represented by the formula:

= ( ∑ n_{i i} ) / ∑ n_i

Merits of Arithmetic Mean:

The advantages of AM are as follows:

• Arithmetic means are strictly defined.
• Arithmetic means are easy to understand and easy to evaluate.
• Arithmetic means are based upon all the observations.
• Arithmetic means are conformable to algebraic treatment.
• Among all the averages, arithmetic means are least affected by fluctuations of sampling.

Thus, we observe that arithmetic means satisfy all the properties laid down by Professor Yule for an ideal average.