Economic: Mathematics Used in Economics | Measure of Central Tendency

Measures of Central Tendency

In statistics, the first task is to collect a data from the field of investigation which is also known as the universe or population. After this process of presentation data in a table or graph or any diagram it analysis starts. We need to analyze the data to draw conclusion and inferences. Those inferences are then forecasted to use in practice for different purposes. The central values of the data are those values or items of the data, around which all other items tends to gather. Such central values are the representatives of the whole data. Calculation of representative values of data is called the measure of central tendency.

According to Crum & Smith, “An average is sometimes called a measure of central tendency because individual values of variables cluster it.”

Types of Average

The following are the values of a data. These are also called the measure of central location.

a. Arithmetic mean

b. Geometric mean

c. Harmonic mean

d. Median

e. Mode

Arithmetic Mean (A.M.)

The arithmetic mean is calculated by dividing total values of the item in a variable by their number.

Methods of calculating arithmetic mean are as follow:-

1. Direct Method

i. Individual Series

Let x₁, x₂, x₃, ..........x_n be the items of a data. Then their arithmetic mean (A.M) = \(\frac{x_1+x_2+x_3+.........x_n}{n}\)

i.e. \(\overline{x}\) = \(\frac{\sum_{i=1}^n xi}{n}\)

It can be simply written as: \(\overline{x}\) = \(\frac{∑x}{n}\)

where,

n = number of observations

ii. Discrete Series

In case of discrete series

\(\overline{x}\) = \(\frac{\sum_{i=1}^n f_ix_i}{\sum_{i=1}^n f_i}\)

simply, \(\overline{x}\) = \(\frac{∑fx}{∑f}\) = \(\frac{∑fx}{N}\)

where,

∑f = N is the total frequency

iii. Continuous Series

In case of continuous series the formula \(\overline{x}\) = \(\frac{∑fx}{N}\) remains same except that x is the mid value of the class interval.

iv. Weighted Arithmetic Mean

Arithmetic mean gives equal importance to all the items in a series. The relative importance of different items in a series may differ. This relative importance is known as weight.

Weighted arithmetic mean (\(\overline{X}_w\)) = \(\frac{∑wx}{∑w}\)

where,

∑w = total weight

v. Combined mean

Let there be two series of n1 items and n2 items. Let \(\overline{x}_1\) and \(\overline{x}_2\) be their AM's. Let \(\overline{x}_{12}\) be the combined mean of the series.

Then \(\overline{x}_{12}\) = Error!

Where n1 + n2 is the sum of number of items of both series.

2. Shortcut Method or Deviation Method

i. Individual series

\(\overline{X}\) = a + \(\frac{\sum fd}{n}\)where, assumed mean and d = x-a

where, assumed mean and d = x-a

a = assumed mean

d = x-a

ii. Discrete series

\(\overline{X}\) = a + \(\frac{\sum fd}{N}\) where, N = \(\sum\)f, a = assumed mean and d = x-a

where, N = \(\sum\)f, a = assumed mean and d = x-a

N = \(\sum\)f, a = assumed mean and d = x-a

a = assumed mean and d = x-a

d = x-a

iii. Continuous series

In case of continuous series the formula \(\overline{X}\) = a + \(\frac{\sum fd}{N}\) remains the same as in discrete series except that x is the mid value of the class.

3. Step Deviation Method

i. Individual series

The step deviation method, in case of individual series, is given by \(\overline{x}\) = a + \(\frac{\sum d'}{N}\)× h

where, a = assume mean, d' = \(\frac{x-a}{h}\) and h = common factor

ii. Discrete series

In case of discrete series, the step deviation method is used as \(\overline{x}\) =a + \(\frac{\sum d'}{N}\)× h

where, a = assumed mean, d' = \(\frac{x-a}{h}\) and ∑f = N = Total frequency

iii. Continous series

In case of continous series the formula \(\overline{x}\) =a + \(\frac{\sum d'}{N}\)× h remains the same as in discrete series exccept that x is the mid value of the class interval. Here h is the length of the class interval.

Merits of Arithmetic Mean

It is easy to understand and calculate.
It is based on all the observation of the series.
It is least affected by fluctuations of sampling.
It is a calculated value.

Demerits of Arithmetic Mean

It can give a risible result.
it is affected by extreme points.
It cannot be picked up by observation.
It cannot be calculated for the problem related to open and classes.

Geometric Mean (G.M.)

Geometric mean is used to find average rate of the population growth and rate of interest.

i. Individual series

\(\begin{align*} \text{GM} &= \sqrt[n]{x_1 \times x_2 \times x_3 \times .......... \times x_n}\\ &= (x_1 \times x_2 \times x_3 \times ........ \times x_n)^\frac 1n\\ \end{align*} \)
\(\begin{align*} \text{or, log GM} &= \frac 1n log (x_1 \times x_2 \times x_3 \times .......... \times x_n)\\ &= \frac 1n [log x_1 + logx_2 + logx_3 + ...... +logx_n]\\ &= \frac 1n [\sum logx_i]\\ \end{align*} \)
∴ Gm = antilog [\(\frac{∑logx}{n}\)]

ii. Discrete series

GM = antilog [\(\frac{∑logx}{n}\)]

Where, N =∑f

iii. Continous series

In case of continous series the formula GM = antilog [\(\frac{∑logx}{n}\)] applies very well except that x is the mid value of the class interval.

Merits of Geometric Mean

It is rigidly defined.
It is based upon all the observations.
It is suitable for further mathematical treatment.
It is not affected by fluctuations of sampling.
It gives more weight to small items.

Demerits of Geometric Mean

If one of the items is negative or zero, it is difficult to compute.
It gives more importance to smaller items as compared to large items.
It has abstract mathematical characters for a layman.

Harmonic Mean (H.M.)

Harmonic mean is another measure of central tendency and also based on mathematic-like arithmetic mean and geometric mean.The harmonic mean of a number of an observation is the reciprocal of the arithmetic mean of reciprocals of given values. It is denoted by H.M or only H.

i. Individual series

HM = \(\frac{n}{∑\frac{1}{x}}\)

Where, n = number of variate values or items.

ii. Discrete series

HM = \(\frac{N}{∑f× \frac{1}{x}}\) =\(\frac{N}{∑ \frac{f}{n}}\)

where N = ∑f

iii. Continuous series

HM = \(\frac{N}{∑ \frac{f×1}{x}}\) = \(\frac{N}{∑\frac{f}{m}}\)

where N =∑f and 'x' is the mid value of the class interval.

Merits of Harmonic mean

It is based on all the observation.
It is not much affected by the fluctuation of sampling.
It is the most appropriate average while dealing with speed.

Demerits of Hamonic mean

It is not easy to understand.
It is difficult to compute.
It can not be determined when the variate value is zero.

Median

The observation of a data that divides the whole data into two equal parts is called its median.

According to Cantor, "the median is that value of the variable which divides the group into two equal parts, one part comprising all the values greater and other all values less than the median."

i. Individual series

Before calculating the median in individual series, we first put the items in ascending or descending order and then use the following formula:

Median (M_d) = value of (\(\frac{n+1}{2}\))^th

Where n is the number of items. When n is even, median is taken arithmetic mean of two middle values i.e. \(\frac{n}{2}\)th and (\(\frac{n}{2}\) + 1)^th values.

ii. Discrete series

To calculate the median in discrete series, we first put the items in ascending or descending order and add the frequencies from top to bottom to get cumulative frequencies and then after we use given formula:

Median (M_d) = value of (\(\frac{N+1}{2}\))^th item value of item having cf equal to or just greater than (\(\frac{N+1}{2}\))^th item.

iii. Continuous series

In case of continuous series, the following is used to calculate median where median class lies\(\frac{n}{2}\)th item:

Median (M_d) = L + \(\frac{\frac{N}{2} - cf}{f}\)× h

Where, L = lower limit of the class in which the median falls. The median falls in that class where cf is equal or just greater than \(\fra{N}{2}\0. The class is called model class.

N = total frequency
cf = cumulative frequency preceding the median class
f = the frequency of the median class
h= size of the class

Merits of median

It is easy and simple to calculate.
It os rigidly defined.
It is located by a graph.
It is used for qualitative data.
It is computed for open-end classes.

Demerits of median

The arrangement of data according to order is necessary.
It is not based on all the observation.
It cannot be determined exactly for ungrouped data.
it is affected by the fluctuation of data.

Mode

Mode of data is that item or value of a variable which repeats the largest number of time. Mode doest not exist in individual series as it does not carry any repeating number. In case of continuous series mode is calculated using following formula:

Mode (M_o) = L + \(\frac{Δ_1}{Δ_1 +Δ_2}\)× h

Where,

L = lower limit of the model class,

Δ₁ = f₁ -f₀,

Δ₂ = f₁ - f₂,

f₁ = largest frequency,

f₀ = frequency preceding modal class,

f₂ = frequency following the modal class,

h = size of the modal class

Merits of mode