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.”
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
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:-
i. Individual Series
Let x_{1}, x_{2}, x_{3}, ..........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.
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.
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
Demerits of Arithmetic Mean
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
Demerits of Geometric Mean
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
Demerits of Hamonic mean
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
Demerits of median
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,
Δ_{1} = f_{1} -f_{0},
Δ_{2} = f_{1} - f_{2},
f_{1} = largest frequency,
f_{0} = frequency preceding modal class,
f_{2} = frequency following the modal class,
h = size of the modal class
Merits of mode
Demerits of mode
References:
Adhikari, Ramesh Prasad, Economics-XI, Asmita Pustak Prakashan, Kathmandu
Kanel, Navaraj et.al., Principles of Economics-XI, Buddha Prakashan, Kathmandu
Kharel, Khom Raj et.al., Economics In English Medium-XI, Sukunda Pustak Bhawan, Kathmandu
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