Basic Concepts of Statistics

One of the most important objectives of statistical analysis is to get one single value that describes the characteristics of entire mass of unwieldy data such a value is called central value or the average. It is a single value, which represents a group of values.

Classification of Data:

(i) Discrete series (ii) Continuous series

TYPE OF AVERAGES

(a) Mean

(i) Arithmetic (ii) Weighted arithmetic mean

(iii) Geometric Mean (iv) Weighted Geometric Mean

(v) Harmonic Mean (vi) Weighted Harmonic Mean

(b) Median

(c) Mode

The Arithmetic Mean:

The arithmetic mean of a statistical data is defined as the quotient of the sum of all the values of the variable by the total number of items. It is denoted by A.M.

For an individual series

(a) A.M. = $\frac{\Sigma x}{n}$

(b) A.M. = A + $\frac{\Sigma d}{n}$, where d = x – A, A is the assumed mean

For a frequency distribution,

(a) A.M =$\frac{\Sigma fx}{\Sigma f}$ (Mean of grouped data)

(b) Short Cut Method

A.M. = A + $\frac{\Sigma fd}{\Sigma f}$, where d = x – A, where A is the assumed mean.

(c) Step Deviation Method

A.M. = A + $\left(\frac{\Sigma fu}{\Sigma f}\right)$h, where A is the assumed mean and u = $\frac{x-A}{h}$.

Weighted Arithmetic Mean:

If w₁, w₂, w₃, …, w_n are the weights assigned to the values x₁, x₂, x₃, …, x_n respectively, then the weighted average is defined as:

Weighted Arithmetic Mean = $\frac{w_1{x}_1+w_2{x}_2+\dots +w_n{x}_n}{w_1+w_2+\dots +w_n}$.

Illustration 1: A group of 10 items has mean 6. If the mean of 4 of these items is 7.5, then the mean of remaining items is

(A) 6.5 (B) 5.5

(C) 4.5 (D) 5.0

Solution: Sum of all the 10 items = 10 x 6 = 60

Sum of four of these items 4 x 7.5 = 30

sum of the remaining six items = 60 –30 = 30

hence, the mean of remaining six items = $\frac{30}{6}$ = 5

Median:

Median is defined as the middle most or the central value of the variables in a set of observations, when the observations are arranged either in ascending or in descending order of their magnitudes. It divides the arranged series in two equal parts. Median is a position average, whereas, the arithmetic mean is the calculated average. When a series consists of an even number of terms, median is the arithmetic mean of the two central items. It is generally denoted by M.

Case I: When n is odd.

In this case $\frac{n+1}{2}$th value is the median i.e. $M=\frac{n+1}{2}thterm.$

Case II: When n is even.

In this case there are two middle terms $\frac{n}{2}$th and $\left(\frac{n}{2}+1\right)th$. The median is the average of those two terms, i.e. $M=\frac{\frac{n}{2}+\left(\frac{n}{2}+1\right)}{2}$ th term

Case III: When the series is continuous.

In this case the data is given in the form of a frequency table with class-interval, etc., and the following formula is used to calculate the Median.

M = L + $\frac{\frac{n}{2}-C}{f}\times i$, where

L = lower limit of the class in which the median lies

n = total number of frequencies, i.e., n = $\sum f$.

f = frequency of the class in which the median lies

C = cumulative frequency of the class preceding the median class

i = width of the class-interval of the class in which the median lies.

Quartile:

Just as the median divides a set of observations (when arranged in ascending or descending order of magnitudes), into two equal parts, similarly quartile divides the observations into four equal parts. The value of the item midway, between the first item and the median is known as first or lower quartile and is denoted be Q₁. The value of the item midway between the last item and the median is known Third or Upper Quartile and is denoted Q₃. The median is known as the Second Quartile and denoted by Q₂.

Case I: For ungrouped data.

Q₁ = $\frac{n+1}{4}$th item, Q₃ = $\frac{3(n+1)}{4}$the item.

Case II: For a frequency distribution

Q₁ = $L+\frac{(n/4-C)}{f}\times i$, $Q_3=L+\frac{(3n/4-C)}{f}\times i$, where

L = lower limit of the class in which a particular quartile lies,

f = Frequency of the class-interval in which a particular quartile lies

i = Class-interval of the class in which a particular quartile lies,

C = Cumulative frequency of the class preceding the class in which the particular quartile lies

In general, Q_h = L + \dfrac{[(nh/4)-C]}{f}\ xi,\ h = 1, 2, 3, 4

Deciles:

The values which divide the series when arranged in ascending or descending order of magnitudes in ten equal parts are called deciles. There are 9 deciles in all, which are denoted by D₁, D₂, D₃, D₄, D₅, D₆, D₇, D₈, D₉ respectively.

Case I: For ungrouped data.

$D_h=\frac{nxh}{10}$

Case II: When the series is grouped.

$D_h=L+\frac{[nh/10-C]}{f}xi$,

where L, C, f, i have their usual meanings and h = 1, 2, 3, 4, 5, 6, 7, 8, 9.

Percentiles:

The values of the variables which divide the series, when arranged in ascending or descending order, into 100 equal parts are called percentiles. There are 99 percentiles denoted by P₁, P₂, P₃, P₄,…,P₉₉ respectively.

Case I: When the series is ungrouped.

The percentiles are calculated by the following formula:

$P_h=\frac{nxh}{100},$ h = 1, 2, …, 99

Case II: When the series is grouped.

$P_h=L+\frac{[(nh/100-C)]}{f}$xi, h = 1, 2, …, 99, where L, C, f, i have their usual meanings.

Mode:

Mode is defined as that value in a series which occurs most frequently. In a frequency distribution mode is that variate which has the maximum frequency.

Continuous Frequency Distribution:

i) Modal Class: It is that class in grouped frequency distribution in which the mode lies.

Mode = $L+\frac{f_m-f_1}{2f_m-f_1-f_2}\times i$, where

L = the lower limit of the modal class

i = the width of the modal class

f₁ = the frequency of the class preceding modal class

f_m = the frequency of the modal class

f₂ = the frequency of the class succeeding modal class.

Sometimes it so happened that the above formula fails to give the mode. In this case, the modal value lies in a class other than the one containing maximum frequency. In such cases we take the help of the following formula:

Mode = $L+\frac{f_2}{f_1+f_2}\times i$, where L, f₁, f₂, i have usual meanings.

Asymmetrical Distribution:

A distribution in which mean, median and mode coincide is called symmetrical distribution. If the distribution is moderately asymmetrical, then mean, median and mode are connected by the formula.

Mode = 3 Median – 2Mean

Illustration 2: The quartile deviation of the daily wages (in Rs) of 7 persons given below:

12, 7, 15, 10, 17, 19, 25 is

(A) 14.5 (B) 5

(C) 9 (D) 4.5

Solution: Arranging the given observations in ascending order, we have 7, 10, 12, 15, 17, 19, 25.

So, Q₁ = 2^nd observation = 10 and Q₃ = 6^th observation = 19

Hence Q.D = $\frac{Q_3-Q_1}{2}=\frac{19-10}{2}$ = 4.5

Geometric Mean:

If x₁, x₂, x₃, …, x_n are n values of a variable x, none of them being zero, then the Geometric mean G is defined as G = (x₁, x₂, x₃, …, x_n )^1/n.

Geometric Mean for Frequency Distribution:

Geometric mean of n values x₁, x₂, x₃, …, x_n of a variable x, occurring with frequency f₁, f₂, f₃, …, f_n respectively is given by

$G={[x_1^{f_1},x_2^{f_2},x_3^{f_3},\dots ,x_n^{f_n}]}^{1/n}$ or G = antilog $\left[\frac{\sum _{i=1}^n{f}_i\log x_i}{n}\right]$

Harmonic Mean:

The harmonic mean of n items x₁, x₂, x₃,…, x_n is defined as:

Harmonic Mean = $\frac{n}{\frac{1}{x_1}+\frac{1}{x_2}+\frac{1}{x_3}+\dots +\frac{1}{x_n}}$

Harmonic Mean of Frequency Distribution:

Let x₁, x₂, x₃, …, x_n be n items which occur with frequencies f₁, f₂, f₃, …, f_n respectively. Then their Harmonic Mean is given by

Harmonic Mean = $\frac{f_1+f_2+f_3+\dots +f_n}{\frac{f_1}{x_1}+\frac{f_2}{x_2}+\frac{f_3}{x_3}+\dots +\frac{f_n}{x_n}}=\frac{\sum f_i}{\sum f_i\times \frac{1}{x_i}}$

Relation between Arithmetic Mean, Geometric Mean and Harmonic Mean:

The arithmetic mean (A. M.), Geometric mean (G.M.) and Harmonic Mean (H.M.) for a given set of observations of a series are related as under:

A. M $⩾$ G.M $⩾$ H.M