Unit - 4

Statistical techniques

4.1 Statistics: Introduction to correlation and regression, Multiple correlation and its properties, Multiple regression analysis, Regression equation of three variables

Correlation:

So far we have confined our attention to the analysis of observation on a single variable. There are, however, many phenomena where the changes in one variable are related to the changes in the other variable.  For instance, the yield of crop varies with the amount of rainfall, the price of a commodity increases with the reduction in its supply and so on. Such a simultaneous variation, i.e., when the changes in one variable are associated or followed by change in the other, is called correlation. Such a data connecting two variables is called bivariate population.

If an increase (or decrease) in the values of one variable corresponds to an increase (or decrease) in the other, the correlation is said to be positive. If the increase (or decrease) in one corresponds to the decrease (or increase) in other, the correlation is said to be negative.  If there is no relationship indicated between the variables, they are said to be independent or uncorrelated.

When two variables are related in such a way that change in the value of one variable affects the value of the other variable, then these two variables are said to be correlated and there is correlation between two variables.

Example- Height and weight of the persons of a group.

The correlation is said to be perfect correlation if two variables vary in such a way that their ratio is constant always.

Types of correlation:

According to the direction of change in variables there are two types of correlation

1. Positive Correlation

2. Negative Correlation

1. Positive Correlation:

Correlation between two variables is said to be positive if the values of the variables deviate in the same direction i.e. if the values of one variable increase (or decrease) then the values of other variable also increase (or decrease). For example:

1. Heights and weights of group of persons;

2. House hold income and expenditure;

3. Amount of rainfall and yield of crops

2. Negative Correlation:

Correlation between two variables is said to be negative if the values of variables deviate in opposite direction i.e. if the values of one variable increase(or decrease) then the values of other variable decrease (or increase). Some examples of negative correlations are correlation between

1. Volume and pressure of perfect gas;

2. Price and demand of goods;

3. Literacy and poverty in a country

Scatter diagram-

Scatter diagram is a statistical tool for determining the potentiality of correlation between dependent variable and independent variable. Scatter diagram does not tell about exact relationship between two variables but it indicates whether they are correlated or not.

To obtain a measure of relationship between the two variables, we plot their corresponding values on the graph, taking one of the variables along the x-axis and the other along the y-axis.

Correlation measures the nature and strength of relationship between two variables. Correlation lies between +1 to -1. A correlation of +1 indicates a perfect positive correlation between two variables. A zero correlation indicates that there is no relationship between the variables. A correlation of -1 indicates a perfect negative correlation.

Definition-

“Correlation analysis deals with the association between two or more variables.” —Simpson and Kafka

“Correlation is an analysis of the co-variation between two variables.” —A.M. Tuttle

Methods of computing coefficient of correlation

1. Scatter diagram method-

It is the simplest method to study correlation between two variables. The correlations of two variables are plotted in the graph in the form of dots thereby obtaining as many points as the number of observations. The degree of correlation is ascertained by looking at the scattered points over the charts.

The more the points plotted are scattered over the chart, the lesser is the degree of correlation between the variables. The more the points plotted are closer to the line, the higher is the degree of correlation. The degree of correlation is denoted by “r”.

• Perfect positive correlation (r = +1) – All the points plotted on the straight line rising from left to right

• Perfect negative correlation (r=-1) – all the points plotted on the straight line  falling from left to right

• High Degree of +Ve Correlation (r= + High): all the points plotted close to the straight line rising  from left to right

• High Degree of –Ve Correlation (r= – High) - all the points plotted close to the straight line falling from left to right.

• Low degree of +Ve Correlation (r= + Low): all the points are highly scattered to the straight line rising  from left to right

• Low Degree of –Ve Correlation (r= - Low): all the points are highly scattered to the straight line falling from left to right

• No Correlation (r= 0) – all the points are scattered over the graph and do not show any pattern

2.     Karl Pearson’s coefficient of correlation:

Coefficient of correlation measures the intensity or degree of linear relationship between two variables. It was given by British Biometrician Karl Pearson (1867-1936).

Karl Pearson’s Coefficient of Correlation is widely used mathematical method is used to calculate the degree and direction of the relationship between linear related variables. The coefficient of correlation is denoted by “r”.

If X and Y are two random variables then correlation coefficient between Xand Y is denoted by r and defined as-

Karl Pearson’s coefficient of correlation-

Here- and

Note-

1. Correlation coefficient always lies between -1 and +1.

2. Correlation coefficient is independent of change of origin and scale.

3. If the two variables are independent then correlation coefficient between them is zero.

Example: Find the correlation coefficient between Age and weight of the following data-

Sol.

x	Y					( ) )
30	56	-10	100	-4	16	40
44	55	4	16	-5	25	-20
45	60	5	25	0	0	0
43	64	3	9	4	16	12
34	62	-6	36	2	4	-12
44	63	4	16	3	9	12
Sum= 240	360	0	202	0	70	32

Karl Pearson’s coefficient of correlation-

Here the correlation coefficient is 0.27.which is the positive correlation (weak positive correlation), this indicates that the as age increases, the weight also increase.

Short-cut method to calculate correlation coefficient-

Here,

Example: Find the correlation coefficient between the values X and Y of the dataset given below by using short-cut method-

Sol.

X	Y
10	90	-20	400	20	400	-400
20	85	-10	100	15	225	-150
30	80	0	0	10	100	0
40	60	10	100	-10	100	-100
50	45	20	400	-25	625	-500
Sum = 150	360	0	1000	10	1450	-1150

Short-cut method to calculate correlation coefficient-

Example

Psychological tests of intelligence and of engineering ability were applied to 10 students. Here is a record of ungrouped data showing intelligence ratio (I.R) and engineering ratio (E.R). Calculate the co-efficient of correlation.

Solution:

We construct the following table:

Student	Intelligence ratio x	Engineering ratio y	X2	Y2	XY
A B C D E F G H I J	100                      6     104                     5     102                     3     101                     2     100                     1       99                     0       98                    -1       96                    -3       93                    -6       92                    -7	101                 3   103                 5   100                 2    98                  0    95                 -3    96                 -2   104                 6    92                 -6    97                 -1    94                 -4	36 25 9 4 1 0 1 9 36 49	9 25 4 0 9 4 36 36 1 16	18 25 6 0 -3 0 -6 18 6 28
Total	990                    0	980                 0	170	140	92

From this table, mean of i.e., and mean of , i.e.,

Substituting these values in the formula (1), we have

Other examples:

Example-1-Compute Pearsons coefficient of correlation between advertisement cost and sales as per the data given below:

Solution

X	Y					(X -
39	47	-26	676	-19	361	494
65	53	0	0	-13	169	0
62	58	-3	9	-8	64	24s
90	86	25	625	20	400	500
82	62	17	289	-4	16	-68
75	68	10	100	2	4	20
25	60	-40	1600	-6	36	240
98	91	33	1089	25	625	825
36	51	-29	841	-15	225	435
78	84	13	169	18	324	234
650	660		5398		2224	2704

r = (2704)/√5398 √2224 = (2704)/(73.2*47.15) = 0.78

Thus Correlation coefficient is positively correlated

Example 2

Compute correlation coefficient from the following data

X	Y					(X -
8	81	1.3	1.8	2.3	5.4	3.1
8	80	1.3	1.8	1.3	1.8	1.8
6	75	-0.7	0.4	-3.7	13.4	2.4
5	65	-1.7	2.8	-13.7	186.8	22.8
7	91	0.3	0.1	12.3	152.1	4.1
6	80	-0.7	0.4	1.3	1.8	-0.9
40	472		7		361	33

= 40/6 =6.7

= 472/6 = 78.7

r = (33)/√7 √361 = (33)/(2.64*19) = 0.66

Thus Correlation coefficient is positively correlated

Example 3

Calculate coefficient of correlation between X and Y series using Karl pearson shortcut method

Solution

Let assumed mean for X = 15, assumed mean for Y = 14

r = 48/√144*√432 = 0.19

3.     Spearman’s rank correlation coefficient

A group of n individuals may be arranged in order to merit with respect to some characteristic. The same group would give different orders for different characteristics. Considering the order corresponding to two characteristics A and B for that group of individuals.

Let xi, yi be the ranks of the ith individuals in A and B respectively. Assuming that no two individuals are bracketed equal in either case, each of the variables taking the values 1,2,3,…,n, we have

If X,Y be the deviation of x, y from their means, then

Similarly  

Now let di = xi - yi  so that di = (xi - - (yi - = Xi - Yi

Hence the correlation coefficient between these variables is

This is called the rank correlation coefficient (Spearman’s rank corr) and is denoted by ρ.

Spearman’s rank correlation-

When the ranks are given instead of the scores, then we use Spearman’s rank correlation to find out the correlation between the variables.

Spearman’s rank correlation coefficient can be defined as-

Example:

	1	6	5	10	3	2	4	9	7	8
	6	4	9	8	1	2	3	10	5	7

Solution:

If di = xi - yi, then di = -5, 2, -4, 2, 2, 0, 1, -1, 2, 1

Hence

Example

Three judges A, B, C, give the following ranks. Find which pair of judges has common approach

Solution: Here

	Ranks by
1 6 5 10 3 2 4 9 7 8	3 5 8 4 7 10 2 1 6 9	6 4 9 8 1 2 3 10 5 7	-2 1 -3 6 -4 -8 2 8 1 -1	-3 1 -1 -4 6 8 -1 -9 1 2	5 -2 4 -2 -2 0 -1 1 -2 -1	4 1 9 36 16 64 4 64 1 1	9 1 1 16 36 64 1 81 1 4	25 4 16 4 4 0 1 1 4 1
Total			0	0	0	200	214	60

Since is maximum, the pair of judges A and C have the nearest common approach.

Example: Compute the Spearman’s rank correlation coefficient of the dataset given below-

Solution

Person	Rank in test-1	Rank in test-2	d =
A	9	1	8	64
B	10	2	8	64
C	6	3	3	9
D	5	4	1	1
E	7	5	2	4
F	2	6	-4	16
G	4	7	-3	9
H	8	8	0	0
I	1	9	-8	64
J	3	10	-7	49
Sum				280

Example: If X and Yare uncorrelated random variables, the of correlation between and

Solution.

Let and

Then

Now

Similarly

Now

Also

(As and are not correlated, we have )

Similarly

Example 1 –

Solution

Here, highest value is taken as 1

R = 1 – (6*8)/5(52 – 1) = 0.60

Example 2 -

Calculate Spearman rank-order correlation

Solution

Rank by taking the highest value or the lowest value as 1.

Here, highest value is taken as 1

R = 0.67

Therefore, this indicates a strong positive relationship between the ranks individuals obtained in the math and English exam.

Example 3 –

Find Spearman's rank correlation coefficient between X and Y for this set of data:

Solution

R =

Example 4 – Calculation of equal ranks or tie ranks

Find Spearman's rank correlation coefficient:

Solution

R = 1 – (6*81.5)/8(82 – 1) = 0.02

Example 5 –

Solution

R = 1 – (6*24)/10(102 – 1) = 0.85

The correlation between X and Y is positive and very high.

Key takeaways:

1. Positive Correlation: Correlation between two variables is said to be positive if the values of the variables deviate in the same direction
2. Negative Correlation: Correlation between two variables is said to be negative if the values of variables deviate in opposite direction
3. Karl Pearson’s coefficient of correlation:

4.     Correlation coefficient always lies between -1 and +1.

5.     If the two variables are independent then correlation coefficient between them is zero

6.     Short-cut method to calculate correlation coefficient-

7.     Spearman’s rank correlation-

Regression Analysis

As the correlation analysis studies the nature and extent of interrelationship between the two variables X and Y, regression analysis helps us to estimate or approximate the value of one variable when we know the value of other variable. Therefore, we can define the ‘Regression’ as the estimation (prediction) of one variable from the other variable when they are correlated to each other. e.g., We can estimate the Demand of the commodity if we know it’s Price.

Why are there two regressions?

When the variables X and Y are correlated there are two possibilities,

(i)Variable X depends on variable y. In this case we can find the value of x if know the value of y. This is called regression of x on .

(ii)Variable depends on variable X. We can find the value of y if know the value of X. This is called regression of y on x. Hence there are two regressions,

Regression of X on Y; (b) Regression of X on Y.

Regression-

If the scatter diagram indicates some relationship between two variables and , then the dots of the scatter diagram will be concentrated round a curve. This curve is called the curve of regression. Regression analysis is the method used for estimating the unknown values of one variable corresponding to the known value of another variable.

Regression is the measure of average relationship between independent and dependent variable

Regression can be used for two or more than two variables.

There are two types of variables in regression analysis.

1. Independent variable

2. Dependent variable

The variable which is used for prediction is called independent variable.

It is known as predictor or regressor.

The variable whose value is predicted by independent variable is called dependent variable or regressed or explained variable.

The scatter diagram shows relationship between independent and dependent variable, then the scatter diagram will be more or less concentrated round a curve, which is called the curve of regression.

When we find the curve as a straight line then it is known as line of regression and the regression is called linear regression.

Note- regression line is the best fit line which expresses the average relation between variables.

LINE OF REGRSSION

When the curve is a straight line, it is called a line of regression. A line of regression is the straight line which gives the best fit in the least square sense to the given frequency.

Equation of the line of regression-

Let

y = a + bx …………..  (1)

Is the equation of the line of y on x.

Let be the estimated value of for the given value of .

So that, According to the principle of least squares, we have the determined ‘a’ and ‘b’ so that the sum of squares of deviations of observed values of y from expected values of y,

That means-

Or

   …….. (2)

Is minimum.

Form the concept of maxima and minima, we partially differentiate U with respect to ‘a’ and ‘b’ and equate to zero.

Which means

And

These equations (3) and (4) are known as normal equation for straight line.

Now divide equation (3) by n, we get-

This indicates that the regression line of y on x passes through the point
.

We know that-

The variance of variable x can be expressed as-

Dividing equation (4) by n, we get-

From the equation (6), (7) and (8)-

Cov (x, y) +

Multiply (5) by, we get-

Subtracting equation (10) from equation (9), we get-

Since ‘b’ is the slope of the line of regression y on x and the line of regression passes through the point (), so that the equation of the line of regression of y on x is-

This is known as regression line of y on x.

Note-

1. are the coefficients of regression.
2. 

Example: Two variables X and Y are given in the dataset below, find the two lines of regression.

Sol.

The two lines of regression can be expressed as-

And

x	y			Xy
65	66	4225	4356	4290
66	68	4356	4624	4488
67	65	4489	4225	4355
67	69	4489	4761	4623
68	74	4624	5476	5032
69	73	4761	5329	5037
70	72	4900	5184	5040
71	70	5041	4900	4970
Sum = 543	557	36885	38855	37835

Now-

And

Standard deviation of x-

Similarly-

Correlation coefficient-

Put these values in regression line equation, we get

Regression line y on x-

Regression line x on y-

Regression line can also be find by the following method-

Example: Find the regression line of y on x for the given dataset.

Sol.

Let y = a + bx is the line of regression of y on x, where ‘a’ and ‘b’ are given as-

We will make the following table-

x	Y	Xy
4.3	12.6	54.18	18.49
4.5	12.1	54.45	20.25
5.9	11.6	68.44	34.81
5.6	11.8	66.08	31.36
6.1	11.4	69.54	37.21
5.2	11.8	61.36	27.04
3.8	13.2	50.16	14.44
2.1	14.1	29.61	4.41
Sum = 37.5	98.6	453.82	188.01

Using the above equations we get-

On solving these both equations, we get-

a = 15.49 and b = -0.675

So that the regression line is –

y = 15.49 – 0.675x

Example: Show that the geometric mean of the coefficients of regression is the coefficient of correlation.

Sol.

We know that the coefficients of regression are-

Then-

Example: Prove that arithmetic mean of the coefficients of regression is greater than the coefficient of correlation.

Sol.

We know that the coefficients of regression are-

Here we need to prove that- A.M. > r

So that-

Which is true.

Note – Standard error of predictions can be find by the formula given below-

Difference between regression and correlation-

1. Correlation is the linear relationship between two variables while regression is the average relationship between two or more variables.

2. There are only limited applications of correlation as it gives the strength of linear relationship while the regression is to predict the value of the dependent varibale for the given values of independent variables.

3. Correlation does not consider dependent and independent variables while regression consider one dependent variable and other indpendent variables.

Example 3:

The following data give the experience of machine operators and their performance rating given by the number of good parts turned out per 100pieces.

Obtain the two regression equations and estimate the performance rating of an operator who has put 15 years in service.

Solution: We define the variables,

X: Experience		Y: Performance rating Table of calculations:
X	Y	Xy	X2	Y2
16 12 18 4 3 10 5 12	87 88 89 68 78 80 75 83	1392 1056 1602 272 232 800 375 996	256 144 324 16 9 100 25 144	7569 7744 7921 4624 6084 6400 5625 6889

Now the two regression equations are,

(

Where,

      and  

Also,

Co(x,y)

Co(x,y)=30.75

Now we find,

Regression co-efficient od X on Y

  and

Regression xo-efficient of X on Y

Now substituting the values of x , y , bxy and byx in the regression

Equations we get,

(x-10) = 0.67(y-81) -------x on y (i)

(y-81) =1.13(x-10) ------- y on x (ii)

As the two regression equations.

Now to estimate Performance rating (y) when Experience (x) = 15, we use

The regression equation of y on x

 (y-81) =1.13(15-10)

 y = 81+ 5.65 = 86.65

Hence the estimated performance rating for the operator with 15 years of

Experience is approximately 86.65 i.e. approximately 87

Regression coefficients in terms of correlation coefficient

We can also obtain the regression coefficients bxy and byx from standard deviations, and correlation coefficient ‘r’ using the formulas.

            and   

Also consider,

        i.e.

Hence the correlation coefficient ‘r’ is the geometric mean of the regression coefficients, bxy and byx

Example 4: Find the marks of a student in the Subject of Mathematics who have scored 65 marks in Accountancy Given,

Coefficient of correlation between the marks of Mathematics and marks of Accountancy is 0.64.

Solution: We define the variables,

X: Marks in Mathematics

Y: Marks in Accountancy

Therefore, we have,

Now we want to approximate the marks in Mathematics (x), we obtain the

Regression equation of x on y, which is given by

Substituting the values, we get,

i.e.                                   

Therefore, when marks in Accountancy (Y) = 65

x- 70 = 0.57(65-80)

   x = 70-2.85 = 67.15                   i.e. 67 approx.

Substitute the value of a and b in the equation. Regression line of X on Y is      

Example 5:

Find the means values of x,y and r from the two regression equations.

3x+2y-26=0 and sx+y-31=0. Also find sx when sy = 3.

Solution: The two regression equations are,

3x+2y-26=0 -------- (i)

6x+y-31=0 ----------(ii)

Now for x and y we solve the two equations as the simultaneous equations.

On solving (1) and (2), we get-

x = 4 and y = 7.

Now to find ‘r’ we express the equations in the form y=a+bx

So, from eqns (i) and (ii)

        and         

Since, b1 < b2 (i.e., b1 is smaller in number irrespective of sign + or -)

... Equation (i) is regression of on and

Hence, eqn (ii) is regression of on and - 1/6 = - 0.16

Now we find,

Note: The sign of ‘r’ is same as the sign of regression coefficients

Now to find 6x when 6y = 3, we use the formula,

Hence means

Example 6: From the following data obtain the two regression equations:

:	6	2	10	4	8
:	9	11	5	8	7

Solution:          

Computation of Regression Equation

X	Y	Xy	x2	y2
6 2 10 4 8	9 11 5 8 7	54 22 50 32 56	36 4 100 16 64	81 121 25 64 49

Regression line of y and X is expressed by the equation of the form

To determine the values of a and b, the following two normal equations are solved

Substituting the value, we get

Multiplying equation (i) by 6, we get

Deduct equation (iv) from (iii)

Substitute the value of b in equation (i)

Substitute the value of a and b in the equation

Regression line of Y on X is

Regression line X on Y is

The corresponding normal equations are

Substituting the values

Multiply equation (i) by 8

Deduct equation (iv) from (iii)

Substitute the value of b in equation) i)

Substitute the value of a and b in the equation. Regression line of X on Y is      

Example 7 Calculate the regressive coefficients from the data given below:

	Series	Series
Average	25	22
Standard deviation	4	2	r=0.8

Solution: The coefficient of regression of y on x is

The coefficient of regression of y on x is

Example 8: The following scores were worked out from a test in Mathematics and English in an annual examination.   Scores in    Mathematics (x)   English (y)   Mean 39.5 47.5   Standard deviation 10.8 16.8 R=+0.42   Find both the regression equations. Using these regression estimate finds the value of Y for X=50 and the value of x for Y=30.

and

Regression Coefficients : X on Y

Y on X

Regression equation: X on Y

Substituting the values

X-75= 0.652 (Y-115)= 0.652Y-74.98

Or               

Where       

Regression equation: Y on X

Y-115= 0.78 (X-75) =0.78 X-58.5

Example 10:

From the following regression equation, find means x , y ,and d ‘r’ 3x-2y-10 = 0, 24x-25y+145 = 0

Solution: The two regression equations are,

3x-2y-10=0 (i)

24x-25y+145=0 (ii)

Now for x and y we solve the two equations as the simultaneous equations.

Therefore, by (i) x 8 and (ii) x1, we get

24x-16y-80 = 0

24x-25y+145 = 0

9y-225=0 y = 25

Putting y = 25 in eqn (i), we get

3x-2(25)-10 = 0

3x – 60=0

X = 20

Hence x= 20 and y= 25.

Now to find ‘r’ we express the equations in the form y=a+bx So, from eqns (i) and (ii)

B1=1.5   b2=0.96

Equation (ii) is regression of y on x and byx = 0.9

Hence eqn (i) is regression of x on y and  bxy = 1/1.5 =0.67

Now we find, r =  √ bxyXbyx i.e. r=√0.67x0.96 = + 0.84

Regression line of least Square Method

It is a mathematical method and with it gives a fitted trend line for the set of data in such a manner that the following two conditions are satisfied.

The sum of the deviations of the actual values of Y and the computed values of Y is zero.

The sum of the squares of the deviations of the actual values and the computed values is least.

This method gives the line which is the line of best fit. This method is applicable to give results either to fit a straight-line trend or a parabolic trend.

The method of least squares as studied in time series analysis is used to find the trend line of best fit to a time series data.

Secular Trend Line

The secular trend line (Y) is defined by the following equation:

Y = a + b X

Where, Y = predicted value of the dependent variable

a = Y-axis intercept i.e. the height of the line above origin (when X = 0, Y = a)

b = slope of the line (the rate of change in Y for a given change in X)

When b is positive the slope is upwards, when b is negative, the slope is downwards

X = independent variable (in this case it is time)

To estimate the constants a and b, the following two equations have to be solved simultaneously:

ΣY = na + b ΣX

ΣXY = aΣX + bΣX2

To simplify the calculations, if the midpoint of the time series is taken as origin, then the negative values in the first half of the series balance out the positive values in the second half so that ΣX = 0. In this case, the above two normal equations will be as follows:

ΣY = na

ΣXY = bΣX2

Q1) Fit the straight line to the following data.

A1)

The normal equation are:

Σy = aΣx + nb

And

Σxy = aΣx2 + bΣx

Now,

X	Y		XY
1	1	1	1
2	2	4	4
3	3	9	9
4	4	16	16
5	5	25	25

15 = 15a + 4b and 55 = 55a + 15b

Solving these two equations,

We get a=1 and b=0,

Therefore the required straight-line equation is y=x.

Regression line Least Square Method

Q2) Fit the straight-line curve to the following data.

A2) First drawing the table,

X	Y		XY
75	82	5625	1
80	78	6400	4
93	86	8349	9
65	72	4225	16
87	91	7569	25
71	80	5041
98	95	9605
68	72	4624
84	89	7056
77	74	5929

The normal equation are:

Σy = aΣx + nb

And

Σxy = aΣx2 + bΣx.

Substituting the values, we get,

819 = 798a + 10b

66045 = 64422a + 798b

Solving, we get

a = 0.9288 and b = 7.78155

Therefore, the straight line equation is:

y = 0.9288x + 7.78155.

Key takeaways:

1. If the scatter diagram indicates some relationship between two variables and , then the dots of the scatter diagram will be concentrated round a curve. This curve is called the curve of regression.
2. Regression line is the best fit line which expresses the average relation between variables.
3. are the coefficients of regression.
4. 
5. The geometric mean of the coefficients of regression is the coefficient of correlation.
6. The arithmetic mean of the coefficients of regression is greater than the coefficient of correlation
7. Standard error of predictions can be find by the formula given below-

4.2 Measures of central tendency and dispersion: Mean, Median, Quartile, Decile, Percentile, Mode, Mean deviation, Standard deviation

Measures of central tendency

Professor Bowley defines the average as-

“Statistical constants which enable us to comprehend in a single effort the significance of the whole”

An average is a single value which is the best representative for a given data set.

Measures of central tendency show the tendency of some central values around which data tend to cluster.

The following are the various measures of central tendency-

1. Arithmetic mean

2. Median

3. Mode

4. Weighted mean

5. Geometric mean

6. Harmonic mean

Arithmetic mean or mean-

Arithmetic mean is a value which is the sum of all observation divided by total number of observations of the given data set.

If there are n numbers in a dataset- then arithmetic mean will be-

If the numbers along with frequencies are given then mean can be defined as-

Example-1: Find the mean of 26, 15, 29, 36, 35, 30, 14, 21, 25 .

Sol.

Example-2: Find the mean of the following dataset.

Sol.

We have the following table-

Then Mean will be-

Direct method to find mean-

Example: Find the arithmetic mean of the following dataset-

Sol.

We have the following distribution-

Mean =

Short cut method to find mean-

Suppose ‘a’ is assumed mean, and ‘d’ is the deviation of the variate x form a, then-

Example: Find the arithmetic mean of the following dataset.

Sol.

Let the assumed mean (a) = 25,

Step deviation method for mean-

Where

Median-

Median is the mid value of the given data when it is arranged in ascending or descending order.

1. If the total number of values in data set is odd then median is the value of item.

Note-The data should be arranged in ascending r descending order

2. If the total number of values in data set is even then median is the mean of the item.

Example: Find the median of the data given below-

7, 8, 9, 3, 4, 10

Sol.

Arrange the data in ascending order-

3, 4, 7, 8, 9, 10

So there total 6 (even) observations, then-

Median for grouped data-

Here,

Example: Find the median of the following dataset-

Sol.

So that median class is 20-30.

Now putting the values in the formula-

So that the median is 28.57

Mode-

A value in the data which is most frequent is known as mode.

Example: Find the mode of the following data points-

Sol. Here 6 has the highest frequency, so that the mode is 6.

Mode for grouped data-

Here,

Example: Find the mode of the following dataset-

Sol.

Here highest frequency is 9. So that the modal class is 40-50,

Put the values in the given data-

Hence the mode is 42.86

Note-

Mean – Mode = [Mean - Median]

Geometric Mean-

If are the values of the data, then the geometric mean-

Harmonic mean-

Harmonic mean is the reciprocal of the arithmetic mean-

It can be defined as-

Concept of partition values:

Partition values-

The values divide the distribution into certain number of equal parts are called partition values.

Data should be arranged in ascending order descending order.

Quartile, deciles and percentile are the partition values.

Note-

1. Quartile divides the data into four equal parts.
2. Deciles and percentiles divide the distribution into ten and hundred equal parts, respectively

Quartile-

There are three quartiles, i.e. Q1, Q2 and Q3 which divide the total data into four equal parts when it has been orderly arranged. Q1, Q2 and Q3 are termed as first quartile, second quartile and third quartile or lower quartile, middle quartile and upper quartile, respectively. The first quartile, Q1, separates the first one-fourth of the data from the upper three fourths and is equal to the 25th percentile. The second quartile, Q2, divides the data into two equal parts (like median) and is equal to the 50th percentile. The third quartile, Q3, separates the first three-quarters of the data from the last quarter and is equal to 75th percentile.

For ungrouped data, we find the quartiles as follows-

For grouped data, we find the quartiles as follows-

i’th quartile can be find as-

l = lower class limit of i'th quartile class,

h = width of the ith quartile class,

N = total frequency,

C = cumulative frequency of pre ith quartile class, and

f = frequencies of ith quartile class.

Deciles

Deciles divide whole distribution in to ten equal parts. There are nine deciles.

For ungrouped data, we find the deciles as follows-

For grouped data, we find the deciles as follows-

i’th decile can be find as-

Percentile-

Percentiles divide whole distribution in to 100 equal parts. There are ninety nine percentiles.

For ungrouped data, we find the percentile as follows-

For grouped data, we find the percentile as follows-

i’th percentile can be find as-

Example: Calculate the first and third quartile of the following data-

Sol.

Here N/4 = 28/4 = 7

The 7th observation falls in the class 10-20. So, this is the first quartile class. 3N/4 = 21th observation falls in class 30-40, so it is the third quartile class.

For first quartile l = 10, f = 5, C = 3, N = 28

We know that-

For third quartile l = 30, f = 9, C = 15

Measures of dispersions:

As the name suggests, the measure of dispersion shows the scatterings of the data. It tells the variation of the data from one another and gives a clear idea about the distribution of the data. The measure of dispersion shows the homogeneity or the heterogeneity of the distribution of the observations.

According to Spiegel, the degree to which numerical data tend to spread about an average value is called the variation or dispersion of data.

Classification of Measures of Dispersion

There are two basic kinds of a measure of dispersion-

1. Absolute measures
2. Relative measures

Following are the different types of measures of dispersion-

According to Spiegel-

“The degree to which numerical data tend to spread about an average value is called the variation or dispersion of data”

The different measures of dispersion are-

1. Range

2. Quartile deviation

3. Mean deviation

4. Standard deviation

5. Variance

Significance of measures of dispersion-

Measures of variation are pointed out as to how far an average is representative of the entire data. When variation is less, the average closely represents the individual values of the data and when variation is large; the average may not closely represent all the units and be quite unreliable.

Another purpose of measuring variation is to determine the nature and causes of variations in order to control the variation itself. Measurements of dispersion are helpful to control the causes of variation.

Many powerful statistical tools in statistics such as correlation analysis, the testing of hypothesis, the analysis of variance, techniques of quality control, etc. are based on different measures of dispersion.

Range

Range is the simplest measure of dispersion. Range is the difference between the maximum value of the variable and the minimum value of the variable in the distribution.

Example: Find the range of the distribution 4, 22, 14, 12, 16, 8, 13, 17, 21, 6, 5, 26.

Sol.

For the given distribution, the maximum value is 26 and the minimum value is 4, so that the range of the distribution is –

Example- Find the range of the data- 8, 5, 6, 4, 7, 10, 12, 15, 25, 30

Sol. Here the maximum value is 30 and the minimum value is 4 so that the range is-

30 – 4 = 26

Coefficient of range-

The coefficient of range can be calculated as follows-

Coefficient of Range =

Advantages of range-

1. It is very simple to calculate
2. It has useful applications in areas like order statistics and statistical quality control.

Disadvantages of range-

1. It utilizes only the maximum and the minimum values of variable in the series and gives no importance to other observations
2. It is affected by fluctuations of sampling
3. If a single value lower than the minimum or higher than the maximum is added or if the maximum or minimum value is deleted range is seriously affected

Quartile Deviation

The quartiles divide a data set into quarters. The first quartile, (Q1) is the middle number between the smallest number and the median of the data. The second quartile, (Q2) is the median of the data set. The third quartile, (Q3) is the middle number between the median and the largest number.

Quartile deviation or semi-inter-quartile deviation is

Relative measure of Q.D. Known as Coefficient of Q.D. And is defined as

Example: Find the quartile deviation of the following data.