How to Compute Variance

In probability theory and statistics, the variance is used as one of several descriptors of a distribution. In particular, the variance is one of the moments of a distribution. The variance is a parameter describing a theoretical probability distribution, while a sample of data from such a distribution can be used to construct an estimate of this variance: in the simplest cases this estimate can be the sample variance. (Source: Wikipedia)

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How to Compute Variance - Examples

Example 1: In class 7 student’s height are as follows 144, 154, 175, 180, 165, 160, 170 centimeters. Compute the variance of given data. 

Solution:

Mean = Sum of all the elements in a data set / total number of elements in a data set

by adding and dividing by 7 to get

       `barx` = 1148 / 7 = 164

Table for getting the variance

x	x – 164	(x - 164 )2
144	-20	400
154	-10	100
175	11	121
180	16	256
165	1	1
160	-4	16
170	6	36
Total		930

Formula to compute the variance is

`s^2 = 1/(N-1) sum_(i=1)^n (x_i-barx)^2`

Put all the values in the formula to get

930 / (7-1) = 155

Therefore variance is 155.

Example 2: In a class 9 student’s weight are 45, 50, 61, 85, 62, 72, 66, 75, 78 kilograms. Compute the variance of given data. 

Solution:

Mean = Sum of all the elements in a data set / total number of elements in a data set

Mean by adding and dividing by 9 to get

        x = 594 / 9 = 66

Table for getting the variance:

x	x – 66	(x - 66 )2
45	-21	441
50	-16	256
61	-5	25
85	19	361
62	-4	16
72	6	36
66	0	0
75	9	81
78	12	144
Total		1360

Formula to find the variance is

`s^2 = 1/(N-1) sum_(i=1)^n (x_i-barx)^2` 

Put all the values in the formula to get

1360 / (9-1) = 170       

Therefore variance is 170

How to Compute Variance - Practice

Problem 1: In class 7 student’s height are as follows 154, 164, 185, 190, 175, 170, 180 centimeters. Compute the variance of given data. 

Answer: 155

Problem 2: In a class 9 student’s weight are 55, 60, 71, 95, 72, 82, 76, 85, 88 kilograms. Compute the variance of given data. 

Answer: 170

Problem 3: 9 person's age are 25, 35, 30, 42, 45, 60, 39, 14, 52 years. Compute the variance of given data. 

Answer: 195.5

Subdivision In Math

Algebra is a subdivision in math, which comprises of infinite number of operations on equations, polynomials, inequalities, radicals, rational numbers, logarithms, etc. Sketching graphs for algebraic equations or function is also a part of algebra. Graphing is nothing but the pictorial view of the given function or equation to study their characteristics, it may be a line, parabola, hyperbola, curve, circle, etc. Graphing negative square root is also done in algebra. The procedure to graph negative square root  function with the graph is explained in the following sections.

                                             

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Procedure for graphing negative square root:

The procedure for graph negative square root function is given below,

Step 1: Consider the equations as y = `sqrtx` , 

Step 2:  y =`sqrtx` . Since the given equation is a function of x, let y =f(x).

Step 3: Therefore f(x) =` sqrtx`

Step 4: Substitute various values for ‘x’ and find corresponding f(x).

Step 5: Tabulate the values as columns x & f(x).  The values of x as -6, -5, -4, -3, -2, -1, 0, and for f(x), their corresponding  values. Positive values cannot be used, since they tend to become imaginary.

Step 6: The values in the table are the co-ordinates, graph them.

Step 7: Connect the points to find the shape of the function.

Graph for negative square root:

The graphs for the negative square root function is shown below,
1. Convert the given equation as

y = `sqrt(-x)`

Since the given equation is a function of y,

Let y =f(x).

Therefore

f(x) =`sqrt(-x)`

Substitute various values for ‘y’ and find corresponding f(y).

When x= -9

f(-3) = `sqrt(-(-9))`

3

 therefore the co-ordinates are (-9,3)

When x= -4

f(-2) = `sqrt(-(-4))` ,

2,

therefore the co-ordinates are (-4, 2)

When x= -1,

f(-1) = `sqrt(-1)` ,

1,

1, therefore the co-ordinates are (-1, 1)

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When x= 0

f(0) =` sqrt(0)` ,

0,

0, therefore the co-ordinates are (0, 0)

Following the above procedure and the co-ordinates are,

                      
The graph for the above table is shown below,