VARIANCE Sum of the square of deviation, divided by the total number of observations, is known as Variance. Mathematically, V(x) = δ² = Σ ( x – x )² n OR V(x) = δ² = Σx² - Σx ² n n

Where, x = Variables or Observations x = Arithmetic Mean (Sum of all observations divided by the total number of observations) n = Total number of observations ( x – x ) = Deviation (The difference between individual value about its mean) Σx = Sum of all observations Σx² = Sum of the square of all observations

1 st PROPERTY OF VARIANCE Explained by: MARIA SAHER KHAN

The properties of variance are as follows: 1) The variance of a constant is always be zero. Mathematically, V(c) = 0 Short Explanation: If we calculate the variance of constant or same variables the answer will be equal to zero.

Suppose we have the data: x x²V(x) = δ² = Σx² - Σx ² 5 25 n n V(x) = δ² = ² Σx= 15 Σx²= V(x) = δ² = 0 Proved.

2 nd PROPERTY OF VARIANCE Explained by: NIDA SOHAIL

2) The second property of variance is that, it is not affected by change of origin. Mathematically, V(x + a) = V(x) Where a = constant no. Short Explanation: If we add or subtract any constant number from given data, their calculated variance is same as that of the variance calculated by the original data.

Let the data be : x x² y= (x + 10) y² z = (x - 10) z² Σx = 30 Σx² = 270 Σy = 70 Σy² = 1270 Σz = -10 Σz² = 70 We have to prove that, V(x) = V(y) = V(z)

Calculating V(x):- By using the formula, V(x) = δ² = Σx² - Σx ² n n V(x) = δ² = 270 – 30 ² 4 4 V(x) = δ² = 11.25

Calculating V(y):- By using the formula, V(y) = δ² = Σy² - Σy ² n n V(y) = δ² =1270 – 70 ² 4 4 V(y) = δ² = 11.25

Calculating V(z):- By using the formula, V(z) = δ² = Σz² - Σz ² n n V(z) = δ² = 70 – -10 ² 4 4 V(z) = δ² = Hence it is proved that,V(x) = V(y) = V(z)

3 RD PROPERTY OF VARIANCE Explained by: ANUM FAROOQ

3) The third property of variance is that, it affects by change of scale. Mathematically, V(ax) = a² V(x) Where a = constant no. Short Explanation: Multiplication of data by some quantity and then take its variance will be different from the variance calculated by the original data. To overcome this problem we multiply the square of same constant with the variance of original data so that we get the same result.

Let the data be : x x² y = (4x) y² Σx = 24 Σx² = 164 Σy = 96 Σy² = 2624

Calculating V(x):- By using the formula, V(x) = δ² = Σx² - Σx ² n n V(x) = δ² = 164 – 24 ² 4 4 V(x) = δ² = 5 According to the condition we will have to multiply the square of constant term with the variance in order to calculate the accurate result. a² V(x) = 4² x 5 a² V(x) = 16 x 5 a² V(x) = 80

Calculating V(y):- By using the formula, V(y) = δ² = Σy² - Σy ² n n V(y) = δ² = 2624 – 96 ² 4 4 V(y) = δ² = 80 Hence it is proved that, V(ax) = a² V(x)

 It is a relative measure of dispersion and independent of units and expressed in percentages.  It was introduced by Karl Pearson.  It is used to compare the variability of two sets of data.  The group which has lower value of C.V, is more consistent or more stable, and the group which has higher value of C.V does not have consistency and has large variation.

The Formula is: C.V = δ x 100 X Where, δ is the symbol of Standard Deviation and is defined as, “The square root of the sum of square of deviation, divided by the total number of observations.” Formula is: δ = ∑ ( x – x )² n

Following is the data of two daily use commodities one is Flour taken in Kg and the other is Milk in liter. Find which commodity has higher consistency in its prices and which has greater variation. Months Flour/kg Milk/liter January February March April May June 34 40

(Prices) x/kg ( x – x ) = (x - 28 ) ( x – x )² Σx = 168 Σ( x – x ) = 0 Σ (x – x ) ² = 112

Calculating C.V:- To calculate C.V we have to calculate Standard Deviation and Arithmetic Mean first. δ = Σ (x - x)² X = Σx n n δ = 112 X = δ = 4.32 kg X = 28 kg

Now we will calculate the C.V for Flour: C. V = δ x 100 X C.V = 4.32 x C.V = %

(Prices) x/liter ( x – x ) = (x - 36 ) ( x – x )² Σx = 216 Σ( x – x ) = 0 Σ (x – x ) ² = 42

Calculating C.V:- To calculate C.V we have to calculate Standard Deviation and Arithmetic Mean first. δ = Σ (x - x)² X = Σx n n δ = 42 X = δ = 2.65 liter X = 36 liter

Now we will calculate the C.V for Milk: C. V = δ x 100 X C.V = 2.65 x C.V = 7.34 %

Since the C.V of Milk is less than the C.V of Flour, therefore the prices of Milk are more consistent and more stable. Where as the greater C.V of Flour describes that it has variation it its prices and is not consistent.

Thank you so much  Our honorable teacher Sir Zafar Ali.  The students of BS Commerce (2 nd Semester) to cooperate with us. We wish you all the very best for your future.  Please pardon us if we hurt you throughout the Presentation.