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In Descriptive Statistics, a data can be described if at least a measure of central tendency and a measure of variability is given. The same concept is applied to the distributions of random variable. Stags in Finding the Variance and Standard Deviation 1. Find the mean of the probability distribution. 2. Subtract the mean from each value of the random variable X. 3. Square the results obtained in Step 2. 4. Multiply the results obtained in Step 3 by corresponding probability. 5. Get the sum ofthe results obtained in step 4. Example: The number of cars sold per day at a local car dealership, along with its corresponding probabilities, is shown in the succeeding table. Compute the variance and the standard deviation of the probability distribution by following the given steps. Number of Cars Sold Probability X Pix) 0 i 19 1 i m 2 i 3 i 4 1 Solution: Steps Solution 1. Find the mean of the Number of Cars probability distribution using the formula p= ZX-PUO 2. Subtract the mean from each value of the random variable X. 3. Square the results obtained in Step 2. 4. Multiply the results obtained in Step 3 by the Corresponding Prnhahllitv 10 10 3 0.8 0.64 ONON 4 1.8 3.24 4. Multiply the results X P(x) X.P(X) X - ul ( X - H)2 (X - H)2 . P(X) obtained in Step 3 by O 2 0 -2.2 4.84 0.484 the Corresponding 10 1 2 Probability. 1.44 0.288 10 10 1.2 3 6 2 - 0.2 0.04 0.012 10 10 3 2 6 0.8 0.64 0.128 10 10 8 4 1.8 3.24 0.648 5. Get the sum of the X P(x) X.P(X) X - ul ( X - H)2 (X - H)2 . P(X) results obtained in step O 2 0 -2.2 4.84 0.484 4. The result is the value 10 2 of the variance. So, the 2 -1.2 1.44 0.288 10 10 formula for the variance 3 6 0.04 0.012 N is: 10 10 - 0.2 3 2 6 80 0.64 0.128 10 10 62 = [ (x - 1)2 . P(x) 4 2 8 3.24 0.648 10 10 1.8 62 = E(x - 1)2 . P(x) = 1.56 6. Get the square root of the variance to get the The variance of the probability distribution is 1.56 standard deviation. The standard deviation is o = V1.56 = 1.25. Formula for the Variance and Standard Deviation of a Discrete Probability Distribution The variance of a discrete probability distribution is given by the formula: 62 = [ (x - 1)2 . P(X) The Standard deviation of a discrete probability distribution is given by the formula: " = \\ [ ( x - 14 ) 2 . P ( X ) Where: X = value of the random variable P(X) = probability of the random variable X H = mean of the probability distribution. The variance and standard deviation can be obtained using a shorter formula and procedure. Alternative Procedure in Finding the Variance and Standard Deviation of a Probability Distribution. 1. Find the mean of the probability distribution. 2. Multiply the square of the value of the random variable X by its corresponding probability. 3. Get the sum of the results obtained in step 2 4. Subtract the mean from the results obtained in step 3. Solution: Steps Solution 1. Find the mean of the probability distribution using the formula Number of Probability X . P(X) " = [x . P (X) Cars Sold P(X) X 0 0 1 2 3 4Solution: Steps Solution 1. Find the mean of the probability distribution using the formula Number of Probability X . P(X) " = [x . P (X ) Cars Sold P(X) X O 0 1 2 3 4 H = Ex . P(X) = T 22 = 2.2 2. Multiply the square of the value of the random variable X by its corresponding X P (X ) X . P (X ) X2 . P(X ) probability. 1 0 02 10 0 1 2 2 12 10 10 2 22 10 10 3 2 18 32 S NS NE 10 10 4 32 42 2 3. Get the sum of the results obtained in step 2. X P(X) X . P ( X ) x2 . P(X) 0 O 02 10 = 0 1 IN 12 2 22 10 3 18 32 10 10 4 2 32 42 10 10 1= > x2 . P (X) = 64 10 = 6.4 4. Subtract the square of the mean from the results obtained in step 3 to get the | The Variance is variance. So , the formula for the 62 = x2 . P(X) - 12 variance of a probability distribution is = 6.4- (2.2) 2 = 1.56 62 = [x2 . P(X) - 12 The standard deviation is the square The standard deviation is : root of the variance. Thus, o = VEX2 . P(X) - 12 = V1.56 = 1.25 6 = x2 . P (X) - 1 2 The variance is 1.56 and the standard deviation is 1.25. Alternative Formula for the Variance and Standard Deviation of a Discrete Probability Distribution The variance deviation of a discrete probability distribution is given by the formula: 62 = [x2 . P(X) - 12 The standard deviation of a discrete probability distribution is given by the formula: " = x2 . P(X) - 12 Where: X = value of the random variable P(X) = probability of the random variable X u = mean of the probability distributionThe variance is 1.56 and the standard deviation is 1.25. Alternative Formula for the Variance and Standard Deviation of a Discrete Probability Distribution The variance deviation of a discrete probability distribution is given by the formula: -[ x2 . P ( x ) - 1 2 The standard deviation of a discrete probability distribution is given by the formula: [ x 2 . P( X ) - 1 2 Where: X = value of the random variable P(X) = probability of the random variable X u = mean of the probability distribution Explore your understanding: Answer first the transfer and do the explore. Which formula is easier to use in finding the variance and standard deviation of the probability distribution? Why ? Firm - Up your knowledge: Complete the table below and find the variance and standard deviation of the probability distributions. 1.) X P(X) X . P(X) X2 . P (X) 6 11 HINHINHINHINHIN 16 21 Deepen your understanding: 1.) Find the variance and standard deviation of the probability distribution of the random variable X, which can take only the values 2, 4, 5 and 9, given that P(2) = 20 , P(4) = 20' P(5) = 5, and P(9) = 10. Transfer: Solve the problem in two ways. The number of computers sold per day at a local computer store, along with its corresponding probabilities, is shown in the table. Find the variance and standard deviation of the distribution. Number of Computers Sold X Probability P(X) 0 0.1 0.2 W N H 0.3 0.2 A 0.2