The Standard Deviation of a Discrete Random Variable
The Standard Deviation of a Discrete Random Variable Lecture 24 Section 7. 5. 1 Fri, Oct 20, 2006
The Variance and Standard Deviation Variance of a Discrete Random Variable – The variance of the values that the random variable takes on, in the long run. n This is the average squared deviation of the values that the random variable takes on, in the long run. n Standard Deviation of a Discrete Random Variable – The square root of the variance. n
Why Do We Need the Standard Deviation? Our goal later will be to assign a margin of error to our estimates of a population mean. n To do this, we need a measure of the variability of our estimator x. n This, in turn, requires a measure of the variability of x, namely, the standard deviation. n
The Variance of a Discrete Random Variable The variance of X is denoted by X 2 or Var(X) n The standard deviation of X is denoted by X. n Usually there are no other variables, so we may write 2 and . n
The Variance The variance is the expected value of the squared deviations. n That agrees with the earlier notion of the average squared deviation. n Therefore, n
The Variance n Since the variance of X is the average value of (X – )2, we use the method of weighted averages to compute it.
Example of the Variance n Again, let X be the number of children in a household. x P(x) 0 1 2 0. 10 0. 30 0. 40 3 0. 20
Example of the Variance n Subtract the mean (1. 70) from each value of X to get the deviations. x P(x) x–µ 0 1 2 0. 10 0. 30 0. 40 -1. 7 -0. 7 +0. 3 3 0. 20 +1. 3
Example of the Variance n Square the deviations. x P(x) x–µ (x – µ)2 0 1 2 0. 10 0. 30 0. 40 -1. 7 -0. 7 +0. 3 2. 89 0. 49 0. 09 3 0. 20 +1. 3 1. 69
Example of the Variance n Multiply each squared deviation by its probability. x P(x) x–µ (x – µ)2 P(x) 0 1 2 3 0. 10 0. 30 0. 40 0. 20 -1. 7 -0. 7 +0. 3 +1. 3 2. 89 0. 49 0. 09 1. 69 0. 289 0. 147 0. 036 0. 338
Example of the Variance n Add up the products to get the variance. x P(x) x–µ (x – µ)2 P(x) 0 1 2 3 0. 10 0. 30 0. 40 0. 20 -1. 7 -0. 7 +0. 3 +1. 3 2. 89 0. 49 0. 09 1. 69 0. 289 0. 147 0. 036 0. 338 0. 810 = 2
Example of the Variance n Take the square root to get the standard deviation. x P(x) x–µ (x – µ)2 P(x) 0 1 2 3 0. 10 0. 30 0. 40 0. 20 -1. 7 -0. 7 +0. 3 +1. 3 2. 89 0. 49 0. 09 1. 69 0. 289 0. 147 0. 036 0. 338 0. 810 = 2 0. 9 =
Alternate Formula for the Variance n It turns out that That is, the variance of X is “the expected value of the square of X minus the square of the expected value of X. ” n Of course, we could write this as n
Example of the Variance n One more time, let X be the number of children in a household. x 0 P (x ) 0. 10 1 2 3 0. 30 0. 40 0. 20
Example of the Variance n Square each value of X. x 0 P (x ) 0. 10 x 2 0 1 2 3 0. 30 0. 40 0. 20 1 4 9
Example of the Variance n Multiply each squared X by its probability. x 0 P (x ) 0. 10 x 2 P(x) 0. 00 1 2 3 0. 30 0. 40 0. 20 1 4 9 0. 30 1. 60 1. 80
Example of the Variance n Add up the products to get E(X 2). x 0 P (x ) 0. 10 x 2 P(x) 0. 00 1 2 3 0. 30 0. 40 0. 20 1 4 9 0. 30 1. 60 1. 80 3. 70 = E(X 2)
Example of the Variance Then use E(X 2) and µ to compute the variance. n Var(X) = E(X 2) – µ 2 = 3. 70 – (1. 7)2 = 3. 70 – 2. 89 = 0. 81. n It follows that = 0. 81 = 0. 9. n
TI-83 – Means and Standard Deviations n n n n Store the list of values of X in L 1. Store the list of probabilities of X in L 2. Select STAT > CALC > 1 -Var Stats. Press ENTER. Enter L 1, L 2. Press ENTER. The list of statistics includes the mean and standard deviation of X. Use x, not Sx, for the standard deviation.
TI-83 – Means and Standard Deviations Let L 1 = {0, 1, 2, 3}. n Let L 2 = {0. 1, 0. 3, 0. 4, 0. 2}. n Compute the parameters and . n
Example: Powerball www. powerball. com n Find the standard deviation of the value of a Powerball ticket. n x P (x ) 20000 0. 0000684 4 200000 0. 0000002806 10000 0. 000001711 100 0. 00007015 100 0. 00008384 7 0. 003437 7 0. 001341 4 0. 007881 3 0. 01450 0 0. 9727
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