SECTION 4.1 Probability Distributions 195 Although the mean of the random variable of a probability distribution describes a typical outcome, it gives no information about how the outcomes vary. To study the variation of the outcomes, you can use the variance and standard deviation of the random variable of a probability distribution. The variance of a discrete random variable is s 2 = Σ1x - m2 2P1x2. The standard deviation is s = 2s 2 = 2Σ1x - m2 2P1x2. Variance and Standard Deviation of a Discrete Random Variable Finding the Variance and Standard Deviation The probability distribution for the personality inventory test for passive-aggressive traits discussed in Example 2 is shown below. Find the variance and standard deviation of the probability distribution. Score, x 1 2 3 4 5 Probability, P1x2 0.16 0.22 0.28 0.20 0.14 SOLUTION To find the variance and standard deviation, note that from Example 5 the mean of the distribution before rounding is m = 2.94. (Use this value to avoid rounding until the last calculation.) Use a table to organize your work, as shown below. x P1x2 x − M 1x − M2 2 1x − M2 2P1x2 1 0.16 -1.94 3.7636 0.602176 2 0.22 -0.94 0.8836 0.194392 3 0.28 0.06 0.0036 0.001008 4 0.20 1.06 1.1236 0.224720 5 0.14 2.06 4.2436 0.594104 ΣP1x2 = 1 Σ1x - m2 2P1x2 = 1.6164 Variance So, the variance is s 2 = 1.6164 ≈ 1.6 and the standard deviation is s = 2s 2 = 21.6164 ≈ 1.3. Interpretation “Usual” data values will be within 211.32 = 2.6 of the mean. TRY IT YOURSELF 6 Find the variance and standard deviation of the probability distribution constructed in Try It Yourself 2. Answer: Page A38 EXAMPLE 6 Study Tip An alternative formula for the variance of a probability distribution is s 2 = [Σx2P1x2] - m 2. Tech Tip You can use technology such as Minitab, Excel, StatCrunch, or the TI-84 Plus to find the mean and standard deviation of a discrete random variable. For instance, to find the mean and standard deviation of the discrete random variable in Example 6 on a TI-84 Plus, enter the possible values of the discrete random variable x in L1. Next, enter the probabilities P1x2 in L2. Then, use the 1-Var Stats feature with L1 as the list and L2 as the frequency list to calculate the mean and standard deviation (and other statistics), as shown below. TI-84 PLUS x=2.94 ∑x=2.94 ∑x2=10.26 Sx= ox=1.271377206 n=1 1-Var Stats
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