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AP Statistics 6.1B Notes Standard Deviation (and Variance) of a Discrete Random Variable The variance of a random variable is an average of the squared deviation xi x of the values of the 2 variable X from its mean x . To get the standard deviation of a random variable, we take the square root of the variance. The standard deviation of a random variable X is a measure of how much the values of the variable tend to vary, on average, from the mean x . Suppose that X is a discrete random variable whose probability distribution is Value: … x1 x2 x3 Probability: p1 p2 p3 … and that x is the mean of X. the variance of X is Var(X) x2 x1 x p1 x2 x p2 x3 x p3 ... 2 2 2 = xi x pi 2 The standard deviation of X, x is the square root of the variance. Example A large auto dealership keeps track of sales made during each hour of the day. Let X = the number of cars sold during the first hour of business on a randomly selected Friday. Based on previous records, the probability distribution of X is as follows: Cars sold 0 1 2 3 Probability 0.3 0.4 0.2 0.1 1. Compute and interpret the mean of X. 2. Compute and interpret the standard deviation of X. Continuous Random Variables A continuous random variable X takes all values in an interval of numbers. The probability distribution of X is described by a density curve. The probability of any event is the area under the density curve and above the values of X that make up the event. The probability distribution for a continuous random variable assigns probabilities to intervals of outcomes rather than to individual outcomes. In fact, all continuous probability models assign probability 0 to every individual outcome. Only intervals of values have positive probability.