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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.