Putting the normal distribution to work
Standardising, reading the normal table in both directions, intervals at one, two and three standard deviations, the usual quantiles and the approximation of a binomial by a normal distribution.
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Standard normal distribution: what is it?
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The bell-shaped distribution with mean 0 and standard deviation 1, the one given in the tables.
Any normal distribution reduces to it by a change of variable. It is symmetric: P(Z ≤ −a) = 1 − P(Z ≤ a).
For X normal with mean 200 and standard deviation 20, what is P(X ≤ 180)? Given P(Z ≤ 1) = 0.8413.
0.15870.84130.34130.6826
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0.1587z = (180 − 200)/20 = −1, and by symmetry P(Z ≤ −1) = 1 − 0.8413 = 0.1587. 0.8413 is the opposite side of the bell; 0.3413 is the area between 0 and 1.
If X follows a normal distribution with mean μ and standard deviation σ, then Z = … follows the standard normal distribution.
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If X follows a normal distribution with mean μ and standard deviation σ, then Z = (X − μ) / σ follows the standard normal distribution.
Subtracting μ recentres on 0; dividing by σ brings the width back to 1. The probability does not change: P(X ≤ x) = P(Z ≤ z).
Standardising: what is it?
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Turning a value x into z = (x − μ)/σ: subtract the mean then divide by the standard deviation.
z counts the number of standard deviations separating x from the mean: that is what allows different distributions to be compared and the table to be read.