# The z-distribution # # R's function 'pnorm' calculates the area under the normal distribution # below a z-score. By default, it uses a standard normal distribution # (mean 0, s.d. 1). # # For example, to find the area under the standard normal below z=1, we use: pnorm(1) # Note that the tables in the book and the Excel spreadsheet give you # the areas ABOVE z. To get the values in these tables we need to subtract # the result from 'pnorm' from 1: 1-pnorm(1) # You should see that this value matches the value in the third column in # the standard normal table for z=1. # The area under the standard normal below z = -2 is: pnorm(-2) # The area under the standard normal between z=1 and z=2 can be found # by finding the area below z=2 and subtracting the area below z=1: pnorm(2) - pnorm(1) # The area between z = -2 and z=1 can be caculated similarly: pnorm(1) - pnorm(-2) # To go the other way and find z-scores from areas we use # the function 'qnorm'. # # For example, the z-score for which 95% of the area of the # standard normal falls below is: qnorm(.95) # This is, of course the same as the z-score for which 5% falls above. # To find the z-score for which 10% of the area lies below we use: qnorm(.1) # To find the values of z that bracket the middle 95% is found # by first finding the lower z-score. This is the z-score # for which only 2.5% falls below: qnorm(.025) # Since the z-distribution is symmetric, the upper value of z # is just the positive sign of this: -qnorm(.025) # In general, if you want to find the z scores that bracket # the middle p% we can first set a variable 'p' to some value: p <- .5 # And then use 'c' to concatenate the two z-scores into a single # vector: c(qnorm((1-p)/2),-qnorm((1-p)/2))