# The Normal Distribution
#
# In the tutorial we learned how to convert raw scores to z-scores
# and back again. For example, given a normal distribution with mean 70
# and standard deviation of 10, the z-score for a raw score of 80 is:
z <- (80-70)/10
print(z)
# We can define variables 'mu' and 'sigma' to make this more general:
mu <- 70
sigma <- 10
# and define X to be the raw score:
X <- 80
z <- (X-mu)/sigma
print(z)
# To find the area above this score, we can use 'pnorm':
1-pnorm(z)
# In R there's a shortcut. 'pnorm' (and qnorm) allow you to send in
# means and standard deviations instead of using the default 0 and 1.
# This will give you the same answer:
1-pnorm(80,70,10)
# or, equivalently:
1-pnorm(X,mu,sigma)
# Here are a couple more examples:
# The proportion of raw scores that fall below 55 is:
pnorm(55,mu,sigma)
# The proportion of raw scores that fall above 90 is:
1-pnorm(90,mu,sigma)
# Converting areas to scores is just as easy. For example, for the
# normal distribution with mean 70 and standard deviation 10, we
# can calculate the score for which 5% lies above by first finding
# the z-score using qnorm:
z <- qnorm(1-.05)
print(z)
# And then convert to raw scores using our formula:
X <- z*sigma + mu
print(X)
# As with pnorm, there's a shortcut by giving qnorm the mean and
# standard deviation instead of the default 0 and 1. This should
# give us the same answer:
qnorm(1-.05,mu,sigma)
# The scores that bracket the middle 95% can be found similarly. The
# lower score is:
qnorm(.05/2,mu,sigma)
# and the upper score is:
qnorm(1-.05/2,mu,sigma)
# We can concatenate these into a single vector:
c(qnorm(.05/2,mu,sigma),qnorm(1-.05/2,mu,sigma))