2.2 Standard Normal Calculations

Empirical vs. Standardizing Since all normal distributions share common properties, we have been able to use the Rule to describe the distribution. All normal distributions are the same if measured in units of size  about the mean, µ, as the center. If units aren’t measured like this, we can change the unit, which is called standardizing.

To Standardize or z - score Let x = an observation from a distribution. The standardized value of x is defined by the formula,, which is known as the z-score. The z – score tells us how many standard deviations the observation falls from the mean and in what direction.

Return to the Giraffe Problem The mean, µ = 204 inches with a standard deviation,  = 5.5. So for a giraffe that is 215 inches tall, his z – score, standardized height, would be z=( )/5.5=2. Interpreted as two standard deviations to the right of the mean. If we wanted to know the probability/percentages of giraffes that are up to 215 inches tall, we refer to the standard normal probabilities table, first page (back and front) of your text or yellow packet. We look for and we get.9772 or 97.72%.

Giraffe Problem continued Using the calculator, we go to 2 nd – VARS (DISTR). We select normalcdf(0, 215, 204, 5.5) or normal cdf(0, 215, 2(z-score)). The calc returns with a proportion of.9772, again roughly 97.7%. On average, about 97.7% of giraffes are 215 inches tall or shorter.

Giraffes For a giraffe that is inches tall, his z – score is ________, which means that this height is _____ standard deviations to the _________ of the mean. What is the probability of giraffes being at most this tall? _________. What percentages of giraffes are less than inches? What percentages are taller than inches?

Summary If we standardize a normal distribution, it makes the distribution into a single distribution that is still normal called the standard normal distribution. In this distribution the mean = 0 and the standard deviation is 1, N(0,1). If a variable x has any normal distribution N(µ,), then the standardized variable is z. Classwork: The Length of Pregnancy worksheet. Give each answer as a one-liner in context.