http://www.ck12.org Chapter 7. Sampling Distributions and Estimations
7.2 The z-Score and the Central Limit Theorem
Learning Objectives
- Calculate thez−score of a mean distribution of a random variable in problem situations.
- Understand the Central Limit Theorem and calculate a sampling distribution using the mean and standard
deviation of a normally distributed random variable. - Understand the relationship between the Central Limit Theorem and normal approximation of the binomial
distribution.
Introduction
In the previous lesson you learned that sampling is an important tool for determining the characteristics of a
population. Although the parameters of the population (mean, standard deviation, etc.) were unknown, random
sampling was used to yield reliable estimates of these values. The estimates were plotted on graphs to provide a
visual representation of the distribution of the sample mean for various sample sizes. It is now time to define some
properties of the sampling distribution of the sample mean and to examine what we can conclude about the entire
population based on it.
All normal distributions have the same basic shape and therefore rescaling and recentering can be implemented to
change any normal distributions to one with a mean of zero and a standard deviation of one. This configuration is
referred to as standard normal distribution. In this distribution, the variable along the horizontal axis is called the
z−score. This score is another measure of the performance of an individual score in a population. Thez−score
measures how many standard deviations a score is away from the mean. Thez−score of a termxin a population
distribution whose mean isμand whose standard deviationσis given by:
z=
x−μ
σSinceσis always positive,zwill be positive whenXis greater thanμand negative whenXis less thanμ. Az−score
of zero means that the term has the same value as the mean. For the normal standard distribution, whereμ=0, if we
letx=σ, thenz=1. If we letx= 2 σ,z=2. Thus, a value ofztells the number of standard deviations the given
value ofxis above or below the mean.
Example:On a nationwide math test the mean was 65 and the standard deviation was 10. If Robert scored 81, what
was hisz−score?
Solution: