Download Teaching Assistant Chapter Highlights and Outline

Teaching Assistant Chapter Highlights and Outline
Notes from Chapter 4
Authored by Jeremy Cox
Chapter Outline
4.1 The Normal Distribution
 Data sets with large sample sizes (40 or more) can be approximated with the
normal distribution. Since the normal distribution is understood very well, we
can generate a table for it (Appendix B). By assuming a distribution is
normal, we can carry out MANY calculations and understand a great deal
about it by merely finding the mean and standard deviation.
4.1.1 The Normal Distribution as a Theoretical Distribution
 The normal distribution is theoretical; it is an imaginary, ideal distribution.
 It can never be achieved in reality because there are infinite number of cases
in a normal distribution, and there are points all the way out to infinity.
4.1.2 The Shape of the Normal Distribution
 The normal curve is bell-shaped and symmetrical.
 Symmetry is extremely important in the bell-curve. Particularly, the area
under the curve from –1.5 to -1.0 is the same as from 1.0 to 1.5.
4.1.3 Area Under the Curve and Probability
 The area contained in a region of the normal curve represents the probability
that a randomly drawn value will come from that region’s section of the curve.
4.2 Rules of Probability
 We discuss probability in terms of events (where something happens) and the
outcomes (the different ways that event happens.)
 The probability that a certain outcome will occur is how often that outcome
occurs out of all the outcomes. (Equation 4.1 P(A) = f(A)/N)



Mutually Exclusive outcomes and Independent events
Outcomes that cannot happen at the same time are Mutually Exclusive.
Events that do not influence the outcome of each other are Independent
Events.
4.2.1 The Converse Rule
 The probability that A will occur is equal to 1 minus the probability that A
will not occur.
4.2.2 The Addition Rule

The probability that one of so many outcomes in one event will occur is the
sum of the probabilities of each individual outcome.
4.2.3 The Multiplication Rule
 The probability that in x multiple events, x specific outcomes will occur is
equal to the product of the probabilities of each individual outcome.
4.3 Standard Scores
 Standard deviation is “a yardstick used in measuring distances in a
distribution” (p100). We can convert raw data into standard scores, that
measure the score’s distance from the mean in units of standard deviations.
We call these z-scores.
score  x
z-score =
s
4.3.1 Characteristics of Standard Scores
 When a distribution is converted to z-scores, the shape stays the same, the
mean will be 0, and the variance and standard deviation is always 1.
4.3.2 Some Uses of Standard Scores
 Like percentile ranks, standard scores give an idea of the location of the score
in the distribution. Notice that z-scores to not have a linear order to them like
percentile ranks! Z-scores also are interval data, whereas percentile ranks are
ordinal.
 Standard scores provide a way to compare scores on distributions which have
different units of measure.
4.4 The Standard Normal Distribution
 The standard normal distribution is the distribution resulting from computing
z-scores for a normal distribution.
4.4.1 Characteristics of the Standard Normal Distribution
 Mean = 0, s = s2 = 1
 We can memorize the % spread of scores based on standard deviations
(Figure 4.1, p93).
 We can also use Table 1 of Appendix B to determine areas under parts of the
curve and determine the percentage of the population at those parts of the
curve.
4.4.2 Some Uses of the Standard Normal Distribution
 If we can assume a set of data is normal, we can use the standard normal
distribution to approximate information about that data.
 Estimate percent ranks
 Estimate percentile’s score or z-score
 Estimate percentages between scores
4.5 What if the Distribution Is Not Normally Distributed?
 The main point of this short section is that the normal distribution is an ideal.
We can use it to approximate finites sets of “error-containing” data. In the
future, we will learn how to minimize this error and when an approximation is
appropriate or not.