PSY 307 – Statistics for the Behavioral Sciences Chapter 8 – The Normal Curve, Sample vs Population, and Probability

Demos  How normal distributions are generated:  How changes in the mean and std deviation affect the shape of the normal distribution:  Finding the proportion for a given z score:  Finding the z-score for a given portion of the distribution:

A Family of Normal Curves  A normal curve has a symmetrical, bell-like shape. The lower half (below the mean) is the mirror image of the upper half.  Values for the mean, median and mode are always the same number.  The mean and SD specify the location and shape (steepness) of the normal curve.

Different Normal Curves Same SD but different MeansSame Mean but different SDs

Z-Score  Indicates how many SDs an observation is above or below the mean of the normal distribution.  Formula for converting any score to a z-score: Z = X –  

Properties of Z-Scores  A z-score expresses a specific value in terms of the standard deviation of the distribution it is drawn from. The z-score no longer has units of measure (lbs, inches).  Z-scores can be negative or positive, indicating whether the score is above or below the mean.

Standard Normal Curve  By definition has a mean of 0 and an SD of 1.  Standard normal table gives proportions for z-scores using the standard normal curve.  Proportions on either side of the mean equal.50 (50%) and both sides add up to 1.00 (100%).

Finding Proportions Actually +/-1.96

Finding Exact Proportions  tml tml  ds/Table/Pages/Table1.htm ds/Table/Pages/Table1.htm

Other Distributions  Any distribution can be converted to z-scores, giving it a mean of 0 and a standard deviation of 1.  The distribution keeps its original shape, even though the scores are now z-scores. A skewed distribution stays skewed.  The standard normal table cannot be used to find its proportions.

Why Samples?  Population – any complete set of observations or potential observations.  Sample – any subset of observations from a population. Usually of small size relative to a population. Optimal size depends on variability and amount of error acceptable.

A Sample comes from a Population

Random Samples  To be random, all observations must have an equal chance of being included in the sample. The selection process must guarantee this. Random selection must occur at each stage of sampling.  Casual or haphazard is not the same as “random.”

Techniques for Random Selection  Fishbowl method – all observations represented on slips of paper drawn from a fishbowl. Depends on thoroughness of stirring.  Random number tables – enter the table at a random point then read in a consistent direction. Random digit dialing during polling.

Hypothetical Populations  Cannot be truly randomly sampled because all observations are not available for sampling.  Treated as real populations and sampled using random procedures.  Inferential statistics are applied to samples from hypothetical populations as if they were random samples.

Random Assignment  Random assignment ensures that, except for random differences, groups are similar.  When a variable cannot be controlled, random assignment distributes its effect across groups. Any remaining difference can be attributed to effect, not uncontrolled variables.

How to Assign Subjects  Flip a coin.  Choose even/odd numbers from a random number table.  Assign equal numbers of subjects to each group by pairs: When one subject goes to one group, the next goes to the other group.  Extend the same process to larger numbers of groups.

Probability  The proportion or fraction of times a particular outcome is likely to occur.  Probabilities permit speculation based on observations. Relative frequency of heights also suggests the likelihood of a particular height occurring.  Probabilities of simple outcomes are combined to find complex outcomes

Addition Rule  Used to predict combinations of events.  Mutually exclusive events are events that cannot happen together.  Add the separate probabilities to find out the probability of any one of the outcomes occurring.  Pr(A or B) = Pr(A) + Pr(B)

Addition Rule (Cont.)  When events can occur together, addition must be adjusted for the overlap between outcomes.  Add the probabilities then subtract the amount that is shared (counted twice): Drunk drivers =.40 Drivers on drugs =.20 Both =.12

Multiplication Rule  Used to calculate joint probabilities – events that both occur at the same time. Birthday coincidence  Pr(A and B) = [Pr(A)][Pr(B)]  The events combined must be independent of each other. One event does not influence the other.

Dependent Outcomes  Dependent – when one outcome influences the likelihood of the other outcome.  The probability of the dependent outcome is adjusted to reflect its dependency on the first outcome. The resulting probability is called a conditional probability. Drunk drivers & drug takers example.

Probability and Statistics  Probability tells us whether an outcome is common (likely) or rare (unlikely).  The proportions of cases under the normal curve (p) can be thought of as probabilities of occurrence for each value.  Values in the tails of the curve are very rare (uncommon or unlikely).