The Normal Curve, Sample vs Population, and Probability

PSY 307 – Statistics for the

Behavioral Sciences

Chapter 8 – The Normal Curve, Sample vs Population, and Probability

Demos

How normal distributions are generated: http://www.ms.uky.edu/~mai/java/stat/GaltonMachine.html

How changes in the mean and std deviation affect the shape of the normal distribution: http://onlinestatbook.com/chapter6/varieties_demo.html

Finding the proportion for a given z score: http://onlinestatbook.com/java/normal.html

Finding the z-score for a given portion of the distribution: http://onlinestatbook.com/java/normalshade.html

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 Means Same 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

http://davidmlane.com/hyperstat/z_table.html

http://www.sfu.ca/personal/archives/richards/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 http://www.cut-the-knot.org/do_you_know/coincidence.shtml

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).