Variance Formula

N

X

2

2

1

2

2

n

XXs

Probability

A. The importance of probability

Hypothesis testing and statistical significanceProbabilistic causation - because error always exists

in our sampling (sampling error) we can only deal with probabilities of being correct or incorrect in our conclusions

The Normal Curve

A. What is a Probability Distribution

-How does it differ from a freq. distribution?-Theory vs. Empirical Observations-Examples?

B. The normal curve represents a probability distribution

-Why does it make sense to use the idea of a normal curve?-The area under the curve-Theory v. Reality

The Normal Curve

• The mean and standard deviation, in conjunction with the normal curve allow for more sophisticated description of the data and (as we see later) statistical analysis

• For example, a school is not that interested in the raw GRE score, it is interested in how you score relative to others.

• Even if the school knows the average (mean) GRE score, your raw score still doesn’t tell them much, since in a perfectly normal distribution, 50% of people will score higher than the mean.

• This is where the standard deviation is so helpful. It helps interpret raw scores and understand the likelihood of a score.

• So if I told you if I scored 710 on the quantitative section and the mean score is 591. Is that good?

• It’s above average, but who cares.

• What if I tell you the standard deviation is 148?

• What does that mean?

• What if I said the standard deviation is 5?

• Calculating z-scores

z-scores & conversions

• What is a z-score?– A measure of an observation’s distance from

the mean.– The distance is measured in standard

deviation units.• If a z-score is zero, it’s on the mean.• If a z-score is positive, it’s above the mean.• If a z-score is negative, it’s below the mean.• If a z-score is 1, it’s 1 SD above the mean.• If a z-score is –2, it’s 2 SDs below the mean.

Converting raw scores to z scores

What is a z score? What does it represent

Z = (x-µ) / σ

Z = (710-563)/140 = 147/140 = 1.05

Converting z scores into raw scores

X = z σ + µ [(1.05*140)+563=710]

Examples of computing z-scores

5 3 2 2 1

6 3 3 2 1.5

5 10 -5 4 -1.25

6 3 3 4 .75

4 8 -4 2 -2

X X XX SD SD

XXz

11

The Normal Curve

• A mathematical model or and an idealized conception of the form a distribution might have taken under certain circumstances.

– A sample of means from any distribution has a normal distribution (Central Limit Theorem)

– Many observations (height of adults, weight of children in Nevada, intelligence) have Normal distributions

Stuff you don’t need to know:

pi = ≈3.14159265

e = ≈2.71

Finding Probabilities under the Normal Curve

So what % of GRE takers scored above and below 710? (Z = 1.05)

The importance of Table A in Appendix C

- Why is this important?Infer the likelihood of a result

Confidence Intervals/Margin of Error

Inferential Statistics (to be cont.- ch.6-7)

THE NORMAL CURVE AND PROBABILITY

MIDTERM (OCT. 6th)What’s on it?

Babbie Chapters 1,2,4,5,6Fox/Levin Chapters 1,2,3,4,5Lecture Notes Through Oct. 4th

Statistical Calculations?Calculate the mean, median, modeCalculate the Variance & Standard Deviation

Extra credit: calculate a z-score/using Table A

Computer/SPSS questions? NO

Announcement

• Bring Levin and Fox book for exam

• Bring a calculator that can take the square root (lab computers should have function)

• No office hours on Tuesday, October 4th

• Extended office hours day of midterm– 8:30 to 11:30am Thursday, October 6th