Part IIITaking Chances for Fun and Profit

Chapter 8 Are Your Curves Normal? Probability and

Why it Counts

0900 Quiz #3 N=26

2|1389 3|01112333335669 4|00012334

X-bar=34.62; Median=13th and 14th dp=33 Mode=33;

S=6.03;

1030 Quiz #3 N=33

2|0355678899 3|033334668899 4|00111223455

X-bar=34.73; Median=33+1/2=17th dp=36; Mode=33; s= 7.02;

Frequency distribution: 900 quiz scores

Freq CF RF CRF

21 – 24 2 2 .077 .077

25 – 28 1 3 .038 .115

29 – 32 6 9 .231 .346

33 – 36 8 17 .308 .654

37 – 40 4 21 .154 .808

41 – 44 5 26 .182 1.00

What you will learn in Chapter 7

Understanding probability is basic to understanding statistics

Characteristics of the “normal” curve i.e. the bell-shaped curve

All about z scores Computing them Interpreting them

Why Probability?

Basis for the normal curve Provides basis for understanding probability of

a possible outcome Basis for determining the degree of

confidence that an outcome is “true” Example:

Are changes in student scores due to a particular intervention that took place or by chance along?

The Normal Curve (a.k.a. the Bell-Shaped Curve) Visual representation of a distribution of

scores Three characteristics…

Mean, median, and mode are equal to one another

Perfectly symmetrical about the mean Tails are asymptotic (get closer to horizontal

axis but never touch)

The Normal Curve

Hey, That’s Not Normal!

In general, many events occur right in the middle of a distribution with few on each end.

More Normal Curve 101

More Normal Curve 101 For all normal distributions…

almost 100% of scores will fit between -3 and +3 standard deviations from the mean.

So…distributions can be compared

Between different points on the X-axis, a certain percentage of cases will occur.

What’s Under the Curve?

The z Score

A standard score that is the result of dividing the amount that a raw score differs from the mean of the distribution by the standard deviation.

What about those symbols?

( ),

X Xz

s

The z Score

Scores below the mean are negative (left of the mean) and those above are positive (right of the mean)

A z score is the number of standard deviations from the mean

z scores across different distributions are comparable

What z Scores Represent

The areas of the curve that are covered by different z scores also represent the probability of a certain score occurring.

So try this one… In a distribution with a mean of 50 and a

standard deviation of 10, what is the probability that one score will be 70 or above?

Why Use Z scores?

• Percentages can be used to compare different scores, but don’t convey as much information

• Z scores also called standardized scores, making scores from different distributions comparable; Ex: You get two different scores in two different subjects(e.g Statistics 28 and English 76). They are not yet comparable, so lets turn them into percentages( e.g 28/35=80% and 76/100, 76%). Relatively you did better in statistics.

Percentages Verse Z scores

• How do you compare to others? From percentages alone, you have no way of knowing. Say µ on English exam was =70 with ó of 8 pts, your 76 gives you a z-score of .75, three-fourths of one stand deviation above the mean; Mean on statistics test is 21, with ó of 5 pts; your score of 28 gives a z score of 1.40 standard deviations above mean; Although English and statistics scores were similar, comparing z scores shows you did much better in statistics

Using z scores to find percentiles

• Prof Oh So Wise, scores 142 on an evaluation. What is Wise’s percentile ranking? Assume profs’ scores are normally distributed with µ of 100 and ó of 25.

X-µ 142-100 z= 1.68ó 25Area under curve ‘Small Part’ = .0465, equals those

who scored above the prof;1–.0465= 95.35th percentile. Oh so wise is in top 5% of

all professors. Not bad at all. Never use from ‘mean to z’ to find percentile!! We’re

only concerned with scores above or below a certain rank

Starting with An Area Under Curve and Finding Z and then X… Using the previous parameters of µ of 100

and ó of 25, what score would place a professor in top 10% of this distribution? After some algebra, we have X=µ+z (ó)

100(µ) + 1.28(z)(25)(ó)=132 (X). A score of 132 would place a professor in top 10 %;

What scores place a professor in most extreme 5% of all instructors?

What does ‘most extreme’ mean?

It is not just one end of the distribution, but both ends, or 2.5% at either end;

X= µ + z(ó)= 100+ 1.96(25)= 149 100 +-1.96(25)=51; 51 and 149 place a

professor at the most extreme 5 % of the distribution;

The Difference between z scores

What z Scores Really Represent

Knowing the probability that a z score will occur can help you determine how extreme a z score you can expect before determining that a factor other than chance produced the outcome

Keep in mind… z scores are typically reserved for populations.

Hypothesis Testing & z Scores

Any event can have a probability associated with it. Probability values help determine how

“unlikely” the even might be The key --- less than 5% chance of occurring

and you have a significant result

Some rules regarding normal distribution Percentiles – if raw score is below the mean

use’ small part’ to find percentile ; if raw score is above the mean, use’ big part’ to find percentile; check to see that you’re right by constructing a frequency distribution and identifying cumulative percentage

If raw scores are on opposite sides of the mean, add the areas/percentages. If raw scores are on same side of mean, subtract areas/percentages

Using the Computer

Calculating z Scores

Glossary Terms to Know

Probability Normal curve

Asymptotic Standard Scores

z scores