Six topics in Statistics

Topic 1: Frequency DistributionsPutting scores in order adds meaning

A 40 4 45%39 738 1037 836 15

B 35 8 32%34 833 832 7

C 31 4 16%30 529 728

D 27 5%26 225 124 2

F <24 1 1%

Bar graphs (histograms) are visual representations of frequency distributions.

Topic 2: What is the center of the distribution?

Measures of Central Tendency

Quiz Scores 4 3 5 4 4

Mode --Most common = 4 Mean --Arithmetic avg = 20/5 = 4 Median --Middle score = 4

Central Tendency: Mean vs. Median 1968 TOPPS Baseball Cards

Nolan Ryan $1500

Billy Williams $8

Luis Aparicio $5

Harmon Killebrew $5

Orlando Cepeda $3.50

Maury Wills $3.50

Jim Bunning $3

Tony Conigliaro $3

Tony Oliva $3

Lou Pinella $3

Mickey Lolich $2.50

Elston Howard $2.25

Jim Bouton $2

Rocky Colavito $2

Boog Powell $2

Luis Tiant $2

Tim McCarver $1.75

Tug McGraw $1.75

Joe Torre $1.5

Rusty Staub $1.25

Curt Flood $1

With Ryan:Median=$2.50Mean=$74.14

Without Ryan:Median=$2.38Mean=$2.85

The median is a better measure of central tendency than the mean when there are extreme scores.

Measures of Central Tendency in Dunder Mifflin Salaries

Watch out for extreme scores or outliers. Let’s look at the salaries of the employees of the

Dunder Mifflin Paper Company in Scranton:$25,000-Pam $25,000- Kevin$25,000- Angela$100,000- Andy$100,000- Dwight$200,000- Jim$300,000- Michael The median salary looks good at $100,000.

The mean salary also looks good at about $110,000. But the mode salary is only $25,000. Maybe not the best place to work. Then again, living in Scranton is kind of cheap.

Topic 3: How spread out are the data?Measures of variation

Range• The spread between

the highest number & the lowest number.

• Only considers two numbers

Standard deviation

Calculation Example for Standard Deviation

PuntDistance

36384145

Mean =160/4 =40 yds

Deviationfrom Mean

-4-2+1+5

DeviationSquared

16 4 125

46

46/4 = 11.5 =variance

std. dev. =

Variance =

11.5 = 3.4 yds

Topic 4: Properties of the Normal Curve

In a large, randomly distributed data set• 68% of scores will be within 1 SD of the mean.• 95% of scores will be within 2 SDs of the mean.• 99.7% of scores will be withing 3 SDs of the mean.

Topic 4: Properties of the Normal Curve

Marilyn vos Savant: claimed IQ of 228.Is it more meaningful to express her IQ as points

above average or as standard deviations above average?

Topic 5: Correlation

A measure of the strength of the relationship between two variables.

Can be positive or negative.

Useful for making predictions.

You can calculate correlations with Excel or Google Docs.

Topic 5: Correlation

What does a correlation looks like?

Scatterplots

Positive Correlation Negative Correlation

No Correlation

Topic 5: Correlation

Topic 5: Correlation

How do you express a correlation numerically?

The Correlation Coefficient

Topic 5: Correlation

A strong correlation is not enough to establish a cause and effect relationship.

Example: There is a correlation between TV watching and grades.

Do you think it’s positive, or negative?

From this, what do we know about cause-and-effect.

Topic 5: Correlation

Even correlations that are clearly not cause-and-effect relationships can be used for prediction.

Ex: College entrance exams and freshman GPA.

Ex: Shoe size and vocabulary size in elementary school children.

Ex: Ice cream sales and the rate of violent crimes.

Topic 6: Statistical Significance

Several statistics (e.g., chi square, t-test) can be used to calculate statistical significance, but our students don’t need to know these

They do need to know how to interpret the results of these tests—the p value.

Topic 6: Statistical Significance

P value is an estimate of the probability that a result was caused by chance.

In an experiment, it’s the likelihood that the difference between the experimental and control conditions as measured by the DV was caused by chance.

We want this difference to be caused by our manipulation—the IV—not by chance.

Topic 6: Statistical Significance

• To say that the results of an experiment are statistically significant means that there is a small likelihood that the results were caused by chance; that is, a high likelihood they were caused by the IV.

• The threshold for statistical significance is no more than a 5% likelihood the results were caused by chance.

• We express this: p ≤ .05