Six topics in Statistics

Topic 1: Frequency Distributions Putting scores in order adds meaning Bar graphs (histograms) are visual representations of frequency distributions.

Topic 2: What is the center of the distribution? Measures of Central Tendency Quiz Scores 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=$2.50 Mean=$74.14 Without Ryan: Median=$2.38 Mean=$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 Punt Distance Mean = 160/4 = 40 yds Deviation from Mean Deviation Squared /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 CorrelationNegative Correlation

No 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