Introduction to summary statistics: Sample mean & sample variance

1. Fred Boehm Statistics 224 January 27, 2014 Introduction to summary statistics:Sample mean & sample variance

2. 224 logistics • Website updates: • Revised office hours info • Alyssa & Huikun: 12-2pm today in MSC 1217c • Fred: 6:30-8pm today in Wendt library (room 129) • Electronic survey • Respond by 6pm tonight (Jan 27) • Completion time: ~ 3 minutes • Email me if you can't find the email with hyper-link • Homework 1 due Wednesday at 11am in class

3. Lecture overview • Key terms in statistics • Statistic & Parameter • Random variable • Measures of central tendency • Sample mean as a statistic related to central tendency • Measures of spread • Sample variance as a statistic related to spread of data • Coin flip examples

4. Statistic vs. Parameter • Statistic – observed values, or function of observed values • Coin Flip Example: • For ten coin flips, what is the number of heads? • Parameter – unknown, underlying value that impacts the observed outcomes • Coin Flip Example: • Is the coin fair? • In other words, is the probability of observing heads equal to 0.5?

5. Random variable • Technical definitions use notions from probability theory • For our purposes, we may think of a random variable as an outcome that has more than one possible value • Random variable example: a coin flip • Two possible outcomes (heads or tails)

6. What is a “sample mean” • A statistic (function of observed data) • Intuitively, the 'center' point of your observations • Mathematically, the “average” of your observed values • Written as X with a bar above it • Pronounced “X bar”

7. Batting Average in Baseball • Baseball batting average • What is the maximum possible value of AVG? • What is the minimum possible value of AVG?

8. Sample mean, continued • Coin flips example • Repeat coin flips and record outcomes

9. Coin Flips Activity • Each student flips the penny 5 times • Record the number of heads (between zero and five) • Show of hands for each value of number of heads • Plot the data (as histogram) in R

10. Coin Flips Activity, continued • Do you think that your coin is fair? • Why? • What might you do to better assess the fairness of your coin? • Turn to your neighbor to discuss these three questions

11. Sample variance • Tells you about the 'spread' of the data • Larger sample variance corresponds to data being more spread out • Mathematically, one definition is:

12. Sample variance & coin flips • You've already flipped your penny 5 times • You recorded the number of heads that you saw • Calculate, from your five flips: • Sample mean = Xbar = (number of heads)/(number of flips)

13. Sample variance & coin flips • Now, calculate the sample variance from your five flips • Compare your sample variance with those of your neighbors • Should you have the same sample variance as your neighbors? • Should you be surprised if you and your neighbor have the same sample variance? • Why?

14. Histograms of three random samples • Black: Variance=100; Sample variance=89 • Red: Variance=16; Sample variance=17 • Green: Variance=1; Sample variance=0.95

15. Sibling count histogram • How do we get the sample mean from a histogram? • What is (approximately) the sample mean here?

16. Sibling count histogram • How do we get the sample mean from a histogram? • What is (approximately) the sample mean here? • 1.8 • Data from Stockholm Birth Cohort Study. • http://www.stockholmbirthcohort.su.se/

17. Lecture overview • Key terms in statistics • Statistic & Parameter • Random variable • Measures of central tendency • Sample mean as a statistic related to central tendency • Measures of spread • Sample variance as a statistic related to spread of data • Coin flip examples • Guessing a sample mean from a histogram