1. School of Information - The University of Texas at Austin LIS 397.1, Introduction to Research in Library and Information Science LIS 397.1 Introduction to Research in Library and Information Science The Gaussian Distribution and Some of Its Uses R. E. Wyllys Copyright 2003 by R. E. Wyllys Last revised 2003 Jan 14

2. School of Information - The University of Texas at Austin LIS 397.1, Introduction to Research in Library and Information Science Probability Distributions A probability distribution is the pattern of values, together with the probabilities of those values, taken by a variable Examples –Uniform distribution: tosses of a coin or die –Exponential distribution –Binomial distribution: numbers of heads in 2, 3, 4,..., etc. tosses of a coin –Gaussian (normal) distribution

3. School of Information - The University of Texas at Austin LIS 397.1, Introduction to Research in Library and Information Science Uniform Distribution

4. School of Information - The University of Texas at Austin LIS 397.1, Introduction to Research in Library and Information Science Exponential (Waiting-Time) Distribution

5. School of Information - The University of Texas at Austin LIS 397.1, Introduction to Research in Library and Information Science Binomial Distribution What happens if we toss a coin three times? There are eight possible sequences TTT, TTH, THT, THH, HTT, HTH, HHT, HHH Possible numbers of heads TTT 0 TTH, THT, HTT 1 THH, HTH, HHT 2 HHH 3 Probabilities P(0 H) = 1/8 P(2 Hs) = 3/8 P(1 H) = 3/8P(3 Hs) = 1/8 The above four pairs (each possible number of heads, together with the probability that that number will occur in a set of three tosses) constitute a probability distribution.

6. School of Information - The University of Texas at Austin LIS 397.1, Introduction to Research in Library and Information Science Binomial Distribution

7. School of Information - The University of Texas at Austin LIS 397.1, Introduction to Research in Library and Information Science Binomial Distribution

8. School of Information - The University of Texas at Austin LIS 397.1, Introduction to Research in Library and Information Science Binomial Distribution

9. School of Information - The University of Texas at Austin LIS 397.1, Introduction to Research in Library and Information Science Binomial Distribution

10. School of Information - The University of Texas at Austin LIS 397.1, Introduction to Research in Library and Information Science Binomial Distribution

11. School of Information - The University of Texas at Austin LIS 397.1, Introduction to Research in Library and Information Science Binomial Distribution

12. School of Information - The University of Texas at Austin LIS 397.1, Introduction to Research in Library and Information Science Binomial Distribution

13. School of Information - The University of Texas at Austin LIS 397.1, Introduction to Research in Library and Information Science Binomial Distribution

14. School of Information - The University of Texas at Austin LIS 397.1, Introduction to Research in Library and Information Science Gaussian (Normal) Distribution

15. School of Information - The University of Texas at Austin LIS 397.1, Introduction to Research in Library and Information Science Estimation Using the Gaussian Distribution Estimates of population parameters –Are based on observed sample values –Come in two main types Types of estimates –Point estimate: single number –Confidence Interval estimate: statements of an interval within which the population parameter is believed to lie the probability of the estimate’s lying within that interval

16. School of Information - The University of Texas at Austin LIS 397.1, Introduction to Research in Library and Information Science Estimation Using the Gaussian Distribution Confidence intervals are based on the behavior of means of samples Central Limit Theorem –Describes behavior of sample means –Asserts that The larger a sample is, the more likely is its mean to be close to the population mean If you take a large number of samples of a given size from a given population, the sample means will cluster around the population mean in a Gaussian (bell-shaped curve) fashion.

17. School of Information - The University of Texas at Austin LIS 397.1, Introduction to Research in Library and Information Science Estimation Using the Gaussian Distribution In a real-life problem, you will have one sample (of a given size) from the population of interest. You must view your one sample as just one (the one you happened to get) out of the entire set of possible samples of the same size. For most real-life problems, there will be a myriad of possible samples of a given size.

18. School of Information - The University of Texas at Austin LIS 397.1, Introduction to Research in Library and Information Science One Sample out of a Myriad Is like One Bee out of a Swarm

19. School of Information - The University of Texas at Austin LIS 397.1, Introduction to Research in Library and Information Science Confidence Intervals for the Population Mean Central Limit Theorem: The means of all possible samples of a given size from a given population cluster around the population mean in a Gaussian fashion. For example, –95% of the means will lie within  1.96 standard deviations (SDs) of the set of means of all possible samples on either side of the population mean  The SD of the set of means of all possible samples is called the standard error of the mean

20. School of Information - The University of Texas at Austin LIS 397.1, Introduction to Research in Library and Information Science Confidence Intervals for the Population Mean Standard error of the mean (standard deviation of the set of means of all possible samples of size n) is theoretically In most cases in practice you use

21. School of Information - The University of Texas at Austin LIS 397.1, Introduction to Research in Library and Information Science Confidence Intervals for the Population Mean Confidence intervals are constructed using Confidence factors come from Gaussian or Student’s t distributions –Large samples (31+) may use Gaussian values –Small samples (30-) must use Student’s t values

22. School of Information - The University of Texas at Austin LIS 397.1, Introduction to Research in Library and Information Science Confidence Intervals for the Population Mean An Example

23. School of Information - The University of Texas at Austin LIS 397.1, Introduction to Research in Library and Information Science Somewhere between the boundaries of a confidence interval lies the population mean... probably!