1. R. G. Bias | School of Information | SZB 562BB | Phone: 512 471 7046 | rbias@ischool.utexas.edu i 1 LIS 397.1 Introduction to Research in Library and Information Science Summer, 2003 Day 3

2. R. G. Bias | School of Information | SZB 562BB | Phone: 512 471 7046 | rbias@ischool.utexas.edu i 2 Standard Deviation σ = SQRT(Σ(X - µ) 2 /N) (Does that give you a headache?)

3. R. G. Bias | School of Information | SZB 562BB | Phone: 512 471 7046 | rbias@ischool.utexas.edu i 3 USA Today has come out with a new survey - apparently, three out of every four people make up 75% of the population. –David Letterman (1947 - )

4. R. G. Bias | School of Information | SZB 562BB | Phone: 512 471 7046 | rbias@ischool.utexas.edu i 4 Statistics: The only science that enables different experts using the same figures to draw different conclusions. –Evan Esar (1899 - 1995), US humorist

5. R. G. Bias | School of Information | SZB 562BB | Phone: 512 471 7046 | rbias@ischool.utexas.edu i 5 How many of you...... told your roommate/friend/significant other some interesting thing about statistics this weekend?... found yourself thinking about statistics, when you NEVER would have guessed you ever would?

6. R. G. Bias | School of Information | SZB 562BB | Phone: 512 471 7046 | rbias@ischool.utexas.edu i 6 Last class... We learned about frequency distributions. I asserted that a frequency distribution, and/or a histogram (a graphical representation of a frequency distribution), was a good way to summarize a collection of data. There’s another, even shorter-hand way.

7. R. G. Bias | School of Information | SZB 562BB | Phone: 512 471 7046 | rbias@ischool.utexas.edu i 7 Measures of Central Tendency Mode –Most frequent score (or scores – a distribution can have multiple modes) Median –“Middle score” –50 th percentile Mean - µ (“mu”) –“Arithmetic average” –ΣX/N

8. R. G. Bias | School of Information | SZB 562BB | Phone: 512 471 7046 | rbias@ischool.utexas.edu i 8 Let’s calculate some “averages” From old data.

9. R. G. Bias | School of Information | SZB 562BB | Phone: 512 471 7046 | rbias@ischool.utexas.edu i 9 A quiz about averages 1 – If one score in a distribution changes, will the mode change? __Yes __No __Maybe 2 – How about the median? __Yes __No __Maybe 3 – How about the mean? __Yes __No __Maybe 4 – True or false: In a normal distribution (bell curve), the mode, median, and mean are all the same? __True __False

10. R. G. Bias | School of Information | SZB 562BB | Phone: 512 471 7046 | rbias@ischool.utexas.edu i 10 More quiz 5 – (This one is tricky.) If the mode=mean=median, then the distribution is necessarily a bell curve? __True __False 6 – I have a distribution of 10 scores. There was an error, and really the highest score is 5 points HIGHER than previously thought. a) What does this do to the mode? __ Increases it __Decreases it __Nothing __Can’t tell b) What does this do to the median? __ Increases it __Decreases it __Nothing __Can’t tell c) What does this do to the mean? __ Increases it __Decreases it __Nothing __Can’t tell 7 – Which of the following must be an actual score from the distribution? a) Mean b) Median c) Mode d) None of the above

11. R. G. Bias | School of Information | SZB 562BB | Phone: 512 471 7046 | rbias@ischool.utexas.edu i 11 OK, so which do we use? Means allow further arithmetic/statistical manipulation. But... It depends on: –The type of scale of your data Can’t use means with nominal or ordinal scale data With nominal data, must use mode –The distribution of your data Tend to use medians with distributions bounded at one end but not the other (e.g., salary). (Look at our “Number of MLB games” distribution.) –The question you want to answer “Most popular score” vs. “middle score” vs. “middle of the see-saw” “Statistics can tell us which measures are technically correct. It cannot tell us which are ‘meaningful’” (Tal, 2001, p. 52).

12. R. G. Bias | School of Information | SZB 562BB | Phone: 512 471 7046 | rbias@ischool.utexas.edu i 12 Have sidled up to SHAPES of distributions Symmetrical Skewed – positive and negative Flat

13. R. G. Bias | School of Information | SZB 562BB | Phone: 512 471 7046 | rbias@ischool.utexas.edu i 13 Why...... isn’t a “measure of central tendency” all we need to characterize a distribution of scores/numbers/data/stuff? “The price for using measures of central tendency is loss of information” (Tal, 2001, p. 49).

14. R. G. Bias | School of Information | SZB 562BB | Phone: 512 471 7046 | rbias@ischool.utexas.edu i 14 Note... We started with a bunch of specific scores. We put them in order. We drew their distribution. Now we can report their central tendency. So, we’ve moved AWAY from specifics, to a summary. But with Central Tendency, alone, we’ve ignored the specifics altogether. –Note MANY distributions could have a particular central tendency! If we went back to ALL the specifics, we’d be back at square one.

15. R. G. Bias | School of Information | SZB 562BB | Phone: 512 471 7046 | rbias@ischool.utexas.edu i 15 Measures of Dispersion Range Semi-interquartile range Standard deviation –σ (sigma)

16. R. G. Bias | School of Information | SZB 562BB | Phone: 512 471 7046 | rbias@ischool.utexas.edu i 16 Range Like the mode... –Easy to calculate –Potentially misleading –Doesn’t take EVERY score into account. What we need to do is calculate one number that will capture HOW spread out our numbers are from that Central Tendency. –“Standard Deviation”

17. R. G. Bias | School of Information | SZB 562BB | Phone: 512 471 7046 | rbias@ischool.utexas.edu i 17 New distribution Number of semesters all 2003 UT ISchool graduates spent in graduate school. (I made this up.) Measures of central tendency. Go with mean. So, how much do the actual scores deviate from the mean?

18. R. G. Bias | School of Information | SZB 562BB | Phone: 512 471 7046 | rbias@ischool.utexas.edu i 18 So... Add up all the deviations and we should have a feel for how disperse, how spread, how deviant, our distribution is. Σ(X - µ)

19. R. G. Bias | School of Information | SZB 562BB | Phone: 512 471 7046 | rbias@ischool.utexas.edu i 19 Damn! OK, so mathematicians at this point do one of two things. Take the absolute value or square ‘em. We square ‘em. Σ(X - µ) 2 Then take the average of the squared deviations. Σ(X - µ) 2 /N But this number is so BIG!

20. R. G. Bias | School of Information | SZB 562BB | Phone: 512 471 7046 | rbias@ischool.utexas.edu i 20 OK...... take the square root (to make up for squaring the deviations earlier). σ = SQRT(Σ(X - µ) 2 /N) Now this doesn’t give you a headache, right? I said “right”?

21. R. G. Bias | School of Information | SZB 562BB | Phone: 512 471 7046 | rbias@ischool.utexas.edu i 21 Hmmm... ModeRange Median????? MeanStandard Deviation

22. R. G. Bias | School of Information | SZB 562BB | Phone: 512 471 7046 | rbias@ischool.utexas.edu i 22 We need... A measure of spread that is NOT sensitive to every little score, just as median is not. SIQR: Semi-interquartile range. (Q3 – Q1)/2

23. R. G. Bias | School of Information | SZB 562BB | Phone: 512 471 7046 | rbias@ischool.utexas.edu i 23 To summarize ModeRange-Easy to calculate. -Maybe be misleading. MedianSIQR-Capture the center. -Not influenced by extreme scores. Mean (µ) SD (σ) -Take every score into account. -Allow later manipulations.

24. R. G. Bias | School of Information | SZB 562BB | Phone: 512 471 7046 | rbias@ischool.utexas.edu i 24 Graphs Graphs/tables/charts do a good job (done well) of depicting all the data. But they cannot be manipulated mathematically. Plus it can be ROUGH when you have LOTS of data. Let’s look at your examples.

25. R. G. Bias | School of Information | SZB 562BB | Phone: 512 471 7046 | rbias@ischool.utexas.edu i 25 Some rules...... For building graphs/tables/charts: –Label axes. –Divide up the axes evenly. –Indicate when there’s a break in the rhythm! –Keep the “aspect ratio” reasonable. –Histogram, bar chart, line graph, pie chart, stacked bar chart, which when? –Keep the user in mind.

26. R. G. Bias | School of Information | SZB 562BB | Phone: 512 471 7046 | rbias@ischool.utexas.edu i 26 Who wants to guess...... What I think is the most important sentence in S, Z, & Z (2003), Chapter 2?

27. R. G. Bias | School of Information | SZB 562BB | Phone: 512 471 7046 | rbias@ischool.utexas.edu i 27 p. 19 Penultimate paragraph, first sentence: “If differences in the dependent variable are to be interpreted unambiguously as a result of the different independent variable conditions, proper control techniques must be used.”

28. R. G. Bias | School of Information | SZB 562BB | Phone: 512 471 7046 | rbias@ischool.utexas.edu i 28 http://highered.mcgraw- hill.com/sites/0072494468/student_view0 /statistics_primer.htmlhttp://highered.mcgraw- hill.com/sites/0072494468/student_view0 /statistics_primer.html Click on Statistics Primer.

29. R. G. Bias | School of Information | SZB 562BB | Phone: 512 471 7046 | rbias@ischool.utexas.edu i 29 Homework - Keep reading. See you tomorrow.