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INF 397C Introduction to Research in Library and Information Science Fall, 2003 Day 2. Standard Deviation. σ = SQRT( Σ (X - µ) 2 /N) (Does that give you a headache?). USA Today has come out with a new survey - apparently, three out of every four people make up 75% of the population.
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INF 397CIntroduction to Research in Library and Information ScienceFall, 2003Day 2
Standard Deviation σ = SQRT(Σ(X - µ)2/N) (Does that give you a headache?)
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 - )
Statistics: The only science that enables different experts using the same figures to draw different conclusions. • Evan Esar (1899 - 1995), US humorist
Scales • The data we collect can be represented on one of FOUR types of scales: • Nominal • Ordinal • Interval • Ratio • “Scale” in the sense that an individual score is placed at some point along a continuum.
Nominal Scale • Describe something by giving it a name. (Name – Nominal. Get it?) • Mutually exclusive categories. • For example: • Gender: 1 = Female, 2 = Male • Marital status: 1 = single, 2 = married, 3 = divorced, 4 = widowed • Make of car: 1 = Ford, 2 = Chevy . . . • The numbers are just names.
Ordinal Scale • An ordered set of objects. • But no implication about the relative SIZE of the steps. • Example: • The 50 states in order of population: • 1 = California • 2 = Texas • 3 = New York • . . . 50 = Wyoming
Interval Scale • Ordered, like an ordinal scale. • Plus there are equal intervals between each pair of scores. • With Interval data, we can calculate means (averages). • However, the zero point is arbitrary. • Examples: • Temperature in Fahrenheit or Centigrade. • IQ scores
Ratio Scale • Interval scale, plus an absolute zero. • Sample: • Distance, weight, height, time (but not years – e.g., the year 2002 isn’t “twice” 1001).
Scales (cont’d.) It’s possible to measure the same attribute on different scales. Say, for instance, your midterm test. I could: • Give you a “1” if you don’t finish, and a “2” if you finish. • “1” for highest grade in class, “2” for second highest grade, . . . . • “1” for first quarter of the class, “2” for second quarter of the class,” . . . • Raw test score (100, 99, . . . .). • (NOTE: A score of 100 doesn’t mean the person “knows” twice as much as a person who scores 50, he/she just gets twice the score.)
Critical Skepticism • Remember the Rabbit Pie example from yesterday. • The “critical consumer” of statistics asked “what do you mean by ’50/50’”? • Let’s look at some other situations and claims.
Company is hurting. • We’d like to ask you to take a 50% cut in pay. • But if you do, we’ll give you a 60% raise next month. OK? • Problem: Base rate.
Sale! • “Save 100%” • I doubt it.
Probabilities • “It’s safer to drive in the fog than in the sunshine.” (Kinda like “Most accidents occur within 25 miles of home.” Doesn’t mean it gets safer once you get to San Marcos.) • Navy literature around WWI: • “The death rate in the Navy during the Spanish-American war was 9/1000. For civilians in NYC during the same period it was 16/1000. So . . . Join the Navy. It’s safer.”
Are all results reported? • “In an independent study [ooh, magic words], people who used Doakes toothpaste had 23% fewer cavities.” • How many studies showed MORE cavities for Doakes users?
Sampling problems • “Average salary of 1999 UT grads – “$41,000.” • How did they find this? I’ll bet it was average salary of THOSE WHO RESPONDED to a survey. • Who’s inclined to respond?
Correlation ≠ Causation • Around the turn of the century, there were relatively MANY deaths of tuberculosis in Arizona. • What’s up with that?
Remember . . . • I do NOT want you to become cynical. • Not all “media bias” is intentional. • Just be sensible, critical, skeptical. • As you “consume” statistics, ask some questions . . .
??? • Who says so?(A Zest commercial is unlikely to tell you that Irish Spring is best.) • How does he/she know?(That Zest is “the best soap for you.”) • What’s missing?(One year, 33% of female grad students at Johns Hopkins married faculty.) • Did somebody change the subject?(“Camrys are bigger than Accords.” “Accords are bigger than Camrys.”) • Does it make sense?(“Study in NYC: Working woman with family needed $40.13/week for adequate support.”)
Quote on front of Huff book: • “It ain’t so much the things we don’t know that get us in trouble. It’s the things we know that ain’t so.” Artemus Ward, US author • Being a critical consumer of statistics will keep you from knowing things that ain’t so.
Claims • “Better chance of being struck by lightening than being bitten by a shark.” • Tom Brokaw – Tranquilizers. • What are some claims you all heard/read?
Before the break . . . • 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.
Measures of Central Tendency • Mode • Most frequent score (or scores – a distribution can have multiple modes) • Median • “Middle score” • 50th percentile • Mean - µ (“mu”) • “Arithmetic average” • ΣX/N
Let’s calculate some “averages” • From old data.
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
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 one 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
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).
Have sidled up to SHAPES of distributions • Symmetrical • Skewed – positive and negative • Flat
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).
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.
Measures of Dispersion • Range • Semi-interquartile range • Standard deviation • σ (sigma)
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”
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?
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)
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!
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”?
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
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.
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.
Who wants to guess . . . • . . . What I think is the most important sentence in S, Z, & Z (2003), Chapter 2?
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.”
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.
Homework • LOTS or reading. See syllabus. • Send a table/graph/chart that you’ve read this past week. Send email by noon, Monday, 9/8/2003. See you next week.
