R. G. Bias | School of Information | SZB 562BB | Phone: 512 471 7046 | rbias@ischool.utexas.edu 1

i

INF 397CIntroduction to Research in Information

Studies

Fall, 2005

Day 2

R. G. Bias | School of Information | SZB 562BB | Phone: 512 471 7046 | rbias@ischool.utexas.edu 2

iStandard Deviation

σ = SQRT(Σ(X - µ)2/N)

(Does that give you a headache?)

R. G. Bias | School of Information | SZB 562BB | Phone: 512 471 7046 | rbias@ischool.utexas.edu 3

i• USA Today has come out with a new

survey - apparently, three out of every four people make up 75% of the population. – David Letterman

R. G. Bias | School of Information | SZB 562BB | Phone: 512 471 7046 | rbias@ischool.utexas.edu 4

i• Statistics: The only science that enables

different experts using the same figures to draw different conclusions. – Evan Esar (1899 - 1995), US humorist

R. G. Bias | School of Information | SZB 562BB | Phone: 512 471 7046 | rbias@ischool.utexas.edu 5

iScales (last week)Nominal Ordinal Interval Ratio

Name = = =

Mutually-exclusive

= = =

Ordered = =

Equal interval

=

+ abs. 0Gender, Yes/No

Class rank, ratings

Days of wk., temp.

Inches, dollars

R. G. Bias | School of Information | SZB 562BB | Phone: 512 471 7046 | rbias@ischool.utexas.edu 6

iCritical Skepticism

• Remember the Rabbit Pie example from last week?

• The “critical consumer” of statistics asked “what do you mean by ’50/50’”?

R. G. Bias | School of Information | SZB 562BB | Phone: 512 471 7046 | rbias@ischool.utexas.edu 7

iRemember . . .

• I do NOT want you to become cynical.

• Not all “media bias” (nor bad research) is intentional.

• Just be sensible, critical, skeptical.

• As you “consume” statistics, ask some questions . . .

R. G. Bias | School of Information | SZB 562BB | Phone: 512 471 7046 | rbias@ischool.utexas.edu 8

iAsk yourself. . .

• 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.”)

R. G. Bias | School of Information | SZB 562BB | Phone: 512 471 7046 | rbias@ischool.utexas.edu 9

iWhat were . . .

• . . . some claims you all heard this week?

R. G. Bias | School of Information | SZB 562BB | Phone: 512 471 7046 | rbias@ischool.utexas.edu 10

iLast week . . .

• 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.

• And I asserted there’s another, even shorter-hand way.

R. G. Bias | School of Information | SZB 562BB | Phone: 512 471 7046 | rbias@ischool.utexas.edu 11

iMeasures 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

R. G. Bias | School of Information | SZB 562BB | Phone: 512 471 7046 | rbias@ischool.utexas.edu 12

iOK, 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).

R. G. Bias | School of Information | SZB 562BB | Phone: 512 471 7046 | rbias@ischool.utexas.edu 13

iScales (which measure of CT?)Nominal

(mode)

Ordinal

(mode, median)

Interval

(any)

Ratio

(any)

Name = = =

Mutually-exclusive

= = =

Ordered = =

Equal interval

=

+ abs. 0

Gender, Yes/No

Class rank, ratings

Days of wk., temp.

Inches, dollars

R. G. Bias | School of Information | SZB 562BB | Phone: 512 471 7046 | rbias@ischool.utexas.edu 14

iMean – “see saw” (from Tal, 2001)

R. G. Bias | School of Information | SZB 562BB | Phone: 512 471 7046 | rbias@ischool.utexas.edu 15

iHave sidled up to SHAPES of distributions

• Symmetrical

• Skewed – positive and negative

• Flat

R. G. Bias | School of Information | SZB 562BB | Phone: 512 471 7046 | rbias@ischool.utexas.edu 16

i“Pulling up the mean”

R. G. Bias | School of Information | SZB 562BB | Phone: 512 471 7046 | rbias@ischool.utexas.edu 17

iWhy . . .

• . . . 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).

R. G. Bias | School of Information | SZB 562BB | Phone: 512 471 7046 | rbias@ischool.utexas.edu 18

iNote . . .

• 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.

R. G. Bias | School of Information | SZB 562BB | Phone: 512 471 7046 | rbias@ischool.utexas.edu 19

iMeasures of Dispersion

• Range

• Semi-interquartile range

• Standard deviation– σ (sigma)

R. G. Bias | School of Information | SZB 562BB | Phone: 512 471 7046 | rbias@ischool.utexas.edu 20

iRange

• Highest score minus the lowest score.• 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 measure of Central Tendency.– ‘Cause MANY different distributions of scores can

have the same central tendency!– “Standard Deviation”

R. G. Bias | School of Information | SZB 562BB | Phone: 512 471 7046 | rbias@ischool.utexas.edu 21

iBack to our data – MLB games

• Let’s take just the men in this class

• xls spreadsheet.

• Measures of central tendency.

• Go with mean.

• So, how much do the actual scores deviate from the mean?

R. G. Bias | School of Information | SZB 562BB | Phone: 512 471 7046 | rbias@ischool.utexas.edu 22

iSo . . .

• Add up all the deviations and we should have a feel for how disperse, how spread, how deviant, our distribution is.

• Let’s calculate the Standard Deviation.

• As always, start inside the parentheses.

• Σ(X - µ)

R. G. Bias | School of Information | SZB 562BB | Phone: 512 471 7046 | rbias@ischool.utexas.edu 23

iDamn!

• OK, let’s try it on a smaller set of numbers.

X

2

3

5

6

R. G. Bias | School of Information | SZB 562BB | Phone: 512 471 7046 | rbias@ischool.utexas.edu 24

iDamn! (cont’d.)

• OK, let’s try it on a smaller set of numbers.

X X - µ

2 -2

3 -1

5 1

6 2

Σ = 16 Σ = 0

µ = 4 Hmm.

R. G. Bias | School of Information | SZB 562BB | Phone: 512 471 7046 | rbias@ischool.utexas.edu 25

iOK . . .

• . . . so mathematicians at this point do one of two things.

• Take the absolute value or square ‘em.

• We square ‘em. Σ(X - µ)2

R. G. Bias | School of Information | SZB 562BB | Phone: 512 471 7046 | rbias@ischool.utexas.edu 26

iStandard Deviation (cont’d.)

• Then take the average of the squared deviations. Σ(X - µ)2/N

• But this number is so BIG!

R. G. Bias | School of Information | SZB 562BB | Phone: 512 471 7046 | rbias@ischool.utexas.edu 27

iOK . . .

• . . . 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”?

R. G. Bias | School of Information | SZB 562BB | Phone: 512 471 7046 | rbias@ischool.utexas.edu 28

iHmmm . . .

Mode Range

Median ?????

Mean Standard Deviation

R. G. Bias | School of Information | SZB 562BB | Phone: 512 471 7046 | rbias@ischool.utexas.edu 29

iWe 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

R. G. Bias | School of Information | SZB 562BB | Phone: 512 471 7046 | rbias@ischool.utexas.edu 30

iTo summarize

Mode Range -Easy to calculate.-May be misleading.

Median SIQR -Capture the center.-Not influenced by extreme scores.

Mean

(µ)

SD

(σ)

-Take every score into account. -Allow later manipulations.

R. G. Bias | School of Information | SZB 562BB | Phone: 512 471 7046 | rbias@ischool.utexas.edu 31

iPractice Problems

R. G. Bias | School of Information | SZB 562BB | Phone: 512 471 7046 | rbias@ischool.utexas.edu 32

iWho wants to guess . . .

• . . . What I think is the most important sentence in S, Z, & Z (2003), Chapter 2?

R. G. Bias | School of Information | SZB 562BB | Phone: 512 471 7046 | rbias@ischool.utexas.edu 33

ip. 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.”

R. G. Bias | School of Information | SZB 562BB | Phone: 512 471 7046 | rbias@ischool.utexas.edu 34

i• http://highered.mcgraw-hill.com/sites/007

2494468/student_view0/statistics_primer.html

• Click on Statistics Primer.

R. G. Bias | School of Information | SZB 562BB | Phone: 512 471 7046 | rbias@ischool.utexas.edu 35

iHomework

• LOTS of reading. See syllabus.

• Send a table/graph/chart that you’ve read this past week. Send email by noon, Friday, 9/16/2005.

See you next week.