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Name that tune.

 Song title?  Performer(s)?

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Descriptive Statistics

“Finding New Information”4/5/2010

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Standard Deviation

σ = SQRT(Σ(X - µ)2/N)

(Does that give you a headache?)

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USA Today has come out with a new survey - apparently, three out of every four people make up 75% of the population. – David Letterman

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Statistics: The only science that enables different experts using the same figures to draw different conclusions. – Evan Esar (1899 - 1995), US humorist

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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.

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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.

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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

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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

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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).

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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.)

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Scales (cont’d.)Nominal Ordinal Interval Ratio

Name = = =

Mutually-exclusive

= = =

Ordered = =

Equal interval =+ abs. 0

Gender, Yes/No

Class rank, Survey ans.

Days of wk., Temp.

Inches, Dollars

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Earlier . . . 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.

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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

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More quiz questions about measures of central

tendency4 – True or false: In a normal distribution (bell curve), the mode, median, and mean are all the same? __True __False

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 tellb) What does this do to the median?

__ Increases it __Decreases it __Nothing __Can’t tellc) 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) Meanb) Medianc) Moded) None of the above

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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). – 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).

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Mean – “see saw” (from Tal, 2001)

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Have sidled up to SHAPES of distributions

Symmetrical Skewed – positive and negative Flat

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“Pulling up the mean”

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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).

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Didja hear the one about . . .

the Aggies who were on a march and came to a river? The Aggie captain asked the farmer how deep the river was.”

“Oh, it averages two feet deep.” All the Aggies drowned.

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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.

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Measures of Dispersion Range Semi-interquartile range Standard deviation

– σ (sigma)

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Range 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” -- σ = SQRT(Σ(X -

µ)2/N)

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Let’s do a short example

What if I asked four undergraduates how many cars they’ve owned in their lives and I got the following answers: 1 1 1 1

There would be NO variance. σ = 0. But what if the answers were 0 0 1 3What’s the mode? Median? Mean? Go with mean. So, how much do the actual scores

deviate from the mean?

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So . . . 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)

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Standard Deviation

Score (X) Mean (µ) X-µ

0 1 -1

0 1 -1

1 1 0

3 1 2

Total 0 (damn)

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Damn! OK, let’s try it on

another set of numbers.

X2356

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Damn! (cont’d.) OK, let’s try it on

a smaller set of numbers.

X X - µ2 -23 -15 16 2

Σ = 16 Σ = 0µ = 4 Hmm.

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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

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X X - µ (X - µ)2

2 -2 43 -1 15 1 16 2 4

Σ = 16 Σ = 0 10µ = 4

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Standard Deviation (cont’d.)

Then take the average of the squared deviations. Σ(X - µ)2/N– Remember, dividing by N was the way

we took the average of the original scores.

– 10/4 = 2.5. But this number is so BIG!

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OK . . . . . . take the square root (to make up

for squaring the deviations earlier). σ = SQRT(Σ(X - µ)2/N) SQRT(2.5) = 1.58 Now this doesn’t give you a

headache, right? I said “right”?

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Hmmm . . .Mode Range

Median ?????

Mean Standard Deviation

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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

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To summarizeMode 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.

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Practice Problems I’ll send you some, tonight.

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http://highered.mcgraw-hill.com/sites/0072494468/student_view0/statistics_primer.html

Click on Statistics Primer.

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References Hinton, P. R. Statistics explained. Shaughnessy, Zechmeister, and

Zechmeister. Experimental methods in psychology.

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