PSY 1950 Descriptive Statistics September 17, 2008.
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PSY 1950Descriptive StatisticsSeptember 17, 2008
• “It is well known to your Lordship, that the method practised by astronomers, in order to diminish the errors arising from the imperfections of instruments, and of the organs of sense, by taking the Mean of several observations, has not been so generally received, but that some persons, of considerable note, have been of opinion, and even publicly maintained, that one single observation, taken with due care, was as much to be relied on as the Mean of a great number.– Thomas Simpson to Earl of Macclesfield “On
the Advantage of Taking the Mean of a Number of Observations, in practical Astronomy” (1755)
• “The science of Means may be summed up in two problems; (1) To find how far the difference between any proposed Means (e.g. the average mortalities in different occupations) is accidental, or indicative of a law; (2) To find what is the best Mean, whether for the purpose contemplated by the first problem, The Elimination of Chance, or other purposes.”– F.Y. Edgeworth’s (1885) “Logic of
Statistics”
Population versus Samples
– Everyone born in Scotland in 1932 (n = 87, 498)
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Types of Variables• Discrete versus Continuous• Physical versus social sciences • Make an argument
– Group 1: Perception is discrete– Group 2: Perception is continuous– Group 3: Depression is discrete– Group 4: Depression is continuous
• What data/knowledge would support or even prove your claim?
Types of measurementVariable = academic interest in science
central tendency = modedispersion = bupkis (sort of)
NominalIndividual Interest
Paul ScienceDave Science
Larry Humanities
Adam Humanities
0
1
2
3
4
5
6
Science Humanities
Interest
Activism
Types of measurementVariable = academic interest in science
central tendency = mode, mediandispersion = range, interquartile range
OrdinalIndividual Interest
Paul mediumDave highLarry lowAdam low
01234567
Low Medium High
Interest
Activism
Types of measurementVariable = academic interest in science
central tendency = mode, median, meandispersion = range, interquartile range, std
deviation
IntervalIndividual Interest
Paul 4Dave 8Larry 2Adam 3
No interest Extreme interest
x
Types of measurementVariable = academic interest in science
central tendency = mode, median, meandispersion = range, interquartile range, std
deviation
RatioIndividual Interest
Paul 3Dave 7Larry 1Adam 2
No interest Extreme interest
x
• “The numbers do not remember where they came from.”– Lord, F.M. (1953). On the
statistical treatment of football numbers. The American Psychologist, 8, 750-751.
Correlational and Experimental Methods
• Complementary methods • Ecological validity versus inferential
power• Hypothesis generation versus
hypothesis testing• Others?
Correlational vs. Experimental Methods
• Cronbach, L. (1957). The two disciplines of scientific psychology, American Psychologist, 12, 671-684. (http://psychclassics.yorku.ca/Cronbach/Disciplines/)
– Cattell (1898): experimentalists’ "regard for the body of nature becomes that of the anatomist rather than that of the lover”
– Bartlett’s (1955): correlationists “chanting in unaccustomed harmony the words of the old jingle ‘God has a plan for every man, and he has a plan for you’”
• “We will come to realize that organism and treatment are an inseparable pair and that no psychologist can dismiss one or the other as error variance.”
Statistics and Methodology are Inseparable
• The use and even conception of descriptive statistics varies with experimental/correlational approach– Shape– Central tendency– Dispersion
Shape• Modality• Symmetry• Kurtosis
Shape: Modality
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Shape: Symmetry/Skewness
Shape: Kurtosis
Joanes, D. N., & Gill, C. A. (1998) Comparing measures of sample skewness and kurtosis. Journal of the Royal Statistical Society (Series’s D): The Statistician, 47, 183-189.
DeCarlo, L. T. (1997). On the meaning and use of kurtosis. Psychological Methods, 2, 292-306.
Histogram Bin Size
Shimazaki, H. and Shinomoto, S. (2007) A method for selecting the bin size of a time histogram. Neural Computation, 19(6), 1503-1527.
http://www.ton.scphys.kyoto-u.ac.jp/~hideaki/res/sshist/cur/bin/histogram_appli.html
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Central Tendency• Mean, Median, Mode• Selection criteria
– skew, extreme scores– undetermined values– open-ended distributions– discrete variables– (arithmetic manipulatibility)– (population estimator)
• Mean number of calories burned in an “extremely passionate” one-minute kiss:– 26
• Mean number of calories in a Hershey’s kiss:– 25
• # participants predicted to administer an apparently lethal shock to a stranger: 1%
• # participants who actually administered an apparently lethal shock to a stranger: 65%
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Estimating Population Central Tendency
• Efficiency
(10,000 samples of n=100 from population with µ=30, =5)
Estimating Population Central Tendency
• Resistance
(10,000 samples of n=99 from population with µ=30, =5; 1 outlier added with value of 100)
Dispersion/Variability/Spread
• Range: Xmax-Xmin
• Interquartile range: Q3-Q1• Deviation-based measures
Deviation• Deviation = X - µ
– Direction– Distance
• Mean deviation = (X - µ)/N– Always zero
• Mean absolute deviation (MAD)= (|X - µ|)/N– Advantage: intuitive– Disadvantage: mathmetically
cumbersome
Deviation• Mean squared deviation (variance)
=(X - µ)2/N = SS/N
– Advantage: mathematically practical– Disadvantage: Non-intuitive,
different units
• Standard deviation ()=√variance• Computation vs. definitional
formulae
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Estimating population dispersion• Bias
– See http://www.ruf.rice.edu/~lane/stat_sim/sampling_dist/index.html
Why n-1 for samples?• Population with 2 = 25• 10,000 samples of n = 25
Why n-1 for samples?• degrees of freedom
– the rank of a quadratic form – the number of independent observations in a sample
of data that are available to estimate a parameter of the population from which that sample is drawn
– the number of scores in the sample that are independent and free to vary
Walker, H.W. (1940). Degrees of freedom. Journal of Educational Psychology, 31, 253-269.
Good, I.J. (1973). What are degrees of freedom? The American Statistician, 27, 5, 227–228
http://www.tufts.edu/~gdallal/dof.htmhttp://www.creative-wisdom.com/computer/sas/df.html
Why not other dispersion measures?– MAD (population MAD = 3.99),
range
Choosing a DV: Central Tendency vs. Dispersion
Choosing a DV: Central Tendency vs. Dispersion
• What is the mechanism? – e.g., sex differences in IQ
• Are deviations nuisance or essence?– e.g., heart rate
• Equal and opposite effects?– e.g., subject X treatment interactions
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