Quantitative Methods for Lawyers - Class #8 - Professor Daniel Martin Katz - MSU College of Law

Sampling (Cont.),Skewness, Kurtosis& Data Distributions Quantitative Methods for Lawyers Class #8 Daniel Martin KatzMichigan State University - College of Law

A Quick ReviewFrom Last Time

Survey DataSurvey Methodology is the field thatstudies the sampling of individuals froma population with a view towards makingstatistical inferences about thepopulation using the sample.Key Ideas in Survey Methods: Useful Applications of Survey Research Types of Surveys Representative Sampling Questionaire Design

Experimental Data Experimental Control Random AssignmentKey Ideas: Experimental Manipulation Factorial Design Double Blind

Random Assignment and RepresentativenessRandom Assignment Prob of Being in Treatment or Control Group Should Be Equal Under Ideal Conditions this Eliminates other Confounds that could undermine ValidityRepresentativeness Would like to overall subject group to mirror the Population of interest This is true even prior to the Random Assignment

Sampling

SamplingSampling is concerned with theselection of a subset of individualsfrom within a population to estimatecharacteristics of the wholepopulationWhy Sample? Cost of Getting Full Population Focus Upon Improving Precision v. Size Might Be Impossible to Get the Full Population

Sampling Stages(1) Defining the population of concern(2) Specifying a sampling frame, a set ofitems or events possible to measure(3) Specifying a sampling method forselecting items or events from theframe(4) Determining the sample size(5) Implementing the sampling plan(6) Sampling and data collecting

Sampling (Cont.)

Sampling Stages(1) Defining the population of concern(2) Specifying a sampling frame, a set ofitems or events possible to measure(3) Specifying a sampling method forselecting items or events from theframe(4) Determining the sample size(5) Implementing the sampling plan(6) Sampling and data collecting

Determining the Sample Size Conceptually We Understand that in order to obtain a representative sample we need to acquire somewhere between 1> ? > Full Population But Exactly How Many Observations Do we need?

Random Sampling ErrorWhen You Sample at Random It isPossible to Have a Skewed Set ofObservation in the SampleImagine a Political Pollthe population of interest are actual voters.pollsters take smaller samples that are intended to berepresentative, that is, a random sample of thepopulation.It is possible that pollsters sample 1,013 voters whohappen to vote for Bush when in fact the population isevenly split between Candidate 1 and Candidate 2, butthis is extremely unlikely (p = 2−1013 ≈ 1.1 × 10−305) giventhat the sample is random.

Random Sampling ErrorFor Simple Random Sample on alarge population, the Inverse of theSquare Root of the Sample Size

Random Sampling Error For Simple Random Sample on a large population, the Inverse of the Square Root of the Sample Size VeryTypicallyReported

Random Sampling ErrorFor Simple Random Sample on alarge population, the Inverse of theSquare Root of the Sample SizeExample:a random sample of size 400 will give a margin oferror, at a 95% confidence level, of 0.98/20 or0.049 - just under 5%.

Random Sampling ErrorFor Simple Random Sample on alarge population, the Inverse of theSquare Root of the Sample SizeExample:a random sample of size 400 will give a margin oferror, at a 95% confidence level, of 0.98/20 or0.049 - just under 5%.A random sample of size 1600 will give a margin oferror of 0.98/40, or 0.0245 - just under 2.5%.

Random Sampling ErrorFor Simple Random Sample on alarge population, the Inverse of theSquare Root of the Sample SizeExample:a random sample of size 400 will give a margin oferror, at a 95% confidence level, of 0.98/20 or0.049 - just under 5%.A random sample of size 1600 will give a margin oferror of 0.98/40, or 0.0245 - just under 2.5%. Notice: Double the Precision Requires 4times the Sample Size!

Top portion of this graphic depicts therelative likelihood that the "true" percentageis in a particular area given a reportedpercentage of 50%. The bottom portion shows 95% confidence intervals (horizontal line segments), the corresponding margins of error (on the left), and sample sizes (on the right). In other words, for each sample size, one is 95% confident that the "true" percentage is in the region indicated by the corresponding segment. The larger the sample is, the smaller the margin of error is.

Central Limit Theorem

Thinking of Data asa Distribution: HistogramHistogram - histogram is a graphicalrepresentation showing a visual impression of thedistribution of data(1) consists of tabular frequencies, shown asadjacent rectangles, erected over discreteintervals (bins)(2) The height of a rectangle is also equal to thefrequency density of the interval, i.e., thefrequency divided by the width of the interval(3) Total area of the histogram is equal to thenumber of data

Thinking of Data asa Distribution: HistogramHistogram of travel time, US 2000 census. Area underthe curve equals the total number of cases. Thisdiagram uses Q/width from the table.

Ordinary v. Cumulative Histogram

Thinking of Data as a DistributionTry to Start Thinking of Any Data Set as aDistributionThis allows you take a broader perspectiveabout the observations contained thereinWhen you get a new dataset you shouldgenerate some summary statistics such as (1) Measures of Central Tendency (2) Measures of Variation ( including the first four moments of the distribution)

Thinking of Data as a DistributionMoment 1 = MeanMoment 2 = VarianceMoment 3 = SkewnessMoment 4 = Kurtosis

Statistics OnlineComputational Resource http://www.socr.ucla.edu/ http://www.socr.ucla.edu/htmls/SOCR_Charts.html

Describing the Shape of the Data

Calculating Skew & Kurtosis http://www.youtube.com/watch?v=eKwJUWkD2FQ

Skewnessskewness is a measure of the asymmetry of a distribution

Skewnessa negative skew indicates that the tail on theleft side of the probability density function islonger than the right side and the bulk of thevalues (possibly including the median) lie tothe right of the mean.

SkewnessA positive skew indicates that the tail onthe right side is longer than the left sideand the bulk of the values lie to the left ofthe mean.

Calculating Skewness1. Subtract Meanfrom each Raw Score.Aka, Deviations fromthe mean2. Raise each of thesedeviations from themean to the third powerand sum. Aka: Sum ofthird moment deviations3. Calculate skewness, which isthe sum of the deviations from themean, raised to the third power,divided by number of cases minus 1,times the standard deviationraised to the third power.

Calculating SkewnessTry This Problem: 1. Subtract Mean from each Raw Score. Aka, Deviations from the mean 2. Raise each of these deviations from the mean to the third power and sum. Aka: Sum of third moment deviations 3. Calculate skewness, which is the sum of the deviations from the mean, raised to the third power, divided by number of cases minus 1, times the standard deviation raised to the third power. http://www.indiana.edu/~educy520/sec5982/week_12/skewness_demo.pdf

Calculating SkewnessTry This Problem:http://www.indiana.edu/~educy520/sec5982/week_12/skewness_demo.pdf

Kurtosiskurtosis is any measure of the"peakedness" of a distributionA high kurtosis distribution has asharper peak and longer, fatter tails,while a low kurtosis distribution hasa more rounded peak and shorter,thinner tails.

KurtosisDistributions with zero excess kurtosis are calledmesokurtic, or mesokurtotic. The most prominentexample of a mesokurtic distribution is the normaldistributionA distribution with positive excess kurtosis is calledleptokurtic, or leptokurtotic. "Lepto-" means "slender".In terms of shape, a leptokurtic distribution has amore acute peak around the mean and fatter tails.A distribution with negative excess kurtosis is calledplatykurtic, or platykurtotic. "Platy-" means "broad".In terms of shape, a platykurtic distribution has alower, wider peak around the mean and thinner tails.

Calculating KurtosisThe moment coefficient of kurtosis of a dataset is computed almost the same way as thecoefficient of skewness:kurtosis = m4 / m22 and “excess” kurtosis: = Kurtosis−3 Note: the excess kurtosis is generally used because the excess kurtosis of a normal distribution is 0. where

Calculating KurtosisExample: n = 100 x̄bar = 67.45 variance m2 = 8.5275

Calculating KurtosisExample: n = 100 x̄bar = 67.45 variance m2 = 8.5275 kurtosis is = m4 / m2² = 199.3760/(8.5275)² = 2.7418 and the excess kurtosis is = 2.7418 − 3 = −0.2582

Next Time... Z Scores StandardizationBinomial DistributionNormal Distribution Transformations Outliers

Quantitative Methods for Lawyers - Class #8 - Professor Daniel Martin Katz - MSU College of Law

by Daniel Katz on Feb 06, 2012

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Quantitative Methods for Lawyers - Class #8 - Professor Daniel Martin Katz - MSU College of Law — Presentation Transcript