Date post: | 16-Jan-2016 |
Category: |
Documents |
Upload: | juliet-alexander |
View: | 224 times |
Download: | 3 times |
Chapter 2Characterizing Your Data Set
Allan Edwards:“Before you analyze your data, graph your data
Chapter 2Characterizing Your Data Set
Allan Edwards:“Before you analyze your data, graph your data
Francis Galton, Father of Intelligence Testing: Whenever you can, count!
Frequency TableVariable is Continuous
Grouped Frequency Table & Distribution
Continuous variable,Data from Same 100Subjects
Constant Interval“Class Interval”
Grouped Frequency HistogramFor Continuous Variable
Bars “Touch”, the end of one interval is beginning of nextValue is middle value of IntervalSpatz says the bars don’t touch – Whaaaaaa?????
Bar Chart for Categorical Variable
Bars are separated – a lot of Biology is not almost English
Standard Normal Distribution
The more Extreme your score the more unusual, improbable you areRemember this relationship -- it’s the basis of 90% of statisticsTypical of many characteristics -- E.G., height, intelligence, speed
Rectangular DistributionNever Seen One
Extreme Scores are NOT less usual/frequent/probable
Non-Normal Distribution
Example: Income -- Where is the mean? How would you characterize these data?
Negative Skew
Bimodal Distribution
Is the Mean appropriate/representativeE.G., Mean age of onset for Anorexia is 17yrs
One Peak is at 14yrs -- Onset of PubertyOne Peak is at 18yrs -- Going away to college
Bimodal Distribution, cont.
Characterizing Your DataMeasures of Central Tendency
Characterizing your Data:Shorthand notation for all of your values
Central Tendency:• A representative value• Where Your Scores tend to “Hang Out”• Where you go to find your data
1. Mean -- What is definition & why do you use it?2. Median -- Middle Value
What if you have an even # of values?3. Mode -- Most frequent value
Which Central Tendency is Best?
•MeanRatio Data (People allow Interval Data)Symmetrical Distributions
•MedianSkewed DistributionsOrdinal (Ranked) Data -- A mean cannot be computed
•ModeNominal (Qualitative) DataBimodal Data
If you Had to Guess the Value of Each (Quantitative) Data Point
• Mode: Highest # of correct guesses
• Median: Errors would be symmetricalOverestimations would balance out Underestimations
• Mean: Errors of Estimation will be smallest, overallTwo Unique Properties of the Mean:
1. Deviations are smallest from the meanThan for any other value
2. Deviation scores sum to zero
How Strong Is Your Tendency?Measures of Heterogeneity
(Chapter 3)
Two Data Sets with nearly identical:•Ns•Means•Medians•Modes
Are these two data sets similar?
Are They The Same?
Some Data Sets are More Heterogeneous
Jockeys: Very Low average height Very Homogeneous
Presbyterians: Medium average height Very Heterogeneous
NBA Players: Very High average height Very Homogenous
How do you characterize a data set’s Heterogeneity?The Greater the Heterogeneity, the Weaker the Central Tendency
Quantifying Heterogeneity
Range: Highest Score minus Lowest ScoreVery sensitive to a single Extreme Score
Inter Quartile Range: 75th percentile minus 25th percentileCaptures 50% of the scoresHow wide do you have to go to capture 50% of values?
The wider you have to go the more Heterogeneity
Heterogeneity, cont.
The more Heterogeneity, the more the scores will deviate from The mean
Xi-Xbar Xi-XbarXi Di Xi Di
4 -25 -16 0 6 07 1
8 2
Sum= 0 0Mean = 6 0 6 0
Heterogeneity, cont.
Two Unique properties of the Mean:
1. All deviation scores sum to zero
2. Raw scores Deviate Less from the mean than from any otherValue
This makes the mean the Best Representative of the dataSet If distribution is symmetrical
Heterogeneity, cont.
Problem: •All deviation scores sum to zero no matter how
Heterogeneous the raw scores
•You Cannot average deviations scores to quantify heterogeneity
Solution:Make all deviation scores Positive
Heterogeneity, cont.
Two way to make all deviation scores Positive:
•Take the Absolute Value of the Deviation Scores:Average of absolute values = Average Deviation
Mean +/- AD Captures 50% of raw scores
•Take the Square of the Deviation ScoresAverage of squared deviation scores = Variance
2 for PopulationS2 for SampleS2 -”hat” for estimating Population from Sample
Variance
Population Estimate of Population from Sample
To Describe sample use NS2 = Sample Variance
Problem: Magnitude of Variance is large relative to individualDeviation scores -- Quantifies but not very descriptive
Standard Deviation
Population Sample
Population Estimate
Mean +/- SD captures 68% of Data Points
Standard Deviation, cont.
The Concept
Standard DeviationStandard Deviation from the Mean“Average” Deviation from the MeanExpected Deviation from the Mean
Expect 68% of your data to be within 1 SD of the meanExpect 95% of your data to be within 2 SD of the mean
If your score is beyond 2 SDs of the meanYou are very infrequentYou are very unusualYou are very improbable
Associate: Infrequent with Improbable
Interpreting a Value
Transforming a score to make it more interpretable:
•Comparing two scores:Two tests of Equal Difficulty but of Different Length
Pretend both tests were 100 items longHow many would you have gotten right?
Percent Correct is a Transformed Score
•Comparing one score to everybody else:Pretend there were 100 people, where would rank?
Percentile is a Transformed Score
Z-scores & Z-transformations
Take each score (Xi) and covert it to ZiMean of z-scores = 0Standard Deviation = 1Units of z-scores are in Standard DeviationsZ-score compares Your Deviation (numerator) to the“Average Deviation” (denominator)
Where you are relative to Population
Think Percentile
Interpreting Your Z-Score
Interpreting Your Z-Score, cont.
Interpreting Your Z-Score, cont.
Interpreting Your Z-Score, cont.