Chapter 2
Characterizing Your Data Set
Allan Edwards:
“Before you analyze your data, graph your data
Chapter 2
Characterizing Your Data Set
Allan Edwards:
“Before you analyze your data, graph your data
Francis Galton, Father of Intelligence Testing:
Whenever you can, count!
Frequency Table
Variable is Continuous
Grouped Frequency Table &
Distribution
Continuous variable,
Data from Same 100
Subjects
Constant Interval
“Class Interval”
Grouped Frequency Histogram
For Continuous Variable
Bars “Touch”, the end of one interval is beginning of next
Value is middle value of Interval
Spatz 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 are
Remember this relationship -- it’s the basis of 90% of statistics
Typical of many characteristics -- E.G., height, intelligence, speed
Rectangular Distribution
Never 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/representative
E.G., Mean age of onset for Anorexia is 17yrs
One Peak is at 14yrs -- Onset of Puberty
One Peak is at 18yrs -- Going away to college
Bimodal Distribution, cont.
Characterizing Your Data
Measures 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?
•Mean
Ratio Data (People allow Interval Data)
Symmetrical Distributions
•Median
Skewed Distributions
Ordinal (Ranked) Data -- A mean cannot be computed
•Mode
Nominal (Qualitative) Data
Bimodal Data
If you Had to Guess the Value of
Each (Quantitative) Data Point
•
Mode: Highest # of correct guesses
•
Median: Errors would be symmetrical
Overestimations would balance out Underestimations
•
Mean: Errors of Estimation will be smallest, overall
Two Unique Properties of the Mean:
1. Deviations are smallest from the mean
Than 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 Score
Very sensitive to a single Extreme Score
Inter Quartile Range: 75th percentile minus 25th percentile
Captures 50% of the scores
How 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
5
6
7
Sum=
Mean =
6
Xi-Xbar
Di
-1
0
1
0
0
Xi
Xi-Xbar
Di
4
-2
6
0
8
2
6
0
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 other
Value
This makes the mean the Best Representative of the data
Set
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 Scores
Average of squared deviation scores = Variance
2 for Population
S2 for Sample
S2 -”hat” for estimating Population from Sample
Variance
Population
Estimate of Population from Sample
To Describe sample use N
S2 = Sample Variance
Problem: Magnitude of Variance is large relative to individual
Deviation 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 Deviation
Standard Deviation from the Mean
“Average” Deviation from the Mean
Expected Deviation from the Mean
Expect 68% of your data to be within 1 SD of the mean
Expect 95% of your data to be within 2 SD of the mean
If your score is beyond 2 SDs of the mean
You are very infrequent
You are very unusual
You 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 long
How 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 Zi
Mean of z-scores = 0
Standard Deviation = 1
Units of z-scores are in Standard Deviations
Z-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.