variability quantitative methods in hpels 440:210
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Variability
Quantitative Methods in HPELS
440:210
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Agenda
Introduction Frequency Range Interquartile range Variance/SD of population Variance/SD of sample Selection
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Introduction
Statistics of variability: Describe how values are spread out Describe how values cluster around the middle
Several statistics Appropriate measurement depends on: Scale of measurement Distribution
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Basic Concepts
Measures of variability:FrequencyRange Interquartile rangeVariance and standard deviation
Each statistic has its advantages and disadvantages
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Agenda
Introduction Frequency Range Interquartile range Variance/SD of population Variance/SD of sample Selection
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Frequency
Definition: The number/count of any variable
Scale of measurement: Appropriate for all scalesOnly statistic appropriate for nominal data
Statistical notation: f
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Frequency
Advantages:Ease of determinationOnly statistic appropriate for nominal data
Disadvantages: Terminal statistic
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Calculation of the Frequency Instat Statistics tab Summary tab Group tab
Select groupSelect column(s) of interestOK
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Agenda
Introduction Frequency Range Interquartile range Variance/SD of population Variance/SD of sample Selection
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Range
Definition: The difference between the highest and lowest values in a distribution
Scale of measurement: Ordinal, interval or ratio
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Range
Advantages:Ease of determination
Disadvantages:Terminal statisticDisregards all data except extreme scores
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Calculation of the Range Instat
Statistics tab Summary tab Describe tab
Calculates range automaticallyOK
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Agenda
Introduction Frequency Range Interquartile range Variance/SD of population Variance/SD of sample Selection
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Interquartile Range
Definition: The difference between the 1st quartile and the 3rd quartile
Scale of measurement:Ordinal, interval or ratioExample: Figure 4.3, p 107
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Interquartile Range
Advantages:Ease of determinationMore stable than range
Disadvantages:Disregards all values except 1st and 3rd
quartiles
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Calculation of the Interquartile Range Instat Statistics tab Summary tab Describe tab
Choose additional statisticsChoose interquartile rangeOK
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Agenda
Introduction Frequency Range Interquartile range Variance/SD of population Variance/SD of sample Selection
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Variance/SD Population Variance:
The average squared distance/deviation of all raw scores from the mean
The standard deviation squared Statistical notation: σ2
Scale of measurement: Interval or ratio
Advantages: Considers all data Not a terminal statistic
Disadvantages: Not appropriate for nominal or ordinal data Sensitive to extreme outliers
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Variance/SD Population Standard deviation:
The average distance/deviation of all raw scores from the meanThe square root of the varianceStatistical notation: σ
Scale of measurement: Interval or ratio
Advantages and disadvantages: Similar to variance
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Calculation of the Variance Population
Why square all values? If all deviations from the mean are
summed, the answer always = 0
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Calculation of the Variance Population
Example: 1, 2, 3, 4, 5 Mean = 3 Variations:
1 – 3 = -2 2 – 3 = -1 3 – 3 = 0 4 – 3 = 1 5 – 3 = 2
Sum of all deviations = 0
Sum of all squared deviations
Variations: 1 – 3 = (-2)2 = 4 2 – 3 = (-1)2 = 1 3 – 3 = (0)2 = 0 4 – 3 = (1)2 = 1 5 – 3 = (2)2 = 4
Sum of all squared deviations = 10
Variance = Average squared deviation of all points 10/5 = 2
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Calculation of the Variance Population
Step 1: Calculate deviation of each point from mean
Step 2: Square each deviation Step 3: Sum all squared deviations Step 4: Divide sum of squared deviations
by N
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Calculation of the Variance Population
σ2 = SS/number of scores, where SS =Σ(X - )2
Definitional formula (Example 4.3, p 112) or
ΣX2 – [(ΣX)2] Computational formula (Example 4.4, p 112)
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Computational formula
Step 4: Divide by N
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Computation of the Standard Deviation Population
Take the square root of the variance
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Agenda
Introduction Frequency Range Interquartile range Variance/SD of population Variance/SD of sample Selection
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Variance/SD Sample
Process is similar with two distinctions: Statistical notation Formula
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Statistical Notation DistinctionsPopulation vs. Sample σ2 = s2
σ = s = M N = n
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Formula DistinctionsPopulation vs. Sample s2 = SS / n – 1, where SS =
Σ(X - M)2
Definitional formula
ΣX2 - [(ΣX)2] Computational formula
Why n - 1?
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N vs. (n – 1) First Reason
General underestimation of population variance
Sample variance (s2) tend to underestimate a population variance (σ2)
(n – 1) will inflate s2
Example 4.8, p 121
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Actual population σ2 = 14
Average biased s2 = 63/9 = 7 Average unbiased s2 = 126/9 = 14
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N vs. (n – 1) Second Reason
Degrees of freedom (df)df = number of scores “free” to varyExample:
Assume n = 3, with M = 5 The sum of values = 15 (n*M) Assume two of the values = 8, 3 The third value has to be 4 Two values are “free” to vary df = (n – 1) = (3 – 1) = 2
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Computation of the Standard Deviation of Sample Instat Statistics tab Summary tab Describe tab
Calculates standard deviation automatically OK
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Agenda
Introduction Frequency Range Interquartile range Variance/SD of population Variance/SD of sample Selection
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Selection
When to use the frequency Nominal data With the mode
When to use the range or interquartile range Ordinal data With the median
When to sue the variance/SD Interval or ratio data With the mean
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Textbook Problem Assignment
Problems: 4, 6, 8, 14.