variability quantitative methods in hpels 440:210
TRANSCRIPT
Variability
Quantitative Methods in HPELS
440:210
Agenda
Introduction Frequency Range Interquartile range Variance/SD of population Variance/SD of sample Selection
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
Basic Concepts
Measures of variability:FrequencyRange Interquartile rangeVariance and standard deviation
Each statistic has its advantages and disadvantages
Agenda
Introduction Frequency Range Interquartile range Variance/SD of population Variance/SD of sample Selection
Frequency
Definition: The number/count of any variable
Scale of measurement: Appropriate for all scalesOnly statistic appropriate for nominal data
Statistical notation: f
Frequency
Advantages:Ease of determinationOnly statistic appropriate for nominal data
Disadvantages: Terminal statistic
Calculation of the Frequency Instat Statistics tab Summary tab Group tab
Select groupSelect column(s) of interestOK
Agenda
Introduction Frequency Range Interquartile range Variance/SD of population Variance/SD of sample Selection
Range
Definition: The difference between the highest and lowest values in a distribution
Scale of measurement: Ordinal, interval or ratio
Range
Advantages:Ease of determination
Disadvantages:Terminal statisticDisregards all data except extreme scores
Calculation of the Range Instat
Statistics tab Summary tab Describe tab
Calculates range automaticallyOK
Agenda
Introduction Frequency Range Interquartile range Variance/SD of population Variance/SD of sample Selection
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
Interquartile Range
Advantages:Ease of determinationMore stable than range
Disadvantages:Disregards all values except 1st and 3rd
quartiles
Calculation of the Interquartile Range Instat Statistics tab Summary tab Describe tab
Choose additional statisticsChoose interquartile rangeOK
Agenda
Introduction Frequency Range Interquartile range Variance/SD of population Variance/SD of sample Selection
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
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
Calculation of the Variance Population
Why square all values? If all deviations from the mean are
summed, the answer always = 0
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
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
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)
Computational formula
Step 4: Divide by N
Computation of the Standard Deviation Population
Take the square root of the variance
Agenda
Introduction Frequency Range Interquartile range Variance/SD of population Variance/SD of sample Selection
Variance/SD Sample
Process is similar with two distinctions: Statistical notation Formula
Statistical Notation DistinctionsPopulation vs. Sample σ2 = s2
σ = s = M N = n
Formula DistinctionsPopulation vs. Sample s2 = SS / n – 1, where SS =
Σ(X - M)2
Definitional formula
ΣX2 - [(ΣX)2] Computational formula
Why n - 1?
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
Actual population σ2 = 14
Average biased s2 = 63/9 = 7 Average unbiased s2 = 126/9 = 14
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
Computation of the Standard Deviation of Sample Instat Statistics tab Summary tab Describe tab
Calculates standard deviation automatically OK
Agenda
Introduction Frequency Range Interquartile range Variance/SD of population Variance/SD of sample Selection
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
Textbook Problem Assignment
Problems: 4, 6, 8, 14.