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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: Frequency Range Interquartile range Variance 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 scales Only statistic appropriate for nominal data Statistical notation: f

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Frequency Advantages: Ease of determination Only statistic appropriate for nominal data Disadvantages: Terminal statistic

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Calculation of the Frequency Instat Statistics tab Summary tab Group tab Select group Select column(s) of interest OK

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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 statistic Disregards all data except extreme scores

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Calculation of the Range Instat Statistics tab Summary tab Describe tab Calculates range 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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Interquartile Range Definition: The difference between the 1 st quartile and the 3 rd quartile Scale of measurement: Ordinal, interval or ratio Example: Figure 4.3, p 107

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Interquartile Range Advantages: Ease of determination More stable than range Disadvantages: Disregards all values except 1 st and 3 rd quartiles

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Calculation of the Interquartile Range Instat Statistics tab Summary tab Describe tab Choose additional statistics Choose interquartile range OK

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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: σ 2 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 mean The square root of the variance Statistical 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 σ 2 = SS/number of scores, where SS = Σ(X - ) 2 Definitional formula (Example 4.3, p 112) or ΣX 2 – [(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 Distinctions Population vs. Sample σ 2 = s 2 σ = s = M N = n

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Formula Distinctions Population vs. Sample s 2 = SS / n – 1, where SS = Σ(X - M) 2 Definitional formula ΣX 2 - [(ΣX) 2 ] Computational formula Why n - 1?

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N vs. (n – 1) First Reason General underestimation of population variance Sample variance (s 2 ) tend to underestimate a population variance (σ 2 ) (n – 1) will inflate s 2 Example 4.8, p 121

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Actual population σ 2 = 14 Average biased s 2 = 63/9 = 7Average unbiased s 2 = 126/9 = 14

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N vs. (n – 1) Second Reason Degrees of freedom (df) df = number of scores “free” to vary Example: 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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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.

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