POLI 205

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Lecture4Variabilitytopost.pdf

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12/14/2018

POLI 205

4. Variability

Why Variability

2 classes with the following quiz scores

A. 0, 4, 4, 5, 7, 10

B. 0, 0, 1, 9, 10, 10

What is the mean of A & B?

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Understanding Variability

• Measures of central tendency indicate modality and symmetry of a distribution of scores

– What do the scores have in common?

• Measures of variability indicate variability of a distribution of scores

– How do the scores differ from each other?

Howard T. Tokunaga, Fundamental Statistics for the Social and Behavioral Sciences © SAGE Publications, 2016

Understanding Variability

• Even though these three distributions have the same modality and symmetry, they are clearly different in terms of variability

Howard T. Tokunaga, Fundamental Statistics for the Social and Behavioral Sciences © SAGE Publications, 2016

• Variability quantifies the differences among scores

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The Range

Howard T. Tokunaga, Fundamental Statistics for the Social and Behavioral Sciences © SAGE Publications, 2016

• Based on the two ends of a distribution

Range = Highest score – lowest score

• Range is reported, along with the values of the highest and lowest scores

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The Range

Strengths

• Easy to compute

• Provides some information about the sample

Weaknesses

• Based on only two scores (may not accurately reflect variability of the entire distribution)

• Affected by extreme scores (outliers)

Howard T. Tokunaga, Fundamental Statistics for the Social and Behavioral Sciences © SAGE Publications, 2016

The Interquartile Range

Howard T. Tokunaga, Fundamental Statistics for the Social and Behavioral Sciences © SAGE Publications, 2016

• Range of the middle 50% of scores • Minimizes the effect of outliers

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The Interquartile Range

Strengths

• Reduces influence of outliers by focusing on middle 50% of the distribution

Weaknesses

• Ignores top 25% and bottom 25% of scores (may not accurately reflect variability of the entire distribution)

Howard T. Tokunaga, Fundamental Statistics for the Social and Behavioral Sciences © SAGE Publications, 2016

The Variance (s2)

• Based on all of the scores in the distribution

• Measures variability in terms of extent to which each score differs (deviates) from the mean

Howard T. Tokunaga, Fundamental Statistics for the Social and Behavioral Sciences © SAGE Publications, 2016

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The Variance (s2)

Howard T. Tokunaga, Fundamental Statistics for the Social and Behavioral Sciences © SAGE Publications, 2016

• Definitional formula for variance

The Variance (s2)

Howard T. Tokunaga, Fundamental Statistics for the Social and Behavioral Sciences © SAGE Publications, 2016

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The Variance (s2)

The Standard Deviation (s)

• Variance: Average squared deviation of a score from the mean

• What if we want to measure variability as the average deviation (not the average squared deviation)?

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The Standard Deviation (s)

• Definitional formula

Measures of Variability for Samples vs. Populations

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Measures of Variability: Drawing Conclusions

• Variability quantifies the degree to which scores in a distribution differ from each other.

How Large is S? Empirical Rule 68% fall between Mean +/- S 95% fall between Mean +/- 2S

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With the Empirical Rule

And the mean and standard deviation, you can start to think about the shape of the distribution.

https://www.intmath.com/counting-probability/normal- distribution-graph-interactive.php

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Looking Ahead

• Important to understand a distribution in terms of its shape, central tendency, and variability

• The next chapter examines the normal distribution, a particular type of distribution