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Chapter 3Variability

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• Central tendency tells us about the similarity between scores • Variability tells us about the differences between scores -ie. how spread out are the scores in the distribution? -ie. how close or far from the mean are the scores?

• There are 3 measures of variability: range, standard deviation & variance

Variability

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Variability: Range

•Symbolized by R•It is the measurement of the width of the entire distribution •To calculate: Subtract the lowest value from the highest value•Least useful measure of variability

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Variability: Standard Deviation

•Symbolized as SD

•The average amount that scores in a distribution deviate from the mean.

•The most common descriptive statistic for variability.

•Two ways to calculate -the Deviation Method -the Computational Method

Note: standard deviations are never

less than zero because you can’t

have less than zero variability.

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Variability: Standard Deviation

•To calculate: -find the mean

-subtract the mean of the distribution from each score: (X-M) or x

-square each difference: (X-M)² or x²

-sum the squares

-divide by N

-take the square root

Deviation Method: used as a teaching method to help clearly understand the concept

x=X-Mx is the “deviation score”

Formula:

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Variability: Standard DeviationComputational Method: is a shortcut that is used most often.

-this is what you should use

Formula:

•To calculate:-Column 1: sum the raw scores: ΣΧ-Column 2: square each raw score & then sum the squares: ΣΧ²-divide the sum of the scores (ΣΧ) by N: M-divide the sum of the squares (ΣΧ²) by N & subtract the squared mean (M²)-find the square root

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Variability: Variance

•Symbolized by V•Measure of how spread out a set of scores are•Average of the squared deviations from the mean **Also called the “mean square deviation”•To calculate V: calculate the SD but don’t find the square root **The variance is equal to the SD² •Q:If the variance is just the square of the SD, why use it? -A: some formulas require using the variance rather than the SD

Formula:

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• Percentile: the point on a distribution where a given percentage of scores fall below.

**EX: 95th percentile means A LOT of scores fall below it **EX: 5th percentile means very FEW scores fall below it -Percentiles are used to show various forms of range -Note: The 50th percentile is right in the middle of the distribution so it is always equal to

the median.

• Quartiles: divide a distribution into quarters -1st quartile coincides with the 25th percentile

-2nd quartile coincides with the 50th percentile -3rd quartile coincides with the 75th percentile

Range & Percentiles

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Range & Percentiles

• Deciles: divide a distribution into tenths -1st decile is equivalent to the 10th percentile & so on

-the lowest score would be in the 1st decile & the highest score would be in the 10th decile

• Interquartile Range: find the difference between the 1st & 3rd quartiles -middlemost 50% of the distribution

• Interdecile Range: find the difference between the 1st & 9th deciles -middlemost 80% of the distribution

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Assessing Kurtosis: 1/6th Rule

• Use the 1/6th rule to quickly evaluate the kurtosis of any unimodal symmetrical distribution

• Mesokurtic distribution: standard deviation is approximately 1/6th of the range

-divide the range by 6 to get the approximate standard deviation **EX: R=600 and SD=100• Leptokurtic distribution: the standard deviation will be LESS than 1/6th

of the range **EX: R=600 and SD=50• Platykurtic distribution: the standard deviation will be MORE than 1/6th

of the range **EX: R=600 and SD=200 Pair Share Topic:

What does a standard deviation tell you?