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Chapter 3 Variability Page 1 Variability • 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 Measures of Variability Worksheet (In-Class) Page 2 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 Page 3 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. Page 4 Variability: Standard Deviation Deviation Method: used as a teaching method to help clearly understand the concept Formula: x=X-M x is the “deviation score” •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 Page 5 Variability: Standard Deviation Computational 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 Page 6 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: Measures of Variability Homework due next class Page 7 Range & Percentiles • 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 Page 8 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 Percentiles Worksheet Page 9 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 Pair Share Topic: **EX: R=600 and SD=200 What does a standard deviation tell you? Page 10