Page 1 Chapter 3 Variability

Page 2 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

Page 3 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 4 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 5 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-M x is the “deviation score” Formula:

Page 6 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 7 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:

Page 8 Percentile: the point on a distribution where a given percentage of scores fall below. **EX: 95 th percentile means A LOT of scores fall below it **EX: 5 th percentile means very FEW scores fall below it -Percentiles are used to show various forms of range -Note: The 50 th percentile is right in the middle of the distribution so it is always equal to the median. Quartiles: divide a distribution into quarters -1 st quartile coincides with the 25 th percentile -2 nd quartile coincides with the 50 th percentile -3 rd quartile coincides with the 75 th percentile Range & Percentiles

Page 9 Range & Percentiles Deciles: divide a distribution into tenths -1 st decile is equivalent to the 10 th percentile & so on -the lowest score would be in the 1 st decile & the highest score would be in the 10 th decile Interquartile Range: find the difference between the 1 st & 3 rd quartiles -middlemost 50% of the distribution Interdecile Range: find the difference between the 1 st & 9 th deciles -middlemost 80% of the distribution

Page 10 Assessing Kurtosis: 1/6 th Rule Use the 1/6 th rule to quickly evaluate the kurtosis of any unimodal symmetrical distribution Mesokurtic distribution: standard deviation is approximately 1/6 th 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 1/6 th of the range **EX: R=600 and SD=50 Platykurtic distribution: the standard deviation will be MORE than 1/6 th of the range **EX: R=600 and SD=200 Pair Share Topic: What does a standard deviation tell you?