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Measures of Variation Basic Concepts Other Concepts What is variation? How data is spread or dispersed Range of a data set... Difference between the maximum data value and the minimum data value RANGE (Maximum Data Value) - (Minimum Data Value) What does Range tell us about variation Uses only max and min values therefore... Use the same round off rule as MOC STANDARD DEVIATION (s) MOV most commonly used in stats Measures variation of values about the mean Deviation of sample values away from the mean Finding Standard Deviation of a Data Set Sample Standard Deviation: ( x x ) s n 1 2 Shortcut Formula: n( x ) (x) s n(n 1) 2 2 Characteristics of Standard Deviation? s is usually positive and NEVER negative s is 0 only when all data values are the same number the larger value for (s) the greater amount the data varies s can increase dramatically with the inclusion of outliers the units (minutes, feet, etc...) are the same as the units of original values Procedure for Calculating (s) Find the mean Subtract the mean from each ind. Value Square each result Add all the squares Divide the sum by (n-1) Find the square root Example From Excel STATS SD Calculate the standard deviation of 1, 3 and 14 Using Calculators to Find Standard Deviation Input your set of data into a List by using STAT and EDIT Hit STAT and right arrow to CALC Choose option 1-VarStats and enter L1 (or whatever list you put the data in) and ENTER Find the Standard Deviation Given the sample below, determine the standard deviation: Nitrate deposits as a result of acid rain for Massachusetts from July to Sept.: 6.40 5.53 5.41 5.21 8.23 4.66 5.24 6.96 6.80 5.78 6.00 Finding the Standard Deviation of a Frequency Table In L1 input all the class marks. InL2 input the frequencies . Use STAT, CALC, and 1-VarStats. On your screen you will see 1-VarStats Input L1 , L2 and hit ENTER. Variation and Standard Deviation Understanding Variation Range Rule of Thumb Empirical (68-95-99) Rule for Data Chebyshev’s Theorem Range Rule of Thumb For typical data sets, the range of a set of data is approximately 4 standard deviations wide. To approximate the standard deviation, range standard deviation 4 Range Rule of Thumb If we know, or approximately know, the value of the standard deviation, we can find estimates of the minimum and maximum scores. minimum (mean) 2 (standard deviation) maximum (mean) + 2 (standard deviation) Empirical Rule for Data This rule applies to a data set that is approximately bell-shaped. 68-95-99 Rule: About 68% of all scores fall within 1 standard deviation of the mean. About 95% of all scores fall within 2 standard deviations of the mean. About 99.7% of all scores fall within 3 standard deviations of the mean. Examples Use the range rule of thumb to estimate the standard deviation of 100 credit rating scores. The minimum is 444 while the max is 850. Empirical Rule for Data Chebyshev’s Theorem A proportion (or fraction) of any data set lying within K standard deviations of the mean is always at least 1 1 K 2 where K is any number greater than 1.