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Measures of Dispersion Week 3 What is dispersion? • Dispersion is how the data is spread out, or dispersed from the mean. • The smaller the dispersion values, the more consistent the data. • The larger the dispersion values, the more spread out the data values are. This means that the data is not as consistent. Consider these sets of data: • Grades from Test # 1 = - 81,83,83,82,86,81,87,80,81,86 • Grades from Test # 2 = - 95,74,65,90,87,97,60,81,99,76 • What differences do you see between the two sets? • What are the Mean scores? Ranges? • Do you believe these grades tell a story? Important Symbols to remember = mean • X = an individual value • N = Population size • n = sample population size • i = 1st data value in population Variance • The average of the squares of each difference of a data value and the mean. Standard Deviation • is the measure of the average distance between individual data points and their mean. • It is the square root of the variance. • The lower case Greek letter sigma is used to denote standard deviation. How to Calculate Standard Deviation • Given the data set {5, 6, 8, 9}, calculate the standard deviation. • Step 1: find the mean of the data set Sum of items x1 x2 x3 ... xn Mean Count n 5 6 8 9 28 Mean 7 4 4 How to Calculate Standard Deviation • Step 2: Find the difference between each data point and the mean. 5 7 2 6 7 1 87 1 97 2 How to Calculate Standard Deviation • Step 3: Square the difference between each data point and the mean. 2 4 2 1 1 2 1 1 2 2 4 2 How to Calculate Standard Deviation • Step 4: Sum the squares of the differences between each data point and the mean. 4 1 1 4 10 How to Calculate Standard Deviation • Step 5: Take the square root of the sum of the squares of the differences divided by the total number of data points; 10 10 3.16227766 1.58113883 4 2 2 * The average distance between individual data points and the mean is 1.58113883 units from 7 Standard Deviation • Formula of what we just did: n 1 2 ( Xi X ) n i 1 • For sample S.D. use 1/(n-1) When to use Pop. vs. Sample • When we have the actual entire population (for example our class, 29 students), we would use the Population formula. • If the problem tells us to use a particular formula; Pop. v. Samp. • If we are working with less entire population of a much larger group, we will use the sample formula. • (Which is one taken away from the pop. total) Why is this useful? • It provides clues as to how representative the mean is of the individual data points. • For example, consider the following two data sets with the same means, but different standard deviations. Bowler # 1 {98, 99, 101, 102} X 100 1.58113883 Bowler # 2 {30 ,51, 149, 169} X 100 78.10889834 The mean with the standard deviation provides a better description of the data set. TI-83 to Calculate Standard Deviation. • Step 1. Press STAT,EDIT,1:EDIT • Step 2. Enter your data in the L1 column, pressing enter after every data entry. • Step 3. Press STAT, CALC,1-Var stats • Step 4. Scroll down to the lower case symbol for the Greek letter sigma • calculator help. Let’s try one more by hand: (1) Find the population standard deviation for the following Stats class test grades: 78, 84, 88, 92, 68, 82, 92, 72, 88, 86, 76, 90 (a) How many grades fall within one SD of the mean? (b) What percent fall within one SD of the mean? * Now check it with the calculator!