Measures of Variation, Exercises of Statistics

Download Measures of Variation and more Exercises Statistics in PDF only on Docsity! Chapter 3.2 Measures of Variation Learning Objectives At the end of this lecture, the student should be able to: • State three different measures of variation used in statistics. • Explain how to calculate variance and standard deviation. • Calculate the coefficient of variation and explain its interpretation. • State Chebyshev’s Theorem. Measures of Variation • Range • Variance and Standard Deviation • Sample and population • Coefficient of variation Photograph by Glenys Gaske Range Simplest to Calculate Range • The range is the difference between the maximum and minimum value. • It is easy to calculate • Don’t forget to actually do the subtraction • In the above data, 78 is the maximum. • 21 is the minimum. • 78 minus 21 = 57 • 57 is the range 42 33 21 78 62 Variance & Standard Deviation are Friends • They are friends because this is how you calculate them: 1. First, calculate the variance. 2. Then, take the square root of variance, and that is the standard deviation. • “Variance” • How much the data vary. • Think: how well does the mean represent the spread of the data? • “Standard deviation” • “Standard” – following a standard, same • “deviation” – like a deviated septum Photograph by David Shankbone Formulas for Variance & Standard Deviation • The formulas for sample variance and sample standard deviation are different than those for population variance and population standard deviation • We don’t use the population ones that often • We will concentrate on how to use the sample ones Formulas for Variance & Standard Deviation • The formulas for sample variance and sample standard deviation are different than those for population variance and population standard deviation • We don’t use the population ones that often • We will concentrate on how to use the sample ones • Two different ways of doing the formula (for both sample and for population) – the “defining formula” and the “computational formula” • Both get the same results • I dislike the “computational formula” because I get really confused • Therefore, I will teach you the “defining formula” Column I x Column II x minus x- bar Column III (x minus x-bar)2 Sum of Squares • ∑( x – x )2 is another way of saying sum of squares. • An easy way to explain this (what the “squares” are) is to show an example of how to calculate it. • Imagine a sample of 6 patients presented to the central lab. We asked them how many minutes they had to wait to get service, and got these numbers: 2 3 3 8 10 10 Column I x Column II x minus x- bar Column III (x minus x-bar)2 2 3 3 8 10 10 ∑x = 36 Sum of Squares • ∑( x – x )2 is another way of saying sum of squares. • Each measurement is an x. First, fill in the x’s. • Fill in the sum (∑) of the x’s at the bottom 2 3 3 8 10 10 Column I x Column II x minus x-bar Column III (x minus x- bar)2 2 3 3 8 10 10 ∑x = 36 Colu n I x Colu n II x minus x- bar Colu n III (x minus x-bar)2 2 2 – 6 = -4 3 – 6 = -3 3 3 – 6 = -3 8 8 – 6 = 2 10 10 – 6 = 4 10 10 – 6 = 4 ∑x = 36 Sum of Squares • ∑( x – x )2 is another way of saying sum of squares. • To figure out Column II, the sample mean (x-bar) is needed. • 36/6 (n=6) = a mean of 6. • Now Column II can be filled in. 2 3 3 8 10 10 Variance Formula • ∑( x – x )2 is another way of saying sum of squares. • For Column III, square each value in Column II. Those are the “squares”. • At the bottom, add them up. That’s the sum of squares. • Sum of squares = 70. 2 3 3 8 10 10 • Remember, the sample was from 6 patients. • Therefore, n = 6 • n – 1 = 6 – 1 = 5. ∑( x – x )2 n - 1 Sample variance = s2 = 70 5 = 14 Standard Deviation Formula ∑( x – x )2 n - 1 Sample variance = s2 = 70 5 = 14 ∑( x – x )2 n - 1 Sample standard deviation= s = √ √14 = 3.74 Let’s Look at Formulas Sample Defining Formulas Population Defining Formulas ∑( x – x )2 n - 1 Sample variance = s2 = ∑( x – x )2 n - 1 Sample standard deviation= s = √ ∑( x – μ )2 N Population variance = Ϭ2 = ∑( x – μ )2 N Population standard deviation= Ϭ = √ CV is used for Comparison • It’s hard to explain with only one group of patients. • CV has no units – so you could compare two different ways of measuring a lab value, for example. • The CV is the measure of the spread of the data relative to the average of the data. • In the first sample, the s is only 50% of the mean. • In the second sample, the s is 62% of the mean. Patients: s = 3.74, x-bar = 6 3.74 6 x 100 = 62% Other Patients: s = 4, x-bar = 8 4 8 x 100 = 50% Chebyshev’s Theorem With Explanation of Chebyshev Intervals What Chebyshev Figured Out • First, he started thinking like this: • If you have an x-bar and an s, you can create lower and upper limits by subtracting the s and adding the s to the x-bar. • You can do this with a μ and Ϭ, too – population version • For example, if I had an μ of 100, and an Ϭ of 5: • If I subtracted 1 Ϭ from 100, I’d get 95 as the lower limit • If I add 1 Ϭ to 100, I get 105 as the upper limit. • I could even try this by doing 2 Ϭ – meaning subtracting 10 for the lower limit and adding 10 for the upper limit • He realized if he used some rules along with this, there would be an interpretation of these limits that would be useful. Let’s Try Chebyshev’s Intervals! • Calculate limits for Patient Sample: • 75% limits: 6 +/- (2 * 3.74) = -1.48 to 13.48 • 88.9% limits: 6 +/- (3 * 3.74) = -5.22 to 17.22 • 93.8% limits: 6 +/- (4 * 3.74) = -8.96 to 20.96 • Interpretation • At least 75% of the data are between - 1.48 and 13.48 • At least 88.9% of the data are between -5.22 to 17.22 • At least 93.8% of the data are between -8.96 and 20.96. s or Ϭ % of data in interval 2 75% 3 88.9% 4 93.8% Patient Sample (waiting room minutes) s = 3.74, x-bar = 6 Let’s Try Chebyshev’s Intervals! • Calculate limits for Patient Sample: • 75% limits: 6 +/- (2 * 3.74) = -1.48 to 13.48 • 88.9% limits: 6 +/- (3 * 3.74) = -5.22 to 17.22 • 93.8% limits: 6 +/- (4 * 3.74) = -8.96 to 20.96 • Interpretation • At least 75% of the data are between - 1.48 and 13.48 • At least 88.9% of the data are between -5.22 to 17.22 • At least 93.8% of the data are between -8.96 and 20.96. s or Ϭ % of data in interval 2 75% 3 88.9% 4 93.8% Patient Sample (waiting room minutes) s = 3.74, x-bar = 6 If dataset had 100 patients: • At least 75 patients would have waited between -1.48 and 13.48 minutes • At least 88.9 patients would have waited between -5.22 to 17.22 minutes • At least 93.8 patients would have waited between -8.96 and 20.96 minutes. Take-home Message on Chebyshev Interval • It works for any distribution (normal, skewed, etc.) • Chebyshev intervals tell you that at least a certain % is in the interval • Chebyshev intervals are sometimes non-sensical (negative numbers, very high limits) • They are not very useful and not used in healthcare • The purpose of teaching this is to point out in statistics, we often use the s or Ϭ and add/subtract it from the mean because it is a good way to make lower and upper limits that have special significance.