S1600 #6. Estimates of Spread. January 28, 2016
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1 S1600 #6 Estimates of Spread January 28, 2016
2 Outline 1 Estimates of Spread Estimates of Spread The Sample Standard Deviation Effect of Multiplication/Addition by a Constant (WMU) S1600 #6 S1600, Lecture 6 2 / 11
3 Estimates of Spread (or Uncertainty, Variation) an estimate of spread is a measure of uncertainty, or variation, or give or take when two or more comparable data sets (comparable means data sets are of same type/same unit of numerical measurements) are compared, the one with smallest spread has least uncertainty around the estimate of center (i.e., least scattered) estimates of spread are non-negative (WMU) S1600 #6 S1600, Lecture 6 3 / 11
4 Estimates of Spread (or Uncertainty, Variation) an estimate of spread is a measure of uncertainty, or variation, or give or take when two or more comparable data sets (comparable means data sets are of same type/same unit of numerical measurements) are compared, the one with smallest spread has least uncertainty around the estimate of center (i.e., least scattered) estimates of spread are non-negative (WMU) S1600 #6 S1600, Lecture 6 3 / 11
5 Estimates of Spread (or Uncertainty, Variation) an estimate of spread is a measure of uncertainty, or variation, or give or take when two or more comparable data sets (comparable means data sets are of same type/same unit of numerical measurements) are compared, the one with smallest spread has least uncertainty around the estimate of center (i.e., least scattered) estimates of spread are non-negative (WMU) S1600 #6 S1600, Lecture 6 3 / 11
6 Estimates of Spread (or Uncertainty, Variation) an estimate of spread is a measure of uncertainty, or variation, or give or take when two or more comparable data sets (comparable means data sets are of same type/same unit of numerical measurements) are compared, the one with smallest spread has least uncertainty around the estimate of center (i.e., least scattered) estimates of spread are non-negative (WMU) S1600 #6 S1600, Lecture 6 3 / 11
7 Rent X = 626 The Sample Standard Deviation (SD) Kalamazoo 2-bedroom apartment rental data Step 1. Calculate X. Step 2. Calculate (obs. X), i.e., how much an obs. missed by the mean (i.e., deviation) Step 3. Calculate (obs. X) 2. That is, calculate squared ( missed by ) s (i.e., deviation 2 ) Step 4. Calculate SS, the total squared ( missed by ) s Step 5. Take square-root of SS/(n 1) to get SD (WMU) S1600 #6 S1600, Lecture 6 4 / 11
8 The Sample Standard Deviation (SD) Rent Kalamazoo 2-bedroom apartment rental data Rent X missed by Step 1. Calculate X Step 2. Calculate (obs. X), i.e., how much an obs. missed by the mean (i.e., deviation) Step 3. Calculate (obs. X) That is, calculate squared ( missed by ) s (i.e., deviation 2 ) Step 4. Calculate SS, the total squared ( missed by ) s Step 5. Take square-root of SS/(n 1) to get SD X = 626 (WMU) S1600 #6 S1600, Lecture 6 4 / 11
9 The Sample Standard Deviation (SD) Kalamazoo 2-bedroom apartment rental data Rent Rent X (Rent X) 2 missed by ( missed by ) X = 626 Step 1. Calculate X. Step 2. Calculate (obs. X), i.e., how much an obs. missed by the mean (i.e., deviation) Step 3. Calculate (obs. X) 2. That is, calculate squared ( missed by ) s (i.e., deviation 2 ) Step 4. Calculate SS, the total squared ( missed by ) s Step 5. Take square-root of SS/(n 1) to get SD (WMU) S1600 #6 S1600, Lecture 6 4 / 11
10 The Sample Standard Deviation (SD) Kalamazoo 2-bedroom apartment rental data Rent Rent X (Rent X) 2 missed by ( missed by ) X = 626 SS = Step 1. Calculate X. Step 2. Calculate (obs. X), i.e., how much an obs. missed by the mean (i.e., deviation) Step 3. Calculate (obs. X) 2. That is, calculate squared ( missed by ) s (i.e., deviation 2 ) Step 4. Calculate SS, the total squared ( missed by ) s Step 5. Take square-root of SS/(n 1) to get SD (WMU) S1600 #6 S1600, Lecture 6 4 / 11
11 The Sample Standard Deviation (SD) Kalamazoo 2-bedroom apartment rental data Rent Rent X (Rent X) 2 missed by ( missed by ) X = 626 SS = /(10 1) = Step 1. Calculate X. Step 2. Calculate (obs. X), i.e., how much an obs. missed by the mean (i.e., deviation) Step 3. Calculate (obs. X) 2. That is, calculate squared ( missed by ) s (i.e., deviation 2 ) Step 4. Calculate SS, the total squared ( missed by ) s Step 5. Take square-root of SS/(n 1) to get SD (WMU) S1600 #6 S1600, Lecture 6 4 / 11
12 Interpretation of SD bowling example Games scores Mean SD First 7 games 163, 231, 224, 238, 279, 239, Last 7 games 246, 244, 247, 248, 237, 258, scores of Walter Ray Williams Jr. in 2008 bowling tournament, Indiana games Last 7 First bigger swings (larger SD) in earlier games and scored typically lower (smaller Mean) he played consistently (smaller SD) in later games, and typically with better scores (larger Mean) score (WMU) S1600 #6 S1600, Lecture 6 5 / 11
13 Sample Standard Deviation is Not Robust As an estimate of the spread of a data set, the sample standard deviation is sensitive to outliers. (WMU) S1600 #6 S1600, Lecture 6 6 / 11
14 iclicker Question 6.1 The fuel efficiency (MPG) of 5 Japanese made cars are listed below Ignoring any rounding error, what is the sum of all the deviations (MPG for Japanese made cars) from the mean MPG for Japanese made cars? A B C D E (WMU) S1600 #6 S1600, Lecture 6 7 / 11
15 iclicker Question 6.2 Recall that an estimate is robust if it is insensitive to outliers. Which of the following statements is true. A. The sample mean and the standard deviation are robust. B. The sample mean is robust but the standard deviation is not. C. The sample mean is not robust but the standard deviation is. D. The sample mean and the standard deviation are not robust. E. None of the previous statements is true. (WMU) S1600 #6 S1600, Lecture 6 8 / 11
16 Effect of Multiplication/Addition by a Constant apartment rental example Recall that the mean and SD are $626 ± $104 (± means give or take ) get a roommate and pay half the rent: $323 ± $52 no roommate but has contribution of $100 per month from parents: $526 ± $104 (WMU) S1600 #6 S1600, Lecture 6 9 / 11
17 Effect of Multiplication/Addition by a Constant apartment rental example Recall that the mean and SD are $626 ± $104 (± means give or take ) get a roommate and pay half the rent: $323 ± $52 no roommate but has contribution of $100 per month from parents: $526 ± $104 (WMU) S1600 #6 S1600, Lecture 6 9 / 11
18 Effect of Multiplication/Addition by a Constant apartment rental example Recall that the mean and SD are $626 ± $104 (± means give or take ) get a roommate and pay half the rent: $323 ± $52 no roommate but has contribution of $100 per month from parents: $526 ± $104 (WMU) S1600 #6 S1600, Lecture 6 9 / 11
19 General Rules when a constant is added to/subtracted from each data value, the same thing happens to the average, but the SD remains unchanged. when each data value is multiplied or divided by a positive constant, the same thing happens to both the average and the SD (WMU) S1600 #6 S1600, Lecture 6 10 / 11
20 General Rules when a constant is added to/subtracted from each data value, the same thing happens to the average, but the SD remains unchanged. when each data value is multiplied or divided by a positive constant, the same thing happens to both the average and the SD (WMU) S1600 #6 S1600, Lecture 6 10 / 11
21 General Rules when a constant is added to/subtracted from each data value, the same thing happens to the average, but the SD remains unchanged. when each data value is multiplied or divided by a positive constant, the same thing happens to both the average and the SD (WMU) S1600 #6 S1600, Lecture 6 10 / 11
22 iclicker Question 6.3 Listed below are the annual salaries (in $1,000) of 5 employees in a company The mean and the standard deviation are, respectively, and If each employee is granted a $1,200 bonus (that is, 1.2 $1, 000), what will the mean and the standard deviation be? A. mean = (= ), SD = (= ) B. mean = 44.64, SD = (= ) C. mean = (= ), SD = D. mean = 44.64, SD = E. None of the previous. (WMU) S1600 #6 S1600, Lecture 6 11 / 11
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