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Chapter 2 Descriptive Statistics § 2.4 Measures of Variation Range The range of a data set is the difference between the maximum and minimum date entries in the set. Range = (Maximum data entry) – (Minimum data entry) Example: The following data are the closing prices for a certain stock on ten successive Fridays. Find the range. Stock 56 56 57 58 61 63 63 67 67 67 The range is 67 – 56 = 11. Larson & Farber, Elementary Statistics: Picturing the World, 3e 3 Deviation The deviation of an entry x in a population data set is the difference between the entry and the mean μ of the data set. Deviation of x = x – μ Example: The following data are the closing prices for a certain stock on five successive Fridays. Find the deviation of each price. The mean stock price is μ = 305/5 = 61. Stock x 56 58 61 63 67 Σx = 305 Deviation x–μ 56 – 61 = – 5 58 – 61 = – 3 61 – 61 = 0 63 – 61 = 2 67 – 61 = 6 Σ(x – μ) = 0 Larson & Farber, Elementary Statistics: Picturing the World, 3e 4 Variance and Standard Deviation The population variance of a population data set of N entries is (x μ )2 2 Population variance = . N “sigma squared” The population standard deviation of a population data set of N entries is the square root of the population variance. 2 Population standard deviation = (x μ )2 “sigma” Larson & Farber, Elementary Statistics: Picturing the World, 3e N . 5 Finding the Population Standard Deviation Guidelines In Words In Symbols 1. Find the mean of the population data set. μ x N 2. Find the deviation of each entry. x μ 3. Square each deviation. x μ2 4. Add to get the sum of squares. SS x x μ 5. Divide by N to get the population variance. 6. Find the square root of the variance to get the population standard deviation. 2 Larson & Farber, Elementary Statistics: Picturing the World, 3e x μ 2 2 N x μ 2 N 6 Finding the Sample Standard Deviation Guidelines In Words In Symbols 1. Find the mean of the sample data set. x x n 2. Find the deviation of each entry. x x 3. Square each deviation. x x 2 4. Add to get the sum of squares. SS x x x 5. Divide by n – 1 to get the sample variance. 6. Find the square root of the variance to get the sample standard deviation. x x s n 1 2 s Larson & Farber, Elementary Statistics: Picturing the World, 3e 2 2 x x n 1 2 7 Finding the Population Standard Deviation Example: The following data are the closing prices for a certain stock on five successive Fridays. The population mean is 61. Find the population standard deviation. Always positive! Stock x 56 58 61 63 67 Σx = 305 Deviation x–μ –5 –3 0 2 6 Σ(x – μ) = 0 Squared (x – μ)2 25 9 0 4 36 Σ(x – μ)2 = 74 SS2 = Σ(x – μ)2 = 74 2 x μ 2 N x μ N 74 14.8 5 2 14.8 3.8 3.85 σ $3.85 Larson & Farber, Elementary Statistics: Picturing the World, 3e 8 Interpreting Standard Deviation When interpreting standard deviation, remember that is a measure of the typical amount an entry deviates from the mean. The more the entries are spread out, the greater the standard deviation. 14 12 x=4 s = 1.18 10 8 6 4 Frequency Frequency 14 12 10 8 6 4 2 2 0 0 2 4 Data value 6 x =4 s=0 2 4 Data value Larson & Farber, Elementary Statistics: Picturing the World, 3e 6 9 Standard Deviation for Grouped Data (x x )2f Sample standard deviation = s n 1 where n = Σf is the number of entries in the data set, and x is the data value or the midpoint of an interval. Example: The following frequency distribution represents the ages of 30 students in a statistics class. The mean age of the students is 30.3 years. Find the standard deviation of the frequency distribution. Continued. Larson & Farber, Elementary Statistics: Picturing the World, 3e 10 Standard Deviation for Grouped Data The mean age of the students is 30.3 years. (x – x )2 (x – x )2f – 8.8 77.44 1006.72 8 – 0.8 0.64 5.12 37.5 4 7.2 51.84 207.36 42 – 49 45.5 3 15.2 231.04 693.12 50 – 57 53.5 2 23.2 538.24 1076.48 Class x f 18 – 25 21.5 13 26 – 33 29.5 34 – 41 n = 30 x–x 2988.80 (x x )2f 2988.8 s 103.06 10.2 n 1 29 The standard deviation of the ages is 10.2 years. Larson & Farber, Elementary Statistics: Picturing the World, 3e 11 Homework Page 72, 51 & 52 Read and take notes on Chapter 4 section 4 (74-83) Pg 84-91 # 1-12 Larson & Farber, Elementary Statistics: Picturing the World, 3e 12 Homework Page 72, 51 & 52 Pg 84-91 # 1-12 Page 72, 51 a Mean 6.005 Median 6.01 b Mean 5.945 Median 6.01 c the mean is affected by the data entry error 52 a Mean 29.63 Median 18.3 b Mean 22.34 Median 17.25 c the mean is affected by the Canadian exports Read and take notes on Chapter 4 section 4 (74-83) Pg 84-91 # 1-12 Larson & Farber, Elementary Statistics: Picturing the World, 3e 13 Homework Page 72, 51 & 52 Pg 84-91 # 1-12 Pg 84-91 # 1-12 2 range 10 mean 16.6 variance 10.2 standard deviation 3.2 4 range 19 mean 17.9 variance 59.6 standard deviation 7.7 6 Range 34-24 = 10 8 The deviation is the difference between the value and the mean. The sum of all of the deviations is zero 10 Neither measure can be zero the standard deviation is the square root of the variance 77777 12 3 3 3 7 7 7 Larson & Farber, Elementary Statistics: Picturing the World, 3e 14 Homework Pg 84-91 # 13-14, 17-19 Larson & Farber, Elementary Statistics: Picturing the World, 3e 15 Homework Pg 84-91 # 13-14, 17-19 13. For graph b, the more of the data is in the center and it is less spread out, therefore, it will have the smaller standard deviation (16). Graph a will have the larger standard deviation (24) 14 Graph b has more variability, therefore it would have a standard deviation of 5, graph a has a standard deviation of 2.4 17 Company B, larger standard deviation, more offers at the ends, but you would also be more likely to get an offer less than 29,000 Larson & Farber, Elementary Statistics: Picturing the World, 3e 16 Homework Pg 84-91 # 13-14, 17-19 18. The lower standard deviation suggests less variability…more consistency, so player B is more consistent. 19 Range Variance Standard Deviation Los Angeles 17.6 37.35 6.11 Long Beach 8.7 8.71 2.95 Based on the data it would suggest that the annual salaries in LA are more variable than the salaries in Long Beach Larson & Farber, Elementary Statistics: Picturing the World, 3e 17

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