# VCU STAT 210 - Lecture11 (68 pages)

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## Lecture11

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- Pages:
- 68
- School:
- Virginia Commonwealth University
- Course:
- Stat 210 - Basic Practice of Statistics

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STAT 210 Lecture 11 Measures of Spread September 20 2017 Test 2 Monday September 25 Sections III IV pages 47 95 Combination of multiple choice questions and short answer questions and problems Bring a calculator and writing instrument Practice Problems Pages 94 through 97 Relevant problems IV 3 IV 4 IV 5 IV 6 a and c and 11 c Recommended problems IV 4 IV 6 c and IV 11 c Additional Reading and Examples Pages 90 through 93 Motivating Example A statistics course at a large university provides free to students statistics review sessions that students can use to answer questions with help solving problems and with help studying for tests The course instructor is interested in the number of students who attend each hour of review session and selects a sample of 15 review session hours spread out over a month s time The number of students who attended these 15 review session hours is as follows This data will be used throughout the rest of this chapter 6 1 8 3 1 5 11 7 4 28 12 9 2 10 13 Top Hat 2 Measures of Central Location 1 Mean Average The population mean is denoted by the Greek letter m read mu and is the sum of all observations divided by how many individuals that there are in the population This is usually an unknown parameter Measures of Central Location The population mean is estimated by the sample mean denoted by X read X bar X S x x1 x2 x3 xn n n This is a statistic Measures of Central Location 2 Median The population median is usually denoted by the Greek letter h read eta and is estimated by the sample median denoted by M Top Hat B Measures of Spread If all the values of a characteristic are the same then the characteristic is a constant both the mean and median are the constant value and there is no spread in the data If however all the values are not the same then the characteristic is called a variable and of interest is to measure the amount of spread or dispersion or variability around the central value Measures of Spread Data set 1 10 10 10 Data set 2 8 10 12 Data set 3 5 10 15 Data set 4 0 10 20 Measures of Spread 1 Range maximum value minimum value The range is a measure of overall variation not variation around the central value The range can be heavily influenced by outliers Example 17 Without 391 The smallest observation is 113 and the largest observation is 222 So the range is 222 113 109 Example 17 With 391 The smallest observation is 113 and the largest observation is 391 So the range is 391 113 278 The range went from 109 to 278 due to the existence of one outlier B Measures of Spread 2 Standard Deviation The standard deviation is a measure of variability about the mean A deviation is the amount that an observation is from the mean x X B Measures of Spread 2 Standard Deviation The standard deviation is a measure of variability about the mean The population standard deviation is denoted by s read sigma Measures of Spread 2 Standard Deviation The standard deviation is a measure of variability about the mean The population standard deviation is denoted by s Since all subjects of the population are rarely known the population standard deviation is usually unknown and must be estimated by the sample standard deviation denoted S Sample Standard Deviation S S xi X 2 n 1 Sample Standard Deviation 1 Calculate the sample mean X Sample Standard Deviation 1 Calculate the sample mean X 2 Compute the n deviations from the mean x X The sum of the deviations x X will always equal 0 S x X 0 Sample Standard Deviation 1 Calculate the sample mean X 2 Compute the n deviations from the mean x X 3 Square each deviation x X 2 Sample Standard Deviation 1 Calculate the sample mean X 2 Compute the n deviations from the mean x X 3 Square each deviation x X 2 4 Sum the squared deviations S x X 2 Sample Standard Deviation 1 Calculate the sample mean X 2 Compute the n deviations from the mean x X 3 Square each deviation x X 2 4 Sum the squared deviations S x X 2 5 Divide this sum by n 1 S x X 2 n 1 The divisor n 1 is called the degrees of freedom Sample Standard Deviation 1 Calculate the sample mean X 2 Compute the n deviations from the mean x X 3 Square each deviation x X 2 4 Sum the squared deviations S x X 2 5 Divide this sum by n 1 S x X 2 n 1 6 Take the square root of the above number Variance A measure of spread around the mean that is related to the standard deviation is the variance The population variance is denoted by s2 read sigma squared and since the entire population is usually unknown the population variance is estimated using the sample variance s2 s2 S x X 2 n 1 Variance The population variance is estimated using the sample variance s2 s2 S x X 2 n 1 The standard deviation is preferred to the variance because while the standard deviation is measured in the units of the original data the variance is measured in the units squared Example 18 From example 13 the sample mean is X 162 625 Example 18 x x X x X 2 128 128 162 625 34 625 1198 89 150 150 162 625 12 625 159 39 183 183 162 625 20 375 415 14 222 222 162 625 59 375 3525 39 113113 162 625 49 625 2462 64 154 154 162 625 8 625 74 39 201 201 162 625 38 375 1472 64 150 150 162 625 12 625 159 39 0 9467 875 Example 18 x x X x X 2 128 128 162 625 34 625 1198 89 150 150 162 625 12 625 159 39 183 183 162 625 20 375 415 14 222 222 162 625 59 375 3525 39 113113 162 625 49 625 2462 64 154 154 162 625 8 625 74 39 201 201 162 625 38 375 1472 64 150 150 162 625 12 625 159 39 0 9467 875 Example 18 s S xi x 2 9467 875 1352 55 36 777 n 1 8 1 Standard Deviation The standard deviation is measured in the units of the original data So the answer to example 18 is 36 777 pounds Standard Deviation The standard deviation is the most often used measure of spread and together with the mean will be used in many of the inference problems that we consider later in the course However like the mean the standard deviation and the variance are influenced by outliers Example 19 From example 14 the sample mean is X 188 Example 19 x x X x X 2 128 128 188 60 3600 150 150 188 38 …

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