Chapter 3: Averages and Variation Name Section 3.2: Measures of Variation Date Objectives: Find the range, variance, and standard deviation. Compute the coefficient of variation from raw data. This worksheet will walk you through the steps to create the range, variance, standard deviation, and coefficient of variation. These measures of central variation tell you about the dispersion, or spread, of a data set. Measures of central variation are also used in calculating other statistical measures and in inferential statistics. Instructions: Compute (A) the range and (B) the variance, standard deviation, and coefficient of variation of the mallard duck data set in Problem 19 in the textbook, which is reproduced here: A. Compute the range. Chapter 3: Averages and Variation Section 3.2: Measures of Variation Example: Find the range of the data set in part (f) of the Focus Problem at the beginning of Chapter 3, which is reproduced below. (Note: We will use this data set for the all of the Examples on this worksheet.) Note that the data is given in cubic meters. Make sure to consider this when stating and interpreting your answer. Range = Largest Data Value – Smallest Data Value = 5.81 – 3.81 =2 The range is 2 × 108 cubic meters. Instructions: Find the range of the mallard duck data set in Problem 19 in the textbook, which has been reproduced on Page 1 of this worksheet. Chapter 3: Averages and Variation Section 3.2: Measures of Variation B. Find the variance, standard deviation, and coefficient of variation. The main expressions and formulas are the mean, the sum of squares, the sample variance, and the sample standard deviation. Chapter 3: Averages and Variation Section 3.2: Measures of Variation The above procedure may seem very lengthy, but that’s because it has a lot of definitions and explanations, and in some cases, it gives two different forms of the defined expression. Focus on finding the mean, the sum of squares, the sample variation, and the sample standard deviation. You will work through these examples step-by-step. Once you get the hang of the procedure and the formulas, you may wish to use the defining or computational formulas to find the sample variation and sample standard deviation. STEP 1: Compute the mean. The mean is used to calculate the variance and the standard deviation. Example: Find the mean of the data set in part (f) of the Focus Problem at the beginning of Chapter 3. x x 3.83 3.81 4.01 4.84 5.81 5.50 4.31 5.81 4.31 4.67 n 42.09 10 4.209 4.21 10 The mean is 4.21 × 108 cubic meters. Instructions: Find the mean of the mallard duck data set in Problem 19 in the textbook, which has been reproduced on Page 1 of this worksheet. Chapter 3: Averages and Variation Section 3.2: Measures of Variation STEP 2: Compute the sum of squares. Example: Find the sum of squares of the data set in part (f) of the Focus Problem at the beginning of Chapter 3. Use the mean from Step 1. Sum of Squares Table Column I x Column II xx x x 2 3.83 3.83 – 4.21 = -0.38 (-0.38 × 108)2 = 0.1444 × 1016 3.81 3.81 – 4.21 = -0.4 (-0.4 × 108)2 = 0.16 × 1016 4.01 4.01 – 4.21 = -0.2 (-0.2 × 108)2 = 0.04 × 1016 4.84 4.84 – 4.21 = 0.63 (0.63× 108)2 = 0.3969 × 1016 5.81 5.81 – 4.21 = 1.6 (1.6 × 108)2 = 2.56 × 1016 5.50 5.50 – 4.21 = 1.29 (1.29 × 108)2 = 1.6641 × 1016 4.31 4.31 – 4.21 = 0.1 (0.1 × 108)2 = 0.01 × 1016 5.81 5.81 - 4.21 = 1.6 (1.6 × 108)2 = 2.56 × 1016 4.31 4.31 – 4.21 = 0.1 (0.1 × 108)2 = 0.01 × 1016 4.67 4.67 – 4.21 = 0.46 (0.46 × 108)2 = 0.2116 × 1016 x 42.09 Note: These are all × 108. This was given with the units in the original problem. For the mean, you could just stick this back on at the end of the calculation, since you were just adding and dividing. However, because the sum of squares involves squaring, you need to square each difference with 108 as part of the number so that the sum of squares and variance are correct. . Column III x x 2 7.757 × 1016 The sum of squares is 7.757 × 1016. Chapter 3: Averages and Variation Section 3.2: Measures of Variation Instructions: Using the table to guide you, find the sum of squares of the mallard duck data set in Problem 19 in the textbook, which has been reproduced at the beginning of this worksheet. (Note: You may wish to write each percentage from the problem in decimal form before calculating the sum of squares so that the decimal point is in the correct place after you square the differences.) Write the given percentages as decimal numbers so that you calculate the sum of squares and variance correctly. Sum of Squares Table Column I x x Column III Column II x x xx x x 2 2 Chapter 3: Averages and Variation Section 3.2: Measures of Variation STEP 3: Calculate the variance. In some cases, the variance is your target statistic; in other cases, you need to calculate it in order to find another statistic, such as the standard deviation. Example: Find the variance of the data set in part (f) of the Focus Problem at the beginning of Chapter 3. Use the sum of squares from Step 2. s 2 x x 2 n 1 7.757 1016 10 1 8.619 1015 There are 10 data values, so n = 10. The variance is 8.619 × 1015. Instructions: Find the variance of the mallard duck data set in Problem 19 in the textbook, which has been reproduced on Page 1 of this worksheet. Chapter 3: Averages and Variation Section 3.2: Measures of Variation STEP 4: Calculate the standard deviation. The variance is often your target statistic; in other cases, you need to calculate it in order to find another statistic, such as the standard deviation. Example: Find the standard deviation of the data set in part (f) of the Focus Problem at the beginning of Chapter 3. Use the variance from Step 3. s x x 2 n 1 8.619 1015 9.28 107 To find the standard deviation, take the square root of the variance, which you found in Step 3. The standard deviation is 9.28 × 107 cubic meters. Instructions: Find the standard deviation of the mallard duck data set in Problem 19 in the textbook, which has been reproduced on Page 1 of this worksheet. Chapter 3: Averages and Variation Section 3.2: Measures of Variation STEP 5: Compute the coefficient of variation. The coefficient of variation does not depend on units of measurement. It can therefore be a better tool to use than the standard deviation for comparing measurements from different populations. Example: Find the coefficient of variation of the data set in part (f) of the Focus Problem at the beginning of Chapter 3. Use the mean from Step 1 and the standard deviation from Step 4. CV s 100% x 9.28 107 100% 4.21 108 To find the coefficient of variation, divide the standard deviation, which you found in Step 5, by the mean, which root of the variance, which you found in Step 1. Then multiply the result by 100 and append a percent sign to the result. 0.22 22% The coefficient of variation is 22%. Note that it is expressed as a decimal or percent. Because both the mean and standard deviation have the same units, the units cancel out when they are divided. Instructions: Find the standard deviation of the mallard duck data set in Problem 19 in the textbook, which has been reproduced on Page 1 of this worksheet.