Measure of Variation

Transcription

1 Measure of Variation Variation is the spread of a data set. The simplest measure is the range. Range the difference between the maximum and minimum data entries in the set. To find the range, the data must be quantitative. Range = (Maximum data entry) (Minimum data enty) Two corporations each hired 10 graduates. The starting salaries for each graduate are shown. Find the range of starting salaries for Corporation A. Starting salaries for Corporation A (in thousands of dollars) SALARY For corporation A the maximum is 47 and minimum is 37. So the range would be = 10 thousand. Starting salaries for Corporation B (in thousands of dollars) SALARY Find the range of salaries for corporation B. Compare your answers to Corporation A. Which corporation would you rather work for? 1

2 The deviation of any entry x in a population data set is the difference between the entry and the mean μ of the data set. Going back to the example with Corporation A, the mean starting salary is Thus the deviation of starting salaries for Corporation A is: Salary Deviation x Notice that the sum of the deviations is 0. It does not make sense to find the average of the deviations and 0 divided by 10 is zero. To solve this problem, take the square of each deviation. The sum of the squares of the deviations, or sum of squares, is denoted by SS x. In a population data, the average of the squares of the deviation is the population variance. Variance is a measure based on the difference or distance each data value is from the mean. sum of the deviation 2

3 Population Variance of a population data set of N entries is The symbol σ is the lowercase Greek letter sigma. As a measure a variation, one disadvantage with the variance is that its units are different from the data set. Starting salaries vs. Starting Salaries squared. To overcome this issue, take the square root of the variance to get the standard deviation. The population standard deviation of a population data set N entries is the square root ot the population variance. Here are some observations about standard deviation. The standard deviation measures the variation of the data set about the mean and has the same units of measure as the data set. The standard deviation is always greater than or equal to 0. When σ = 0, the data set has no variation and all entries have the same value. As the entries get farther from the mean (that is, more spread out), the value of σ increases. 3

4 Guidelines for Finding Population Variance and Standard Deviation IN WORDS 1. Find the mean of the population data set IN SYMBOLS 2. Find the deviation of each entry. 3. Square each deviation 4. Add to get the sum of squares. 5. Divide by N to get the population variance. 6. Find the square root of the variance to get the population standard deviation. Exmple 2 P. 84 4

5 The sample variance and standard deviation of a sample data set of n entries are listed below. Sample Variance = Sample Standard Deviation = Dividing by n 1 gives us an unbiased sample standard deviation. Finding the Sample Variance and Standard Deviation IN WORDS 1. Find the mean of the sample data set. IN SYMBOLS 2. Find the deviation of each entry. 3. Square each deviation. 4. Add to get the sum of squares. 5. Divide by n 1 to get the sample variance. 6. Find the square root of the variance to get the sample sample standard deviation. 5

6 Example 3 on page 85. Try it yourself 3 page 86 2ND LIST L 1 STAT EDIT (type in data in L 1 ) STAT 1 VAR STATS 2nd LIST L 1 ENTER ENTER ENTER x bar is the sample mean and S x is the sample standard deviation. Please read Interpreting Standard Deviation on P. 87 Many real life data sets have distributions that are approximately symmetric and bell shaped. Empirical Rule or Rule For data sets with distributions that are approximately symmetric and bell shaped, the standard deviation has these characteristics. 1. About 68% of the data lie with one standard deviation of the mean. 2. About 95% of the data lie within two standard deviations of the mean. 3. About 99.7% of the data lie within three standard deviations of the mean. 6

7 Example 6 on page 88. a. How many standard deviations is 67.1 to the right of 64.2? b. Use the Empirical Rule to estimate the percent of the data between 64.2 and c. Interpret the result in the context of the data. The Empirical Rule only applies to symmetric Bell shaped distributions. The next theorem gives an inequality statement that applies to all distributions. CHEBYCHEV'S THEOREM The portion of any data set within k standard deviations (k > 1) of the mean is at least k = 2: In any data set, at least deviations of the mean. or 75%, of the data lies within 2 standard k = 3: In any data set, at least or 88.9% of the data lies within 3 standard deviations of the mean. 7

8 On page 89, Example 7 is an age distribution for New York and Alaska shown in histograms. Apply Chebychev's Theorem, if k = 2. What does this mean? If σ = 22.5 and that is the standard deviation, then if k = 2, that means that or 75% of the population falls within 2 standard deviations of the mean (22.5) = (22.5) = 6.7 Cannot have someone with a negative age, so we will cap it at 0. 75% of the population of New York is between 0 and 83.8 years old. Try It Yourself 7 on P. 89 8

9 Standard Deviation for Grouped Data When we have a large data set and it is represented by a frequency distribution, we will use this formula for the sample standard deviation where n = Σf is the number of entries in the data set. Using the table on the left side of P. 90, we can form the table: x f xf Σ=50 Σ=91 Σ=

10 The sample standard deviation is. When a frequency distribution has classes, you can estimate the sample mean and standard deviation by using the midpoint of each class. P. 91 When we want to compare variations of data sets with different units of measure or different means, we will use the coefficient of variation(cv). The coefficient of variation(cv) of a data set describes the standard deviation as a percent of the mean. Population: Sample: 10

11 Heights and Weights of a Basketball Team Heights Weights To find the CV Since 9.4% is larger than 4.5%, this means that the weights are more variable than the heights. 11

12 P odds, odds 12