Refer to Ex 3-18 on page Record the info for Brand A in a column. Allow 3 adjacent other columns to be added. Do the same for Brand B.

Transcription

1 Refer to Ex 3-18 on page Record the info for Brand A in a column. Allow 3 adjacent other columns to be added. Do the same for Brand B.

2 Test on Chapter 3 Friday Sept 27 th. You are expected to provide you own calculator on the test.

3 3-2 Measures of Variation How Can We Measure Variability? Range Variance Standard Deviation Coefficient of Variation Chebyshev s Theorem Empirical Rule (Normal) 3

4 Measures of Variation: Range The range is the difference between the highest and lowest values in a data set. R Highest Lowest 4

5 Chapter 3 Data Description Section 3-2 Example 3-18/19 Page #123 5

6 Example 3-18/19: Outdoor Paint Two experimental brands of outdoor paint are tested to see how long each will last before fading. Six cans of each brand constitute a small population. The results (in months) are shown. Find the mean and range of each group. Brand A Brand B

7 Example 3-18: Outdoor Paint Brand A Brand B X 210 Brand A: N 6 R X 210 Brand B: 6 N R The average for both brands is the same, but the range for Brand A is much greater than the range for Brand B. Which brand would you buy? 7

8 Measures of Variation: Variance & Standard Deviation The variance is the average of the squares of the distance each value is from the mean. The standard deviation is the square root of the variance. The standard deviation is a measure of how spread out your data are. 8

9 Uses of the Variance and Standard Deviation To determine the spread of the data. To determine the consistency of a variable. To determine the number of data values that fall within a specified interval in a distribution (Chebyshev s Theorem). Used in inferential statistics. 9

10 Measures of Variation: Variance & Standard Deviation (Population Theoretical Model) The population variance is 2 X 2 N The population standard deviation is X 2 N 10

11 Chapter 3 Data Description Section 3-2 Example 3-21 Page #125 11

12 Example 3-21: Outdoor Paint Find the variance and standard deviation for the data set for Brand A paint. 10, 60, 50, 30, 40, 20 Months, X µ X - µ (X - µ) X 2 n Population Variance Population Standard Deviation 12

13 Measures of Variation: Variance & Standard Deviation (Sample Theoretical Model) The sample variance is s 2 X X 2 n 1 The sample standard deviation is s X X 2 n 1 13

14 Measures of Variation: Variance & Standard Deviation (Sample Computational Model) Is mathematically equivalent to the theoretical formula. Saves time when calculating by hand Does not use the mean Is more accurate when the mean has been rounded. 14

15 Measures of Variation: Variance & Standard Deviation (Sample Computational Model) The sample variance is s 2 n X 2 2 n n1 X The sample standard deviation is s s 2 15

16 Chapter 3 Data Description Section 3-2 Example 3-23 Page #129 16

17 Example 3-23: European Auto Sales Find the variance and standard deviation for the amount of European auto sales for a sample of 6 years. The data are in millions of dollars. 11.2, 11.9, 12.0, 12.8, 13.4, 14.3 X X s s 2 s n X 2 2 n n X / s s

18 Finding Variance and Standard deviation of Grouped Data Find the variance and the standard deviation for the frequency distribution of the data in the next slide. The data represents the number of miles that 20 runners ran during one week.

19 A Class B Frequency (f)

20 A B C D E Class Frequency (f) Midpoint (X m ) f X m f X m

21 s 2 f X 2 m f X m 2 n n 1 13,

22 Example 3-23 Find the sample variance and standard deviation for the amount of European auto sales for a sample of 6 years shown. The data are in millions of dollars. 11.2, 11.9, 12.0, 12.8, 13.4, 14.3

23 Example 3-24 Find he variance and the standard deviation for the frequency distribution of the data in Example 2-7. the data represent

24 Uses for standard deviation. 1. Spread of data 2. Consistency 3. Determine the number (or %) of data within an interval. 4. Many other used to be discussed fully second semester!

25 Measures of Variation: Coefficient of Variation The coefficient of variation is the standard deviation divided by the mean, expressed as a percentage. CVAR s X 100% Use CVAR to compare standard deviations when the units are different. 25

26 Chapter 3 Data Description Section 3-2 Example 3-25 Page #132 26

27 Example 3-25: Sales of Automobiles The mean of the number of sales of cars over a 3-month period is 87, and the standard deviation is 5. The mean of the commissions is $5225, and the standard deviation is $773. Compare the variations of the two. 5 CVar 100% 5.7% Sales CVar 100% 14.8% Commissions 5225 Commissions are more variable than sales. 27

28 Measures of Variation: Range Rule of Thumb The Range Rule of Thumb approximates the standard deviation as Range s 4 when the distribution is unimodal and approximately symmetric. 28

29 Measures of Variation: Range Rule of Thumb Use X value and 2s X to approximate the lowest 2s highest value in a data set. to approximate the Example: X 10, Range LOW s 3 4 HIGH

30 Using Symbols Let x =20 and s=3, calculator the following: 1. x + s = 2. x 2s = 3. x + 3s = 4. x ± 2.5s = 30

31 Measures of Variation: Chebyshev s Theorem The proportion of values from any data set that fall within k standard deviations of the mean will be at least 1 1 k 2, where k >1 k is not necessarily an integer. 31

32 Measures of Variation: Chebyshev s Theorem # of standard deviations, k Minimum Proportion within k standard deviations Minimum Percentage within k standard deviations 2 1-1/4=3/4 75% 3 1-1/9=8/ % 4 1-1/16=15/ % 32

33 Measures of Variation: Chebyshev s Theorem # of standard deviations, k Minimum Proportion within k standard deviations Minimum Percentage within k standard deviations 33

34 Measures of Variation: Chebyshev s Theorem 34

35 Chapter 3 Data Description Section 3-2 Example 3-27 Page #135 35

36 Example 3-27: Prices of Homes The mean price of houses in a certain neighborhood is $50,000, and the standard deviation is $10,000. Find the price range for which at least 75% of the houses will sell. Chebyshev s Theorem states that at least 75% of a data set will fall within 2 standard deviations of the mean. 50,000 2(10,000) = 30,000 50, (10,000) = 70,000 At least 75% of all homes sold in the area will have a price range from $30,000 and $75,

37 Chapter 3 Data Description Section 3-2 Example 3-28 Page #135 37

38 Example 3-28: Travel Allowances A survey of local companies found that the mean amount of travel allowance for executives was $0.25 per mile. The standard deviation was Using Chebyshev s theorem, find the minimum percentage of the data values that will fall between $0.20 and $ / / k / k 11/ At least 84% of the data values will fall between $0.20 and $

39 The Empirical Rule The empirical rule is only valid for bell-shaped (normal) distributions. The following statements are true. Approximately 68% of the data values fall within one standard deviation of the mean. Approximately 95% of the data values fall within two standard deviations of the mean. Approximately 99.7% of the data values fall within three standard deviations of the mean. The empirical rule will be revisited later in the chapter on normal probabilities.

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41 Measures of Variation: Empirical Rule (Normal) 41

42 Measures of Variation: Empirical Rule (Normal) The percentage of values from a data set that fall within k standard deviations of the mean in a normal (bell-shaped) distribution is listed below. # of standard deviations, k Proportion within k standard deviations 1 68% 2 95% % 42

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45 Homework Section 3-2 Page all, 7-17 every other odd, 19, every other odd 45

46 Application of Empirical Rule Given a data set comprised of 5057 measurements that is bell-shaped with a mean of 177. It has a standard deviation of 55. What percentage of the data should lie between 67 and 287? 46