3.3. Section. Measures of Central Tendency and Dispersion from Grouped Data. Copyright 2013, 2010 and 2007 Pearson Education, Inc.

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1 Section 3.3 Measures of Central Tendency and Dispersion from Grouped Data

2 Objectives 1. Approximate the mean of a variable from grouped data 2. Compute the weighted mean 3. Approximate the standard deviation of a variable from grouped data 3-2

3 Objective 1 Approximate the Mean of a Variable from Grouped Data 3-3

4 We have discussed how to compute descriptive statistics from raw data, but often the only available data have already been summarized in frequency distributions (grouped data). Although we cannot find exact values of the mean or standard deviation without raw data, we can approximate these measures using the techniques discussed in this section. 3-4

5 Approximate the Mean of a Variable from a Frequency Distribution Population Mean Sample Mean x i f i f i x x i f i f i x 1 f 1 x 2 f 2... x n f n f 1 f 2... f n x 1 f 1 x 2 f 2... x n f n f 1 f 2... f n where x i is the midpoint or value of the i th class f i is the frequency of the i th class n is the number of classes 3-5

6 EXAMPLE Approximating the Mean from a Relative Frequency Distribution The National Survey of Student Engagement is a survey that (among other things) asked first year students at liberal arts colleges how much time they spend preparing for class each week. The results from the 2007 survey are summarized below. Approximate the mean number of hours spent preparing for class each week. Hours Frequency Source: 3-6

7 Time Frequency x i x i f i xf i i 14,750 f i x xf i f i i 14,

8 Objective 2 Compute the Weighted Mean 3-8

9 The weighted mean, x w, of a variable is found by multiplying each value of the variable by its corresponding weight, adding these products, and dividing this sum by the sum of the weights. It can be expressed using the formula x w w i x i w i w 1x 1 w 2 x 2... w n x n w 1 w 2... w n where w is the weight of the i th observation x i is the value of the i th observation 3-9

10 EXAMPLE 3-10 Computed a Weighted Mean Bob goes to the Buy the Weigh Nut store and creates his own bridge mix. He combines 1 pound of raisins, 2 pounds of chocolate covered peanuts, and 1.5 pounds of cashews. The raisins cost $1.25 per pound, the chocolate covered peanuts cost $3.25 per pound, and the cashews cost $5.40 per pound. What is the cost per pound of this mix? 1($1.25) 2($3.25) 1.5($5.40) x w $15.85 $

11 Objective 3 Approximate the Standard Deviation of a Variable from Grouped Data 3-11

12 Approximate the Standard Deviation of a Variable from a Frequency Distribution Population Standard Deviation Sample Standard Deviation x i 2 f i f i s x i x 2 f i f i 1 where x i is the midpoint or value of the i th class f i is the frequency of the i th class 3-12

13 An algebraically equivalent formula for the population standard deviation is x i 2 f i x i f 2 f i f i 3-13

14 EXAMPLE Approximating the Standard Deviation from a Relative Frequency Distribution The National Survey of Student Engagement is a survey that (among other things) asked first year students at liberal arts colleges how much time they spend preparing for class each week. The results from the 2007 survey are summarized below. Approximate the standard deviation number of hours spent preparing for class each week. Hours Frequency Source: 3-14

15 Frequ Time ency x i x i x x i x f i , s s , hours , x i x f i 65,687.5 f i s 2 x i x f i 65, fi

16 Section 3.4 Measures of Position and Outliers

17 Objectives 1. Determine and interpret z-scores 2. Interpret percentiles 3. Determine and interpret quartiles 4. Determine and interpret the interquartile range 5. Check a set of data for outliers 3-17

18 Objective 1 Determine and Interpret z-scores 3-18

19 The z-score represents the distance that a data value is from the mean in terms of the number of standard deviations. We find it by subtracting the mean from the data value and dividing this result by the standard deviation. There is both a population z-score and a sample z-score: Population z-score Sample z-score z x z x x σ s The z-score is unitless. It has mean 0 and standard deviation

20 EXAMPLE Using Z-Scores The mean height of males 20 years or older is 69.1 inches with a standard deviation of 2.8 inches. The mean height of females 20 years or older is 63.7 inches with a standard deviation of 2.7 inches. Data is based on information obtained from National Health and Examination Survey. Who is relatively taller? Kevin Garnett whose height is 83 inches Candace Parker whose height is 76 inches or 3-20

21 z kg z cp Kevin Garnett s height is 4.96 standard deviations above the mean. Candace Parker s height is 4.56 standard deviations above the mean. Kevin Garnett is relatively taller. 3-21

22 Interpret Percentiles Objective

23 The kth percentile, denoted, P k, of a set of data is a value such that k percent of the observations are less than or equal to the value. 3-23

24 EXAMPLE Interpret a Percentile The Graduate Record Examination (GRE) is a test required for admission to many U.S. graduate schools. The University of Pittsburgh Graduate School of Public Health requires a GRE score no less than the 70th percentile for admission into their Human Genetics MPH or MS program. (Source: 01.) Interpret this admissions requirement. 3-24

25 EXAMPLE Interpret a Percentile In general, the 70 th percentile is the score such that 70% of the individuals who took the exam scored worse, and 30% of the individuals scores better. In order to be admitted to this program, an applicant must score as high or higher than 70% of the people who take the GRE. Put another way, the individual s score must be in the top 30%. 3-25

26 Objective 3 Determine and Interpret Quartiles 3-26

27 Quartiles divide data sets into fourths, or four equal parts. The 1 st quartile, denoted Q 1, divides the bottom 25% the data from the top 75%. Therefore, the 1 st quartile is equivalent to the 25 th percentile. The 2 nd quartile divides the bottom 50% of the data from the top 50% of the data, so that the 2 nd quartile is equivalent to the 50 th percentile, which is equivalent to the median. The 3 rd quartile divides the bottom 75% of the data from the top 25% of the data, so that the 3 rd quartile is equivalent to the 75 th percentile. 3-27

28 Finding Quartiles Step 1 Arrange the data in ascending order. Step 2 Determine the median, M, or second quartile, Q 2. Step 3 Divide the data set into halves: the observations below (to the left of) M and the observations above M. The first quartile, Q 1, is the median of the bottom half, and the third quartile, Q 3, is the median of the top half. 3-28

29 EXAMPLE Finding and Interpreting Quartiles A group of Brigham Young University Idaho students (Matthew Herring, Nathan Spencer, Mark Walker, and Mark Steiner) collected data on the speed of vehicles traveling through a construction zone on a state highway, where the posted speed was 25 mph. The recorded speed of 14 randomly selected vehicles is given below: 20, 24, 27, 28, 29, 30, 32, 33, 34, 36, 38, 39, 40, 40 Find and interpret the quartiles for speed in the construction zone. 3-29

30 EXAMPLE Finding and Interpreting Quartiles Step 1: The data is already in ascending order. Step 2: There are n = 14 observations, so the median, or second quartile, Q 2, is the mean of the 7 th and 8 th observations. Therefore, M = Step 3: The median of the bottom half of the data is the first quartile, Q 1. 20, 24, 27, 28, 29, 30, 32 The median of these seven observations is 28. Therefore, Q 1 = 28. The median of the top half of the data is the third quartile, Q 3. Therefore, Q 3 =

31 Interpretation: 25% of the speeds are less than or equal to the first quartile, 28 miles per hour, and 75% of the speeds are greater than 28 miles per hour. 50% of the speeds are less than or equal to the second quartile, 32.5 miles per hour, and 50% of the speeds are greater than 32.5 miles per hour. 75% of the speeds are less than or equal to the third quartile, 38 miles per hour, and 25% of the speeds are greater than 38 miles per hour. 3-31

32 Objective 4 Determine and Interpret the Interquartile Range 3-32

33 The interquartile range, IQR, is the range of the middle 50% of the observations in a data set. That is, the IQR is the difference between the third and first quartiles and is found using the formula IQR = Q 3 Q

34 EXAMPLE Determining and Interpreting the Interquartile Range Determine and interpret the interquartile range of the speed data. Q 1 = 28 Q 3 = 38 IQR Q 3 Q The range of the middle 50% of the speed of cars traveling through the construction zone is 10 miles per hour. 3-34

35 Suppose a 15 th car travels through the construction zone at 100 miles per hour. How does this value impact the mean, median, standard deviation, and interquartile range? Without 15 th car With 15 th car Mean 32.1 mph 36.7 mph Median 32.5 mph 33 mph Standard deviation 6.2 mph 18.5 mph IQR 10 mph 11 mph 3-35

36 Objective 5 Check a Set of Data for Outliers 3-36

37 Checking for Outliers by Using Quartiles Step 1 Determine the first and third quartiles of the data. Step 2 Compute the interquartile range. Step 3 Determine the fences. Fences serve as cutoff points for determining outliers. Lower Fence = Q 1 1.5(IQR) Upper Fence = Q (IQR) Step 4 If a data value is less than the lower fence or greater than the upper fence, it is considered an outlier. 3-37

38 EXAMPLE Determining and Interpreting the Interquartile Range Check the speed data for outliers. Step 1: The first and third quartiles are Q 1 = 28 mph and Q 3 = 38 mph. Step 2: The interquartile range is 10 mph. Step 3: The fences are Lower Fence = Q 1 1.5(IQR) = (10) = 13 mph Upper Fence = Q (IQR) = (10) = 53 mph Step 4: There are no values less than 13 mph or greater than 53 mph. Therefore, there are no outliers. 3-38

39 Section 3.5 The Five-Number Summary and Boxplots

40 Objectives 1. Compute the five-number summary 2. Draw and interpret boxplots 3-40

41 Objective 1 Compute the Five-Number Summary 3-41

42 The five-number summary of a set of data consists of the smallest data value, Q1, the median, Q3, and the largest data value. We organize the five-number summary as follows: 3-42

43 EXAMPLE Obtaining the Five-Number Summary Every six months, the United States Federal Reserve Board conducts a survey of credit card plans in the U.S. The following data are the interest rates charged by 10 credit card issuers randomly selected for the July 2005 survey. Determine the five-number summary of the data. 3-43

44 EXAMPLE Obtaining the Five-Number Summary Institution Rate Pulaski Bank and Trust Company 6.5% Rainier Pacific Savings Bank 12.0% Wells Fargo Bank NA 14.4% Firstbank of Colorado 14.4% Lafayette Ambassador Bank 14.3% Infibank 13.0% United Bank, Inc. 13.3% First National Bank of The Mid-Cities 13.9% Bank of Louisiana 9.9% Bar Harbor Bank and Trust Company 14.5% Source:

45 EXAMPLE Obtaining the Five-Number Summary First, we write the data in ascending order: 6.5%, 9.9%, 12.0%, 13.0%, 13.3%, 13.9%, 14.3%, 14.4%, 14.4%, 14.5% The smallest number is 6.5%. The largest number is 14.5%. The first quartile is 12.0%. The second quartile is 13.6%. The third quartile is 14.4%. Five-number Summary: 6.5% 12.0% 13.6% 14.4% 14.5% 3-45

46 Objective 2 Draw and Interpret Boxplots 3-46

47 Drawing a Boxplot Step 1 Determine the lower and upper fences. Lower Fence = Q 1 1.5(IQR) Upper Fence = Q (IQR) where IQR = Q 3 Q 1 Step 2 Draw a number line long enough to include the maximum and minimum values. Insert vertical lines at Q 1, M, and Q 3. Enclose these vertical lines in a box. Step 3 Label the lower and upper fences. 3-47

48 Drawing a Boxplot Step 4 Draw a line from Q 1 to the smallest data value that is larger than the lower fence. Draw a line from Q 3 to the largest data value that is smaller than the upper fence. These lines are called whiskers. Step 5 Any data values less than the lower fence or greater than the upper fence are outliers and are marked with an asterisk (*). 3-48

49 EXAMPLE Obtaining the Five-Number Summary Every six months, the United States Federal Reserve Board conducts a survey of credit card plans in the U.S. The following data are the interest rates charged by 10 credit card issuers randomly selected for the July 2005 survey. Construct a boxplot of the data. 3-49

50 EXAMPLE Obtaining the Five-Number Summary Institution Rate Pulaski Bank and Trust Company 6.5% Rainier Pacific Savings Bank 12.0% Wells Fargo Bank NA 14.4% Firstbank of Colorado 14.4% Lafayette Ambassador Bank 14.3% Infibank 13.0% United Bank, Inc. 13.3% First National Bank of The Mid-Cities 13.9% Bank of Louisiana 9.9% Bar Harbor Bank and Trust Company 14.5% Source:

51 Step 1: The interquartile range (IQR) is 14.4% - 12% = 2.4%. The lower and upper fences are: Lower Fence = Q 1 1.5(IQR) = (2.4) = 8.4% Upper Fence = Q (IQR) = (2.4) = 18.0% Step 2: * [ ] 3-51

52 Use a boxplot and quartiles to describe the shape of a distribution. The interest rate boxplot indicates that the distribution is skewed left. 3-52