x Measures of Central Tendency for Ungrouped Data Chapter 3 Numerical Descriptive Measures Example 3-1 Example 3-1: Solution

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1 Chapter 3 umerical Descriptive Measures 3.1 Measures of Central Tendency for Ungrouped Data 3. Measures of Dispersion for Ungrouped Data 3.3 Mean, Variance, and Standard Deviation for Grouped Data 3.4 Use of Standard Deviation 3.5 Measures of Position 3. Box-and-Whisker Plot Mean 3.1 Measures of Central Tendency for Ungrouped Data The mean for ungrouped data is obtained by dividing the sum of all values by the number of values in the data set Mean for population data: x Median Mean for sample data: The median is the value of the middle term in a data set that has been ranked in increasing order Mode The mode is the value that occurs with the highest frequency in a data set Relationships among the Mean, Median, and Mode x x n 1 Example 3-1 Example 3-1: Solution Table 3.1 lists the total cash donations (rounded to millions of dollars) given by eight U.S. companies during the year 010 x x1 x x 3 x 4 x 5 x x 7 x x 111 x $139. 5million n 8 Thus, these eight companies donated an average of $139.5 million in 010 for charitable purposes. Find the mean of cash donations made by these eight companies 3 4 1

2 Example 3- The following are the ages (in years) of all eight employees of a small company: Find the mean age of these employees The population mean is x years Thus, the mean age of all eight employees of this company is 45.5 years, or 45 years and 3 months Example 3-3 Table 3. lists the total number of homes lost to foreclosure in seven states during 010 ote that the number of homes foreclosed in California is very large compared to those in the other six states. Hence, it is an outlier. Show how the inclusion of this outlier affects the value of the mean 5 Examples 3-3 Solution Median If we do not include the number of homes foreclosed in California (the outlier), the mean of the number of foreclosed homes in six states is Mean with out the outlier 49,73 01,9 0,35 10,84 40,911 18,038 1,848 33,1 ow, to see the impact of the outlier on the value of the mean, we include the number of homes foreclosed in California and find the mean number of homes foreclosed in the seven states. Mean with the outlier 173,175 49,73 0,35 10,84 40,911 18,038 1, ,871 53, How to find the median Rank the data set in increasing order. Find the middle term. The value of this term is the median. Example of weight lost: 10, 5, 19, 8, 3 Rank the data: 3, 5, 8, 10, 19 Find the median: 3, 5, 8, 10, 19 What if there are numbers: 3, 5, 8, 10, 13, 19 The median gives the center of a histogram, with half the data values to the left of the median and half to the right of the median. The advantage of using the median as a measure of central tendency is that it is not influenced by outliers. Consequently, the median is preferred over the mean as a measure of central tendency for data sets that contain outliers. 8

3 Example 3-4 Refer to the data on the number of homes foreclosed in seven states given in Table 3. of Example 3.3. Data: 173,175 49,73 0,35 10,84 40,911 18,038 1,848 Find the median for these data. Solution Rank the data: 10,84 18,038 0,35 40,911 49,73 1, ,175 Locate the middle term 10,84 18,038 0,35 40,911 49,73 1, ,175 Thus, the median number of homes foreclosed in these seven states was 40,911 in 010. Example 3-5 Table 3.3 gives the total compensations (in millions of dollars) for the year 010 of the 1 highest-paid CEOs of U.S. companies. Data: 3.9,.9, 8., 84.5, 1., 8.0, 7.1, 5., 3., 70.1,.5, 1.7 Find the median Solution Rank the data Locate the middle term Thus, the median is ( ) / = Mode The mode is the value that occurs with the highest frequency in a data set Example 3- The following data give the speeds (in miles per hour) of eight cars that were stopped on I-95 for speeding violations Find the mode. A major shortcoming of the mode is that a data set may have none or may have more than one mode, whereas it will have only one mean and only one median. Unimodal: A data set with only one mode. Bimodal: A data set with two modes. Multimodal: A data set with more than two modes. Relationships among Mean, Median, & Mode For a symmetric histogram and frequency curve with one peak, the values of the mean, median, and mode are identical, and they lie at the center of the distribution

4 Relationships among Mean, Median, & Mode For a histogram and a frequency curve skewed to the right, the value of the mean is the largest, that of the mode is the smallest, and the value of the median lies between these two. (otice that the mode always occurs at the peak point.) The value of the mean is the largest in this case because it is sensitive to outliers that occur in the right tail. These outliers pull the mean to the right. Relationships among Mean, Median, & Mode If a histogram and a distribution curve are skewed to the left, the value of the mean is the smallest and that of the mode is the largest, with the value of the median lying between these two. In this case, the outliers in the left tail pull the mean to the left Measures of Dispersion for Ungrouped Data Range Range = maximum value minimum value Variance Deviation from mean ( & ) Kind of average of squared deviation Population variance Sample variance Why divided by n-1, instead of by n for sample variance Standard deviation = square root of variance Why do we need both variance and standard deviation? Population parameters and sample statistics Range = largest value smallest value Example of two students test scores in a class A: ; B: Disadvantages The range, like the mean has the disadvantage of being influenced by outliers. Consequently, the range is not a good measure of dispersion to use for a data set that contains outliers. Its calculation is based on two values only: the largest and the smallest. All other values in a data set are ignored when calculating the range

5 Variance and Standard Deviation The standard deviation is the most used measure of dispersion. The value of the standard deviation tells how closely the values of a data set are clustered around the mean. In general, a lower value of the standard deviation for a data set indicates that the values of that data set are spread over a relatively smaller range around the mean. In contrast, a large value of the standard deviation for a data set indicates that the values of that data set are spread over a relatively large range around the mean. The standard deviation is obtained by taking the positive square root of the variance The variance calculated for population data is denoted by σ² (read as sigma squared), and the variance calculated for sample data is denoted by s². The standard deviation calculated for population data is denoted by σ, and the standard deviation calculated for sample data is denoted by s. 17 Calculation of Variance Get deviations from the mean & then square the deviations Kind of average of the squared deviations x (x - ) (x - ) = 5 5 = = 0 0 = = 0 0 = = 0 0 = = -5 (-5) = 5 18 Book s formula Way Basic Formulas for the Variance and Standard Deviation x Short-cut Formulas x x x and x x x x and s s n 1 n 1 x x x and x x x and s s n n 1 n n 1 Example 3-1 Until about 009, airline passengers were not charged for checked baggage. Around 009, however, many U.S. airlines started charging a fee for bags. According to the Bureau of Transportation Statistics, U.S. airlines collected more than $3 billion in baggage fee revenue in 010. The following table lists the baggage fee revenues of six U.S. airlines for the year 010. (ote that Delta s revenue reflects a merger with orthwest. Also note that since then United and Continental have merged; and American filed for bankruptcy and may merge with another airline.) Find the variance and standard deviation for these data next slide

6 Formula Solution of Example 3-1 Two Observations 1. The values of the variance and the standard deviation are never negative.. The measurement units of variance are always the square of the measurement units of the original data. The standard deviation, not the variance, is the most used measure of dispersion. s x,854 x 1,74,098 n n 1 1 1,74,098 1,357,55.7 s 77, , Example 3-13 Following are the 011 earnings (in thousands of dollars) before taxes for all six employees of a small company Calculate the variance and standard deviation for these data. ( x) x , Population Parameters and Sample Statistics A numerical measure such as the mean, median, mode, range, variance, or standard deviation calculated for a population data set is called a population parameter, or simply a parameter. A summary measure calculated for a sample data set is called a sample statistic, or simply a statistic. Slight difference in formula of population variance and sample variance $

7 3.3 Mean, Variance & Standard Deviation for Grouped Data Grouped data in frequency table Midpoint of each group (class) as approximate data point, and frequency as the number of such approximate points Then follow the conventional definitions for mean, variance, and standard deviation Mean for Grouped Data Mean for population data mf mf x n Mean for sample data where m is the midpoint and f is the frequency of a class Variance and Standard Deviation for Grouped Data Example 3-14 Table 3.8 gives the frequency distribution of the daily commuting times (in minutes) from home to work for all 5 employees of a company. Calculate the mean of the daily commuting times. Equivalent to data set: 5, 5, 5, 5; 15, 15, 15, 15, 15, 15, 15, 15, 15; 5, 5, 5, 5, 5, 5; 35, 35, 35, 35; 45, 45 Mean = 535 / 5 = Example 3-15 Table 3.10 gives the frequency distribution of the number of orders received each day during the past 50 days at the office of a mail-order company. Calculate the mean of orders Equivalent to data set: Mean = 83 / 50 = 1.4 Variance and Standard Deviation for Grouped Data Basic formulas f m f m x and s n 1 Short-cut formulas ( ) mf mf m f m f and s n n 1 where σ² is the population variance, s² is the sample variance, and m is the midpoint of a class. In either case, the standard deviation is obtained by taking the positive square root of the variance 7 8 7

8 Example 3-1 Example 3-1: Solution The following data, reproduced from Table 3.8 of Example 3-14, give the frequency distribution of the daily commuting times (in minutes) from home to work for all 5 employees of a company. Calculate the variance and standard deviation. m ( f mf ) (535 ) 14, minutes Thus, the standard deviation of the daily commuting times for these employees is 11. minutes. Recall the equivalent data set 9 30 Examples of Variance for Grouped Data Example 3-17: Table 3.10 gives the frequency distribution of the number of orders received each day during the past 50 days at the office of a mail-order company. Looks like we have to count on the formulas not really Mean is integer Computer packages: Excel, Minitab, etc. Illustrative example modified from example 3.1 data Time f m mf m f (m-mean) f 0 to < to < to < to < to < Sum Formula way Concept way σ = 3800/4 = Use of Standard Deviation Chebyshev s Theorem For any number k greater than 1, at least (1 1/k²) of the data values lie within k standard deviations of the mean. Empirical Rule For a bell shaped distribution approximately 8% of the observations lie within one standard deviation of the mean 95% of the observations lie within two standard deviations of the mean 99.7% of the observations lie within three standard deviations of the mean 3 8

9 Chebyshev s Theorem Example 3-18 The average systolic blood pressure for 4000 women who were screened for high blood pressure was found to be 187 with a standard deviation of. Using Chebyshev s theorem, find at least what percentage of women in this group have a systolic blood pressure between 143 and 31. μ = 187 and σ = k = (3), area 75% (89%) 143 μ = k = 44/ = The percentage is at least k () 4.75 or 75% Illustration of the Empirical Rule Example 3-19 The age distribution of a sample of 5000 persons is bellshaped with a mean of 40 years and a standard deviation of 1 years. Determine the approximate percentage of people who are 1 to 4 years old. 1. Compare the numbers with the mean. Check how many standard deviations the number is away from the mean

10 Example 3-19? The age distribution of a sample of 5000 persons is bellshaped with a mean of 40 years and a standard deviation of 1 years. Determine the approximate percentage of people who are 8 to 4 years old. Determine the approximate percentage of people who are 1 to 5 years old. Determine the approximate percentage of people who are 5 years or old. Determine the approximate percentage of people who are 8 years or young. 3.5 Measures of Position Quartiles: Q1, Q, Q3 Divide the ranked data into four equal parts Interquartile range = Q3 Q1 Box-and-Whisker Plot Percentile: P1, P,, P99 Divide the ranked data into 100 equal parts Q1 = P5, Q = P50 = median, Q3 = P75 Standard score Use for comparison of different subjects Quartiles and Percentiles Example 3-0 Table 3.3 in Example 3-5 gave the total compensations (in millions of dollars) for the year 010 of the 1 highest-paid CEOs of U.S. companies. That table is reproduced on the next slide. Find the values of the three quartiles. Where does the total compensation of Michael D. White (CEO of DirecTV) fall in relation to these quartiles? Find the interquartile range

11 Rank the data first Example 3-0: Solution By looking at the position of $3.9 million (total compensation of Michael D. White, CEO of DirecTV), we can state that this value lies in the bottom 75% of the 010 total compensation. This value falls between the second and third quartiles. IQR = Interquartile range = Q3 Q1 = = $7.45 million Example 3-1 The following are the ages (in years) of nine employees of an insurance company: Find the values of the three quartiles. Where does the age of 8 years fall in relation to the ages of the employees? Find the interquartile range. Solution Rank the data: Q1 = 30.5, Q = 37, Q3 = 49 The age of 8 falls in the lowest 5% of the ages IQR = Interquartile range = Q3 Q1 = = 18.5 Inclusive or exclusive in the calculations see Excel 41 4 Percentiles and Percentile Rank 3. Box-and-Whisker Plot Calculating Percentiles The (approximate) value of the k th percentile, denoted by P k, is kn P k Value of the th term in a ranked data set 100 where k denotes the number of the percentile and n represents the sample size 0 th percentile Ex 3-: Finding Percentile Rank of a Value umber of valuesless than xi Percentilerank of xi 100 Totalnumber of valuesin thedata set Examples 3- & 3-3 Different packages might give you different answers Excel Minitab Five-umber Summary Min, Q1, Q=Median, Q3, Max A plot that shows the center, spread, and skewness of a data set. It is constructed by drawing a box and two whiskers that use the median, the first quartile, the third quartile, and the smallest and the largest values in the data set between the lower and the upper inner fences. Formal BW plot Also used to detect potential outliers Box = Q1, Q, Q3 Whisker Inner fences: 1.5 times of IQR away from Q1(Q3) Outer fences: 3 times of IQR away from Q1(Q3)

12 Example 3-4 The following data are the incomes (in thousands of dollars) for a sample of 1 households Construct a box-and-whisker plot for these data. Step 1: rank the data and calculate Q1, Q, Q3 & IQR and draw box Step : calculate 1.5 x IQR and find the lower (upper) inner fence = Q1 (Q3) (+) 1.5 x IQR go beyond the box Step 3: locate the smallest (largest) values within the two inner fences Step 4: draw whiskers Step 5: uses and misuses Detecting outliers is a challenging problem Example 3-4 Q1 = 77, Q = 87, Q3 = 101, IQR = x IQR = 1.5 x 4 = 3 Lower inner fence = Q1 3 = 41, upper = Q3 + 3 = 137 Two whiskers: 9 (smallest value > 41) & 11 (largest < 137) 45 4 Standard Score Technology Instruction Standard score defined as Use for comparison An example: compare Mike s height (78 ins) with Rebecca s height (7 ins) BA players: average height = 9 ins & s =.8 ins Mike Jordan s std height is WBA players: average height = 3. ins & s =.5 ins Rebecca Lobo s std height is Who has advantage in height when they play? TI 84 / 83 plus Minitab Excel