3.1 Measure of Center

Transcription

1 3.1 Measure of Center Calculate the mean for a given data set Find the median, and describe why the median is sometimes preferable to the mean Find the mode of a data set Describe how skewness affects these measures of center 3.1-1

2 Measure of Center Measure of Center the value at the center or middle of a data set The three common measures of center are the mean, the median, and the mode

3 Mean the measure of center obtained by adding the data values and then dividing the total by the number of values What most people call an average also called the arithmetic mean

4 Notation Greek letter sigma used to denote the sum of a set of values. x n is the variable usually used to represent the data values. represents the number of data values in a sample

5 Example of summation If there are n data values that are denoted as: x, x2,, 1 x n Then: x x x 1 2 x n 3.1-5

6 data Then: Example of summation 21,25,32,48,53,62,62,64 x

7 Sample Mean x is pronounced x-bar and denotes the mean of a set of sample values x = x n 3.1-7

8 Example of Sample Mean data Then: 21,25,32,48,53,62,62,64 x

9 Notation µ Greek letter mu used to denote the population mean N represents the number of data values in a population

10 Population Mean µ = x N Note: here x represents the data values in the population

11 Advantages Mean Is relatively reliable: means of samples drawn from the same population don t vary as much as other measures of center Takes every data value into account

12 Mean Disadvantage Is sensitive to every data value, one extreme value can affect it dramatically; is not a resistant measure of center Example: 21,25,32,48,53,62,62,64 x ,25,32,48,53,62,62,300 x

13 Median Median the measure of center which is the middle value when the original data values are arranged in order of increasing (or decreasing) magnitude

14 Finding the Median First sort the values (arrange them in order), the follow one of these 1. If the number of data values is odd, the median is the number located in the exact middle of the list. Its position in the list is: n 2 1 th

15 Finding the Median 2. If the number of data values is even, the median is found by computing the mean of the two middle numbers which are those that lie on either side of the data value in the position: n 2 1 th

16 Example of Median 6 data values: Sorted data: (even number of values no exact middle) median

17 Example of Median 7 data values: Sorted data: median

18 Median Median is not affected by an extreme value - is a resistant measure of the center Example: 21,25,32,48,53,62,62,64 21,25,32,48,53,62,62,300 Median is 50.5 for both data sets

19 Median From Example 3.3, page

20 Mode the value that occurs with the greatest frequency Data set can have one, more than one, or no mode

21 Mode Bimodal two data values occur with the same greatest frequency Multimodal more than two data values occur with the same greatest frequency No Mode no data value is repeated Mode is the only measure of central tendency that can be used with nominal data

22 Mode - Examples a) b) c) Mode is 1.10 Bimodal - 27 & 55 No Mode

23 These data values represent weight gain or loss in kg for a random sample of18 college freshman (negative data values indicate weight loss) Do these values support the legend that college students gain 15 pounds (6.8 kg) during their freshman year? Explain

24 Sample Mean x n Median kg median (1 2) / kg Mode:

25 CONCLUSION All of the measures of center are below 6.8 kg (15 pounds) Based on measures of center, these data values do not support the idea that college students gain 15 pounds (6.8 kg) during their freshman year

26 Mean/Median with Graphing Calculator First, enter the list of data values Then select 2nd STAT (LIST) and arrow right to MATH option 3:mean( or 4: median( and input the desired list

27 Example of Computing the Mean Using Calculator Sorted amounts of Strontium-90 (in millibecquerels) in a simple random sample of baby teeth obtained from Philadelphia residents born after 1979 Note: this data is related to Three Mile Island nuclear power plant Accident in x = x n =

28 Example of Computing the Mean Using Calculator Median is

29 Skewed and Symmetric Symmetric distribution of data is symmetric if the left half of its histogram is roughly a mirror image of its right half Skewed distribution of data is skewed if it is not symmetric and extends more to one side than the other

30 Skewed Left or Right Skewed to the left (also called negatively skewed) have a longer left tail, mean and median are to the left of the mode Skewed to the right (also called positively skewed) have a longer right tail, mean and median are to the right of the mode

31 Distribution Skewed Left Mean is smaller than the median

32 Symmetric Distribution Mean, median, mode approximately equal

33 Distribution Skewed Right Mean is larger than the median

34 Example data set Mean: x n x Median: Distribution is skewed right

35 3.2 Measures of Variability The range What is a deviation? The standard deviation and the variance

36 Why is it important to understand variation? A measure of the center by itself can be misleading Example: Two nations with the same median family income are very different if one has extremes of wealth and poverty and the other has little variation among families (see the following table)

37 Example of variation Data Set A Data Set B 50,000 10,000 60,000 20,000 70,000 70,000 80, ,000 90, ,000 MEAN 70,000 70,000 MEDIAN 70,000 70,000 Data set B has more variation about the mean

38 Histograms: example of variation Data set B has more variation about the mean (Target)

39 How do we quantify variation?

40 Definition The range of a set of data values is the difference between the maximum data value and the minimum data value. Range = (maximum value) (minimum value)

41 Example of range. Data: Range = 30-6 =

42 Range (cont.) Ignoring the outlier of 6 in the previous data set gives data Range = = This shows that the range is very sensitive to extreme values; therefore not as useful as other measures of variation

43 Deviation The deviation for a given data value is the distance between the data value and the mean, except that the deviation can be negative while a distance is always positive

44 Deviation A deviation for a given data value is the difference between the data value and the mean of the data set. If x is the data value, 1. For a sample, the deviation of x is x x 2. For a population, the deviation of x is x

45 Deviation The deviation can be positive, negative, or zero. 1. If the data value is larger than the mean, the deviation will be positive. 2. If the data value is smaller than the mean, the deviation will be negative. 3. If the data value equals the mean, the deviation will be zero

46 Example: data 8,5,12,8,9,15,21,16,3 Mean x x n

47 Data Value Deviation

48 Population Variance The population variance is the mean of the squared deviations in the population x 2 N

49 Population Standard Deviation The population standard deviation is the square root of the population variance. x N

50 Sample Variance The sample variance is s 2 x x 2 n

51 Sample Variance Note that the sample variance is only approximately the mean of the squared deviations in the sample because we use n-1 instead of n

52 Sample Variance A statistic is an unbiased estimator of a parameter if its mean value equals the parameter it is trying to estimate. Using n-1 instead of n makes the sample variance an unbiased estimator of the population variance

53 Sample Standard Deviation The sample standard deviation is the square root of the sample variance. x x 2 s n

54 Steps to calculate the sample standard deviation 1. Calculate the sample mean 2. Find the squared deviations from the sample mean for each sample data value: ( x x) 2 3. Add the squared deviations 4. Divide the sum in step 3 by n-1 5. Take the square root of the quotient in step 4 x

55 Example: Standard Deviation Given the data set: 8, 5, 12, 8, 9, 15, 21, 16, 3 Find the standard deviation

56 Example: Standard Deviation Find the mean x x n

57 Data Value Squared Deviations From the Mean ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) (310.78)

58 Example: Standard Deviation Add the squared deviations (last column in the table above)

59 Example: Standard Deviation Divide the sum by 9-1=8: / Take the square root: s

60 Sample Standard Deviation (Computational Formula) x 2 x 2 / n s n

61 Example: Standard Deviation Data: Determine the standard deviation using the previous formula

62 Example: Standard Deviation We need to find each the following: n ( x 2 ) x x

63 Data Table (25 data values) TOTALS:

64 Example: Standard Deviation Thus: n 25 ( x 2 ) x x

65 Example: Standard Deviation And: s x 2 2 x / n n /

66 Standard Deviation - Important Properties The standard deviation is a measure of variation of all values from the mean. The value of the standard deviation s is never negative and usually not zero. The value of the standard deviation s can increase dramatically with the inclusion of one or more outliers (data values far away from all others). Unlike variance, the units of the standard deviation s are the same as the units of the original data values

67 Example: page

68 ANSWER Colony A range Colony B range

69 Example: page (b) Which colony has the greater variability according to the range? ANSWER: colony B

70 Use the previous example and calculate the standard deviation for each colony with a calculator

71 Data is stored in Lists. Locate and press the STAT button on the calculator. Choose EDIT. The calculator will display the first three of six lists (columns) for entering data. Simply type your data and press ENTER. Use your arrow keys to move between lists

72 Enter STAT then arrow right to CALC to get then press ENTER

73 Calculator Example When 1-Var Stats appears on the home screen, enter the name of the list containing the data. You can do this by entering List (= 2 nd STAT) and choosing which list has the desired data

74 1-Var Stats NOTE: Previous example data will give different values than these

75 ANSWER Colony A standard deviation = 21.9 Colony B standard deviation =

76 Compare histograms (SPSS)

77 3.3 Working with grouped data Calculate the weighted mean for a given data set Estimate the mean from grouped data Estimate the variance and standard deviation from grouped data

78 Weighted Mean When data values are assigned different weights, we can compute a weighted mean. Data values: x 1, x2, x3,..., x n Corresponding weights: w 1, w2, w3,..., w n

79 Computing the weighted mean Multiply each data value corresponding weight: w i x i by its Sum these products. Divide the result by the sum of the weights

80 Weighted Mean x w w i w x i i w x 1 1 w 1 w 2 x w 2 2 w w n n x n

81 Example: Weighted Mean Suppose homework/quiz average is weighted 10%, 2 exams are weighted 60%, and final exam is weighted 30%. If a student makes homework/quiz average 87, exam scores of 80 and 92, and final exam score 85, compute the weighted average

82 Example: Weighted Mean ANSWER: 0.10(87) 0.30(80) 0.30(92) (85)

83 Example: Weighted Mean

84 Employment Hourly Mean Wage ($) (weight) x (data value) 12, , , ,119, , , ,061, , Weights Data Values x w w i w x i i 5,060, ,180 = $

85 Estimating the mean from grouped data Given a frequency distribution, how do we compute the mean? Heights of 25 Women: HEIGHT (inches) FREQUENCY

86 Estimating the mean from grouped data Assume all sample values are at the class midpoints

87 Estimating the mean from grouped data Assume all sample values are at the class midpoints. HEIGHT (inches) FREQUENCY Class midpoints: 59.5, 61.5, 63.5, 65.5, 67.5, 69.5,

88 Estimating the mean from grouped data Multiply each class midpoint by its corresponding frequency Add the result Divide by the sum of the frequencies (total number of data values)

89 Estimating the mean from grouped data Class midpoints: 59.5, 61.5, 63.5, 65.5, 67.5, 69.5, 71.5 Frequencies: 3, 3, 4, 7, 6, 1, 1 Estimated mean: 3(59.5) 3(61.5) 4(63.5) 7(65.5) 6(67.5) 1(69.5) 1(71.5) / inches

90 Estimating the mean from a frequency distribution: Class midpoint = Frequency = mi fi ˆ m i f i f i m 1 f 1 f 1 m 2 f f 2 2 m f n n f n Note: ˆ is mu-hat where the hat denotes the fact the mu is not exact, but approximate

91 Estimating the variance from a frequency distribution: ( m ˆ) 2 ˆ 2 i f i f i Estimating the standard deviation from a frequency distribution: ˆ ( m i ˆ) f i 2 f i

92 HEIGHT (inches) ˆ 64.9 inches Frequency Class Midpoints ( m ˆ) 2 i f i ˆ ( m i ˆ) f i 2 f i inches

93 3.4 Measures of Position Find percentiles for small and large data sets Calculate z-scores and explain why we use them Use z-scores to detect outliers

94 Percentile Let p be any integer between 0 and 100. The pth percentile of a data set is a value for which p percent of the values in the data set are less than or equal to this value

95 Steps to find the pth percentile for small data sets Sort the data from small to large. If you are finding the pth percentile of a sample of size n, calculate: p i n 100 which is p percent of n

96 Steps to find the pth percentile for small data sets (cont) If i is an integer, the pth percentile is the mean of the data values in positions i and i+1. If i is not an integer, round up and use the value in this position as the pth percentile

97 Example of Finding Percentile Find the 25 th and 75 th percentiles of these 12 data values

98 Example of Finding Percentile 25% of 12 i % of 12 i

99 Example of Finding Percentile The data can be grouped as follows: 3 rd position % of the data is below 38.5 (the mean of 38 and 39). The 25 th percentile is

100 Example of Finding Percentile The data can be grouped as follows: 9th position % of the data is below 55 (the mean of 53 and 57). The 75 th percentile is

101 Example of Finding Percentile Find the 25 th and 75 th percentiles of these 7 data values:

102 Example of Finding Percentile 25% of 7 i % of 7 i

103 Example of Finding Percentile 1.75 round up to position nd position The 25 th percentile is

104 Example of Finding Percentile 5.25 round up to position th position The 75 th percentile is

105 Example of Finding Percentile Page

106 Example of Finding Percentile Data: Note: 15 data values 3.1 -

107 Example of Finding Percentile 16(a) To find position, 5% of 15 i rounded up to1 16(b) To find position, 95% of 15 i rounded up to

108 Example of Finding Percentile Data: (a) 5 th percentile is 2.0 million 16(b) 95 th percentile is 14.7 million 3.1 -

109 Z score z Score (or standardized value) the number of standard deviations that a given value x is above or below the mean 3.1 -

110 Z score Formulas Sample Population z x x z x s 3.1 -

111 Interpreting Z Scores 1. A z-score has no units. 2. Whenever a value is greater than the mean, its z score is positive 3. Whenever a value is less than the mean, its z score is negative 3.1 -

112 Example of Finding Z Score Page

113 Example of Finding Z score Data: Using calculator we get that x 5.1 and s

114 Example of Finding Z score (a) z-score for fish oil (data value is 4.2) z

115 Example of Finding Z score (a) z-score for Ginseng (data value is 8.8) z

116 Outliers An outlier is an extreme data value. We will define a data value as extreme if it is at least three standard deviations from the mean

117 Outliers and z Scores Data values are not unusual (exteme) if 2 z 2 Data values are unusual or outliers if z 3 or z

118 Interpreting Z Scores page 131: bell shaped distribution 3.1 -

119 3.5 Chebyshev s Rule and the Empirical Rule Calculate percentages using Chebyshev s Rule Find percentages and data values using the Empirical Rule 3.1 -

120 Chebyshev s Rule The proportion (or fraction) of any set of data lying within K standard deviations of the mean is always at least 1 1/K 2, where K is any positive number greater than 1. For K = 2, at least 3/4 (or 75%) of all values lie within 2 standard deviations of the mean. For K = 3, at least 8/9 (or 89%) of all values lie within 3 standard deviations of the mean

121 Example of Chebyshev s Rule Page 139, problem 11(a) A data distribution has a mean of 500 and a standard deviation of 100. Suppose we do not know whether the distribution is bell-shaped. (a) Estimate the proportion of data that falls between 300 and

122 Example of Chebyshev s Rule Page 139, problem 11(a) ANSWER: data values obey 300 x 700 First compute k using given x 300 x k s 500 and s 100 x k s k k

123 Example of Chebyshev s Rule Page 139, problem 11(a) ANSWER: Since k =2, k And Chebyshev s rule says that at least 75% of the data falls between 300 and

124 The Empirical Rule For data sets having a distribution that is approximately bell shaped, the following properties apply: About 68% of all values fall within 1 standard deviation of the mean. About 95% of all values fall within 2 standard deviations of the mean. About 99.7% of all values fall within 3 standard deviations of the mean

125 The Empirical Rule For data sets having a distribution that is approximately bell shaped, the following properties apply: About 68% of all data values obey 1 z 1 About 95% of all data values obey 2 z 2 About 99.7% of all data values obey 3 z

126 The Empirical Rule 3.1 -

127 The Empirical Rule 3.1 -

128 The Empirical Rule 3.1 -

129 The Empirical Rule page 136: bell shaped distribution Explain these percentages

130 The Empirical Rule For the green regions: 1. 50% of the data lies to the left of z= % (half of 68%) of the data lies between z=-1 and z=0. Therefore, 16% (=50%-34%) of the data is to the left of z= % (half of 95%) of the data lies between z=-2 and z=0. Therefore, 2.5% (=50%-47.5%) of the data is to the left of z= Subtracting areas gives that 13.5%=16%-2.5% of the data lies between z=-2 and z= Using symmetry, 13.5% of the data also lies between z=1 and z=

131 Example of Empirical Rule Page 139, problem 12(a) A data distribution has a mean of 500 and a standard deviation of 100. Assume that the distribution is bellshaped. (a) Estimate the proportion of data that falls between 300 and

132 Example of Empirical Rule Page 139, problem 12(a) ANSWER: data values obey 300 x 700 As in problem 11, we are given x 500 and s 100 so that the data in the interval are within 2 standard deviations of the mean 3.1 -

133 Example of Empirical Rule Page 139, problem 12(a) ANSWER: Here we are also given that the distribution is bell-shaped. Using the empirical rule, approximately 95% of the data lies between 300 and

134 Empirical Rule vs. Chebyshev s Rule Note the difference in problems 11 and 12: For problem 11 we are not told that the distributioin is bell-shaped and we can only say that at least 75% of data is between 300 and 700 (using Chebyshev s Rule). We cannot use the Empirical Rule in problem

135 Example Page 141, problem 24(a) In San Francisco the mean and standard deviation of the wind speed in January is 7.2 mph and 7.2 mph (Note these are population parameters) Assume that the distribution of the wind speed is bell-shaped. (a) Estimate the proportion of times that the wind speed is between 1.2 mph and 13.2 mph

136 Example Let the variable x represent wind speed so that 1.2 x 13.2 Note that the mean 7.2 is the midpoint of this interval. Calculate how many standard deviations from the mean this interval represents. Use the formula: k numerical value mean standard deviation 3.1 -

137 Example k 0.83 k so that the data in the interval are within 0.83 standard deviations of the mean. ANSWER The empirical rule implies that less than 68% of the time the windspeed is between 1.2 mph and 13.2 mph. Note: convice yourself of this by sketching areas below the bell-shaped distribution

138 Example Page 141, problem 24(b) (b) Estimate the proportion of times that the wind speed is less than 1.2 mph

139 Example Page 141, problem 24(b) 1. Since 0.0 mph is 1 standard deviation below the mean of 7.2 mph, the empirical rule implies that 34% (half of 68%) of the time, the windspeed is between 0.0 mph and 7.2 mph. 2. Subtracting 34% from 50% gives that 16% of the time the windspeed is less than 0.0 mph. 3. Since 1.2 mph is greater than 0.0 mph but less than 7.2 mph, we can say that at least 16% of the time but no more than 50% of the time the windspeed is less than 0.0 mph Note: convice yourself of this by sketching areas below the bell-shaped distribution

140 3.6 Robust Measures Find quartiles and the interquartile range Calculate the five number summary of a data set Construct a boxplot for a given data set Apply robust detection of outliers 3.1 -

141 Quartiles Are measures of location, denoted Q 1, Q 2, and Q 3, which divide a set of data into four groups with about 25% of the values in each group. Q 1 (First Quartile) is the 25 th percentile Q 2 (Second Quartile) is the 50 th percentile or the median Q 3 (Third Quartile) is the 75 th percentile 3.1 -

142 Example of Quartiles Given the 24 data values (sorted): Find Q, Q Q,

143 Example of Quartiles For first quartile (25 th percentile), position: i therefore the first quartile is the mean of the data values in positions 6 and 7 x x Q

144 Example of Quartiles For second quartile, (50 th percentile), position: i therefore the second quartile is the mean of the data values in positions 12 and 13 x x 5354 Q

145 Example of Quartiles For third quartile, (75 th percentile), position: i therefore the third quartile is the mean of the data values in positions 18 and 19 x x Q

146 Interquartile Range The Interquartile Range (IQR) is the difference between the third quartile and the first quartile which measures the spread of the middle 50% of the data: IQR Q Q 3 1 It is considered a robust measure of variability because it is not affected by outliers in the data (bottom 25% and top 25% of data are ignored)

147 Example of IQR Given the 24 data values (sorted): we found that Q and Q IQR Q3 Q

148 Example of IQR Introduce outliers into previous data set: we still have: Q and Q3 IQR Q3 Q

149 5-Number Summary For a set of data, the 5-number summary consists of the minimum value; the first quartile Q 1 ; the median (or second quartile Q 2 ); the third quartile, Q 3 ; and the maximum value

150 Example of Five Number Summary Given Data (sorted):

151 Example of Five Number Summary Minimum data value is 128 First quartile location 25 i round up to get Q x

152 Example of Five Number Summary Second quartile location i round up to get Q x

153 Example of Five Number Summary Third quartile location i round up to get Q x

154 Example of Five Number Summary Max data value is 166 Five Number Summary min Q Q Q max

155 Robust Detection of Outliers Using a five number summary, a data value is an outlier if 1. It is located 1.5(IQR) or more below the first quartile 2. It is located 1.5(IQR) or more above the third quartile 3.1 -

156 Robust Detection of Outliers Given the data set: has a five number summary: IQR Q3 Q

157 Robust Detection of Outliers Calculate: 1.5(IQR)=27 1.5(IQR) below the first quartile: 42-27=15 1.5(IQR) above the third quartile: =87 Therefore, 2,5,100,200 are outliers 3.1 -

158 Boxplot A boxplot (or box-and-whiskerdiagram) is a graph of a data set that consists of a line extending from the minimum value to the maximum value, and a box with lines drawn at the first quartile, Q 1 ; the median; and the third quartile, Q

159 Example of Boxplot 3.1 -

160 Example of Boxplot Sorted amounts of Strontium-90 (in millibecquerels) in a random sample of baby teeth obtained from Philadelphia residents born after 1979 Note: this data is related to Three Mile Island nuclear power plant Accident in

161 Five Number Summary Boxplot? Example of Boxplot Next slide: page 148 constructing a boxplot by hand or calculator or SPSS 3.1 -

162 Constructing a Boxplot Page 148 (see example 3.41) 3.1 -

163 Boxplot Example of Boxplot 3.1 -

164 Calculator Five Number Summary Enter the data in a list: 3.1 -

165 Calculator Five Number Summary Go to STAT - CALC and choose 1-Var Stats 3.1 -

166 Calculator Five Number Summary On the HOME screen, when 1-Var Stats appears, type the list containing the data

167 Calculator Five Number Summary Arrow down to the five number summary (last five items in the list) 3.1 -

168 Calculator Boxplot CLEAR out the graphs under y = (or turn them off). Enter the data into the calculator lists. (choose STAT, #1 EDIT and type in entries) 3.1 -

169 Calculator Boxplot Press 2nd STATPLOT and choose #1 PLOT 1. Be sure the plot is ON, the second box-andwhisker icon is highlighted, and that the list you will be using is indicated next to Xlist

170 Calculator Boxplot To see the box-and-whisker plot, press ZOOM and #9 ZoomStat. Press the TRACE key to see on-screen data about the box-and-whisker plot

171 Boxplot - Symmetric Distribution Normal Distribution: Heights from a Random Sample of Women NOTE: Q Q Q Q (max data value) Q Q2 3 (min data value) 3.1 -

172 Boxplot - Skewed Distribution Skewed Distribution: Salaries (in thousands of dollars) of NCAA Football Coaches 3.1 -