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Understandable Statistics Seventh Edition By Brase and Brase Prepared by: Lynn Smith Gloucester County College Chapter Three Averages and Variation Copyright (C) 2002 Houghton Mifflin Company. All rights reserved . 1 Measures of Central Tendency • Mode • Median • Mean Copyright (C) 2002 Houghton Mifflin Company. All rights reserved . 2 The Mode the value or property that occurs most frequently in the data Copyright (C) 2002 Houghton Mifflin Company. All rights reserved . 3 Find the mode: 6, 7, 2, 3, 4, 6, 2, 6 The mode is 6. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved . 4 Find the mode: 6, 7, 2, 3, 4, 5, 9, 8 There is no mode for this data. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved . 5 The Median the central value of an ordered distribution Copyright (C) 2002 Houghton Mifflin Company. All rights reserved . 6 To find the median of raw data: • Order the data from smallest to largest. • For an odd number of data values, the median is the middle value. • For an even number of data values, the median is found by dividing the sum of the two middle values by two. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved . 7 Find the median: Data: 5, 2, 7, 1, 4, 3, 2 Rearrange: 1, 2, 2, 3, 4, 5, 7 The median is 3. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved . 8 Find the median: Data: 31, 57, 12, 22, 43, 50 Rearrange: 12, 22, 31, 43, 50, 57 The median is the average of the middle two values = 31 43 37 2 Copyright (C) 2002 Houghton Mifflin Company. All rights reserved . 9 The Mean The mean of a collection of data is found by: • summing all the entries • dividing by the number of entries mean sum of all entries number of entries Copyright (C) 2002 Houghton Mifflin Company. All rights reserved . 10 Find the mean: 6, 7, 2, 3, 4, 5, 2, 8 6 7 2 3 4 5 2 8 37 mean 4.625 4.6 8 8 Copyright (C) 2002 Houghton Mifflin Company. All rights reserved . 11 Sigma Notation •The symbol means “sum the following.” • is the Greek letter (capital) sigma. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved . 12 Notations for mean Sample mean “x bar” Population mean x Greek letter (mu) Copyright (C) 2002 Houghton Mifflin Company. All rights reserved . 13 Number of entries in a set of data • If the data represents a sample, the number of entries = n. • If the data represents an entire population, the number of entries = N. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved . 14 Sample mean x x n Copyright (C) 2002 Houghton Mifflin Company. All rights reserved . 15 Population mean x N Copyright (C) 2002 Houghton Mifflin Company. All rights reserved . 16 Resistant Measure a measure that is not influenced by extremely high or low data values Copyright (C) 2002 Houghton Mifflin Company. All rights reserved . 17 Which is less resistant? • Mean • Median Copyright (C) 2002 Houghton Mifflin Company. All rights reserved . The mean is less resistant. It can be made arbitrarily large by increasing the size of one value. 18 Trimmed Mean a measure of center that is more resistant than the mean but is still sensitive to specific data values Copyright (C) 2002 Houghton Mifflin Company. All rights reserved . 19 To calculate a (5 or 10%) trimmed mean • Order the data from smallest to largest. • Delete the bottom 5 or 10% of the data. • Delete the same percent from the top of the data. • Compute the mean of the remaining 80 or 90% of the data. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved . 20 Compute a 10% trimmed mean: 15, 17, 18, 20, 20, 25, 30, 32, 36, 60 • Delete the top and bottom 10% • New data list: 17, 18, 20, 20, 25, 30, 32, 36 • 10% trimmed mean = x 198 24 .8 n 8 Copyright (C) 2002 Houghton Mifflin Company. All rights reserved . 21 Measures of Variation • Range • Standard Deviation • Variance Copyright (C) 2002 Houghton Mifflin Company. All rights reserved . 22 The Range the difference between the largest and smallest values of a distribution Copyright (C) 2002 Houghton Mifflin Company. All rights reserved . 23 Find the range: 10, 13, 17, 17, 18 The range = largest minus smallest = 18 minus 10 = 8 Copyright (C) 2002 Houghton Mifflin Company. All rights reserved . 24 The standard deviation a measure of the average variation of the data entries from the mean Copyright (C) 2002 Houghton Mifflin Company. All rights reserved . 25 Standard deviation of a sample s (x x) n 1 2 mean of the sample n = sample size Copyright (C) 2002 Houghton Mifflin Company. All rights reserved . 26 To calculate standard deviation of a sample • • • • • • Calculate the mean of the sample. Find the difference between each entry (x) and the mean. These differences will add up to zero. Square the deviations from the mean. Sum the squares of the deviations from the mean. Divide the sum by (n 1) to get the variance. Take the square root of the variance to get the standard deviation. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved . 27 The Variance the square of the standard deviation Copyright (C) 2002 Houghton Mifflin Company. All rights reserved . 28 Variance of a Sample ( x x ) 2 s n 1 Copyright (C) 2002 Houghton Mifflin Company. All rights reserved . 2 29 Find the standard deviation and variance x 30 26 22 78 x x 4 0 4 mean= 26 Copyright (C) 2002 Houghton Mifflin Company. All rights reserved . (x x ) Sum = 0 2 16 0 16 ___ 32 30 The variance s 2 ( x x) n 1 Copyright (C) 2002 Houghton Mifflin Company. All rights reserved . 2 = 32 2 =16 31 The standard deviation s= 16 4 Copyright (C) 2002 Houghton Mifflin Company. All rights reserved . 32 Find the mean, the standard deviation and variance mean = 5 x xx (x - x) 4 1 1 5 0 0 5 0 0 7 2 4 4 1 1 25 Copyright (C) 2002 Houghton Mifflin Company. All rights reserved . 6 33 2 The mean, the standard deviation and variance Mean =5 S tan dard deviation 1.5 1.22 Variance Copyright (C) 2002 Houghton Mifflin Company. All rights reserved . 6 1 .5 4 34 Computation formula for sample standard deviation: s SS x n 1 x 2 where SS x x Copyright (C) 2002 Houghton Mifflin Company. All rights reserved . 2 n 35 To find x 2 Square the x values, then add. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved . 36 To find ( x ) 2 Sum the x values, then square. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved . 37 Use the computing formulas to find s and s2 x x2 n=5 4 16 (Sx) 2 = 25 2 = 625 5 25 Sx2 = 131 5 25 SSx = 131 – 625/5 = 6 7 49 s2 = 6/(5 –1) = 1.5 4 25 16 131 Copyright (C) 2002 Houghton Mifflin Company. All rights reserved . s = 1.22 38 Population Mean and Standard Deviation x population mean N population standard deviation 2 x x N where N number of data values in the population Copyright (C) 2002 Houghton Mifflin Company. All rights reserved . 39 COEFFICIENT OF VARIATION: a measurement of the relative variability (or consistency) of data s CV 100 or 100 x Copyright (C) 2002 Houghton Mifflin Company. All rights reserved . 40 CV is used to compare variability or consistency A sample of newborn infants had a mean weight of 6.2 pounds with a standard deviation of 1 pound. A sample of three-month-old children had a mean weight of 10.5 pounds with a standard deviation of 1.5 pounds. Which (newborns or 3-month-olds) are more variable in weight? Copyright (C) 2002 Houghton Mifflin Company. All rights reserved . 41 To compare variability, compare Coefficient of Variation For newborns: For 3month-olds: CV = 16% Higher CV: more variable CV = 14% Lower CV: Copyright (C) 2002 Houghton Mifflin Company. All rights reserved . more consistent 42 Use Coefficient of Variation To compare two groups of data, to answer: Which is more consistent? Which is more variable? Copyright (C) 2002 Houghton Mifflin Company. All rights reserved . 43 CHEBYSHEV'S THEOREM For any set of data and for any number k, greater than one, the proportion of the data that lies within k standard deviations of the mean is at least: 1 1 k Copyright (C) 2002 Houghton Mifflin Company. All rights reserved . 2 44 CHEBYSHEV'S THEOREM for k = 2 According to Chebyshev’s Theorem, at least what fraction of the data falls within “k” (k = 2) standard deviations of the mean? 1 3 At least 1 2 2 4 75 % of the data falls within 2 standard deviations of the mean. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved . 45 CHEBYSHEV'S THEOREM for k = 3 According to Chebyshev’s Theorem, at least what fraction of the data falls within “k” (k = 3) standard deviations of the mean? 1 8 At least 1 3 2 9 88 . 9 % of the data falls within 3 standard deviations of the mean. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved . 46 CHEBYSHEV'S THEOREM for k =4 According to Chebyshev’s Theorem, at least what fraction of the data falls within “k” (k = 4) standard deviations of the mean? 1 15 At least 1 4 2 16 93 . 8 % of the data falls within 4 standard deviations of the mean. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved . 47 Using Chebyshev’s Theorem A mathematics class completes an examination and it is found that the class mean is 77 and the standard deviation is 6. According to Chebyshev's Theorem, between what two values would at least 75% of the grades be? Copyright (C) 2002 Houghton Mifflin Company. All rights reserved . 48 Mean = 77 Standard deviation = 6 At least 75% of the grades would be in the interval: x 2 s to x 2 s 77 – 2(6) to 77 + 2(6) 77 – 12 to 77 + 12 65 to 89 Copyright (C) 2002 Houghton Mifflin Company. All rights reserved . 49 Mean and Standard Deviation of Grouped Data • Make a frequency table • Compute the midpoint (x) for each class. • Count the number of entries in each class (f). • Sum the f values to find n, the total number of entries in the distribution. • Treat each entry of a class as if it falls at the class midpoint. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved . 50 Sample Mean for a Frequency Distribution xf x n x = class midpoint Copyright (C) 2002 Houghton Mifflin Company. All rights reserved . 51 Sample Standard Deviation for a Frequency Distribution s Copyright (C) 2002 Houghton Mifflin Company. All rights reserved . ( x x) f 2 n 1 52 Computation Formula for Standard Deviation for a Frequency Distribution SS x s n 1 where SSx x Copyright (C) 2002 Houghton Mifflin Company. All rights reserved . xf f 2 2 n 53 Calculation of the mean of grouped data Ages: f 30 - 34 x xf 32 128 37 185 42 84 47 xf = 820 423 4 35 - 39 5 40 - 44 2 f = 20 45 - 49 9 Copyright (C) 2002 Houghton Mifflin Company. All rights reserved . 54 Mean of Grouped Data xf xf x n f 820 41 . 0 20 Copyright (C) 2002 Houghton Mifflin Company. All rights reserved . 55 Calculation of the standard deviation of grouped data Ages: f x x– mean (x – mean)2 32 30 - 34 –9 81 37 4 80 –4 16 42 35 - 39 2 1 1 47 5 f = 20 40 - 44 Mean 324 6 Copyright (C) 2002 Houghton Mifflin Company. All rights reserved . 2 (x – mean)2 f 324 36 (x – mean)2 f = 730 56 Calculation of the standard deviation of grouped data x x 730 f = n = 20 2 ( x x) f s 2 n 1 730 20 1 38 . 42 6 . 20 Copyright (C) 2002 Houghton Mifflin Company. All rights reserved . 57 Computation Formula for Standard Deviation for a Frequency Distribution SS x s n 1 where SSx x Copyright (C) 2002 Houghton Mifflin Company. All rights reserved . xf f 2 2 n 58 Computation Formula for Standard Deviation x f xf x2f 32 4 128 4096 5 37 42 47 185 2 9 6845 3528 84 f = 20 xf = 820 423 Copyright (C) 2002 Houghton Mifflin Company. All rights reserved . 19881 x2f = 34350 59 Computation Formula for Standard Deviation for a Frequency Distribution where SS x f 2 x xf n 2 820 2 34350 730 20 SS x 730 s 6 . 20 n1 20 1 Copyright (C) 2002 Houghton Mifflin Company. All rights reserved . 60 Weighted Average Average calculated where some of the numbers are assigned more importance or weight Copyright (C) 2002 Houghton Mifflin Company. All rights reserved . 61 Weighted Average xw Weighted Average w where w the weight of the data value x. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved . 62 Compute the Weighted Average: • • • • • • Midterm grade = 92 Term Paper grade = 80 Final exam grade = 88 Midterm weight = 25% Term paper weight = 25% Final exam weight = 50% Copyright (C) 2002 Houghton Mifflin Company. All rights reserved . 63 Compute the Weighted Average: • Midterm • Term Paper • Final exam x 92 80 88 w .25 .25 .50 1.00 xw 23 20 44 87 xw 87 87 Weighted Average w 1.00 Copyright (C) 2002 Houghton Mifflin Company. All rights reserved . 64 Percentiles For any whole number P (between 1 and 99), the Pth percentile of a distribution is a value such that P% of the data fall at or below it. The percent falling above the Pth percentile will be (100 – P)%. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved . 65 Percentiles 60% of data P 40 Copyright (C) 2002 Houghton Mifflin Company. All rights reserved . Highest value Lowest value 40% of data 66 Quartiles • Percentiles that divide the data into fourths • Q1 = 25th percentile • Q2 = the median • Q3 = 75th percentile Copyright (C) 2002 Houghton Mifflin Company. All rights reserved . 67 Q1 Median = Q2 Q3 Highest value Lowest value Quartiles Inter-quartile range = IQR = Q3 — Q1 Copyright (C) 2002 Houghton Mifflin Company. All rights reserved . 68 Computing Quartiles • Order the data from smallest to largest. • Find the median, the second quartile. • Find the median of the data falling below Q2. This is the first quartile. • Find the median of the data falling above Q2. This is the third quartile. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved . 69 Find the quartiles: 12 23 41 15 24 45 16 25 51 16 30 17 32 18 33 22 33 22 34 The data has been ordered. The median is 24. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved . 70 Find the quartiles: 12 23 41 15 24 45 16 25 51 16 30 17 32 18 33 22 33 22 34 The data has been ordered. The median is 24. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved . 71 Find the quartiles: 12 23 41 15 24 45 16 25 51 16 30 17 32 18 33 22 33 22 34 For the data below the median, the median is 17. 17 is the first quartile. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved . 72 Find the quartiles: 12 23 41 15 24 45 16 25 51 16 30 17 32 18 33 22 33 22 34 For the data above the median, the median is 33. 33 is the third quartile. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved . 73 Find the interquartile range: 12 23 41 15 24 45 16 25 51 16 30 17 32 18 33 22 33 22 34 IQR = Q3 – Q1 = 33 – 17 = 16 Copyright (C) 2002 Houghton Mifflin Company. All rights reserved . 74 Five-Number Summary of Data • • • • • Lowest value First quartile Median Third quartile Highest value Copyright (C) 2002 Houghton Mifflin Company. All rights reserved . 75 Box-and-Whisker Plot a graphical presentation of the fivenumber summary of data Copyright (C) 2002 Houghton Mifflin Company. All rights reserved . 76 Making a Box-and-Whisker Plot • Draw a vertical scale including the lowest and highest values. • To the right of the scale, draw a box from Q1 to Q3. • Draw a solid line through the box at the median. • Draw lines (whiskers) from Q1 to the lowest and from Q3 to the highest values. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved . 77 Construct a Box-and-Whisker Plot: 12 23 41 15 24 45 16 25 51 16 30 17 32 18 33 22 33 Lowest = 12 Q1 = 17 median = 24 Q3 = 33 22 34 Highest = 51 Copyright (C) 2002 Houghton Mifflin Company. All rights reserved . 78 Box-and-Whisker Plot 60 55 50 45 40 35 30 25 20 15 - Lowest = 12 Q1 = 17 median = 24 Q3 = 33 Highest = 51 10 Copyright (C) 2002 Houghton Mifflin Company. All rights reserved . 79