McGraw-Hill/IrwinCopyright © 2009 by The McGraw-Hill Companies, Inc. All Rights Reserved. Chapter 3 Descriptive Statistics: Numerical Methods

3-2 Descriptive Statistics 3.1Describing Central Tendency 3.2Measures of Variation 3.3Percentiles, Quartiles and Box-and- Whiskers Displays

3-3 Describing Central Tendency In addition to describing the shape of a distribution, want to describe the data set’s central tendency –A measure of central tendency represents the center or middle of the data –“Center” means typical or regular in this setting.

3-4 Parameters and Statistics A population parameter is a number calculated from all the population measurements that describes some aspect of the population A sample statistic is a number calculated using the sample measurements that describes some aspect of the sample

3-5 Measures of Central Tendency Mean,  The average or expected value Median, M d The value of the middle point of the ordered measurements Mode, M o The most frequent value

3-6 The Mean Population X 1, X 2, …, X N  Population Mean Sample x 1, x 2, …, x n Sample Mean

3-7 The Sample Mean and is a point estimate, one-number estimate, of the population mean  It is the value to expect, on average and in the long run For a sample of size n, the sample mean is defined as

3-8 Example 3.1: The Car Mileage Case Example 3.1:Sample mean for first five car mileages from Table 3.1: 30.8, 31.7, 30.1, 31.6, 32.1

3-9 The Median The median M d is a value such that 50% of all measurements, after having been arranged in numerical order, lie above (or below) it 1.If the number of measurements is odd, the median is the middlemost measurement in the ordering, or (n+1)/2 th value in the ordered list. 2.If the number of measurements is even, the median is the average of the two middlemost measurements in the ordering, or the average of n/2 th and (n/2 +1) th values in the ordered list.

3-10 Example: Car Mileage Case Example 3.1: First five observations from Table 3.1: 30.8, 31.7, 30.1, 31.6, 32.1 In order: 30.1, 30.8, 31.6, 31.7, 32.1 There is an odd so median is one in middle, or 31.6

3-11 The Mode The mode M o of a population or sample of measurements is the measurement that occurs most frequently –Modes are the values that are observed “most typically” –Sometimes higher frequencies at two or more values If there are two modes, the data is bimodal If more than two modes, the data is multimodal –When data are in classes, the class with the highest frequency is the modal class The tallest box in the histogram

3-12 Suggested Exercise Page , 3.4

3-13 Data set: 1, 2, 3, 4 Mean= 2.5; median=2.5 Data set: 1, 2, 3, 4, 100 Mean= 22 median= 3 Mean Vs Median

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3-15 Mean Vs Median Compare with the mean, the median is resistant to extreme values. The median can resist the influence of the extreme values better than the mean.

3-16 Measures of Variation Data set 1: 4, 5, 6, 7, 8 Mean Data set 2: 1, 4, 6, 8, 11 Mean

McGraw-Hill/IrwinCopyright © 2009 by The McGraw-Hill Companies, Inc. All Rights Reserved. Data set 1: 4, 5, 6, 7, 8 Mean Data set 2: 1, 4, 6, 8, 11 Mean

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3-19 Measures of Variation Knowing the measures of central tendency is not enough Both of the distributions below have identical measures of central tendency

3-20 Measures of Variation RangeLargest minus the smallest measurement VarianceThe average of the squared deviations of all the population measurements from the population mean StandardThe square root of the variance Deviation

3-21 The Range Largest minus smallest Measures the interval spanned by all the data For Figure 3.13, largest repair time is 5 and smallest is 3 Range is 5 – 3 = 2 days

3-22 Variance For a population of size N, the population variance σ 2 is: For a sample of size n, the sample variance s 2 is:

3-23 Standard Deviation Population standard deviation (σ): Sample standard deviation (s):

3-24 Example: Chris’s Class Sizes This Semester Data points for a populaton are: 60, 41, 15, 30, 34 Mean is µ=36 Variance is: Standard deviation is:

3-25 Example: Sample Variance and Standard Deviation Example 3.7: sample data for first five car mileages from Table 3.1 are 30.8, 31.7, 30.1, 31.6, 32.1 The sample mean is 31.26

3-26 An alternative formula for the sample variance

3-27 Data points are: 60, 41, 15, 30, 34 Mean is 36, Sample variance is:

3-28 Percentiles, Quartiles, and Box-and- Whiskers Displays For a set of measurements arranged in increasing order, the p th percentile is a value such that p percent of the measurements fall at or below the value and (100-p) percent of the measurements fall at or above the value The first quartile Q 1 is the 25th percentile The second quartile (or median) is the 50 th percentile The third quartile Q 3 is the 75th percentile The interquartile range IQR is Q 3 - Q 1

3-29 Example: Quartiles 20 customer satisfaction ratings: i=25/100*20=5 i=75/100*20=15 (5.25, 11.25) M d = (8+8)/2 = 8 Q 1 = (7+8)/2 = 7.5 Q 3 = (9+9)/2 = 9 IQR = Q 3  Q 1 = 9  7.5 = 1.5

3-30 Calculating Percentiles 1.Arrange the measurements in increasing order 2.Calculate the index i=(p/100)n where p is the percentile to find 3.(a) If i is not an integer, round up and the next integer greater than i denotes the p th percentile (b) If i is an integer, the p th percentile is the average of the measurements in the i and i+1 positions

3-31 Percentile Example (p=10 th Percentile) i=(10/100)12=1.2 Not an integer so round up to 2 10 th percentile is in the second position so 11,070 Q1? i=(25/100)*12=3, 7,52411,07018,21126,81736,55141,286 49,31257,28372,81490,416135,540190,250

3-32 Percentile Example (p=25 th Percentile) i=(25/100)12=3 Integer so average values in positions 3 and 4 25 th percentile (18,211+26,817)/2 or 22,514 7,52411,07018,21126,81736,55141,286 49,31257,28372,81490,416135,540190,250

3-33 Five Number Summary 1.The smallest measurement 2.The first quartile, Q 1 3.The median, M d 4.The third quartile, Q 3 5.The largest measurement Displayed visually using a box-and- whiskers plot

3-34 Box-and-Whiskers Plots The box plots the: –first quartile, Q 1 –median, M d –third quartile, Q 3 –inner fences –outer fences

3-35 Box-and-Whiskers Plots Continued Inner fences: IQR= Q 3 –Q 1 –Located 1.5  IQR away from the quartiles: Q 1 – (1.5  IQR) Q 3 + (1.5  IQR) (Q 1 – (1.5  IQR), Q 3 + (1.5  IQR) ) ( *1.5, 9+1.5*1.5) (5.25, 11.25) Outer fences –Located 3  IQR away from the quartiles: Q 1 – (3  IQR) Q 3 + (3  IQR)

3-36 Box-and-Whiskers Plots Continued The “whiskers” are dashed lines that plot the range of the data –A dashed line drawn from the box below Q 1 down to the smallest measurement –Another dashed line drawn from the box above Q 3 up to the largest measurement

3-37 Box-and-Whiskers Plots Continued

3-38 Outliers Outliers are measurements that are very different from other measurements –They are either much larger or much smaller than most of the other measurements Outliers lie beyond the fences of the box-and- whiskers plot: less than Q 1 – (1.5  IQR) or greater than Q 3 + (1.5  IQR) –Measurements between the inner and outer fences are mild outliers –Measurements beyond the outer fences are severe outliers