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Chapter 3

Chapter 3. 3-1. Data Description. Outline. 3-2. 3-1 Introduction 3-2 Measures of Central Tendency 3-3 Measures of Variation 3-4 Measures of Position 3-5 Exploratory Data Analysis. Objectives. 3-3.

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Chapter 3

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  1. Chapter 3 3-1 Data Description

  2. Outline 3-2 • 3-1 Introduction • 3-2 Measures of Central Tendency • 3-3 Measures of Variation • 3-4 Measures of Position • 3-5 Exploratory Data Analysis

  3. Objectives 3-3 • Summarize data using the measures of central tendency, such as the mean, median, mode, and midrange. • Describe data using the measures of variation, such as the range, variance, and standard deviation.

  4. Objectives 3-4 • Identify the position of a data value in a data set using various measures of position, such as percentiles, deciles and quartiles.

  5. Objectives 3-5 • Use the techniques of exploratory data analysis, including stem and leaf plots, box plots, and five-number summaries to discover various aspects of data.

  6. 3-2 Measures of Central Tendency 3-6 • Astatisticis a characteristic or measure obtained by using the data values from a sample. • Aparameteris a characteristic or measure obtained by using the data values from a specific population.

  7. 3-2 The Mean (arithmetic average) 3-7 • Themeanis defined to be the sum of the data values divided by the total number of values. • We will compute two means: one for the sample and one for a finite population of values.

  8. 3-2 The Mean (arithmetic average) 3-8 • The mean, in most cases, is not an actual data value.

  9. 3-2 The Sample Mean 3-9 T h e s y m b o l X r e p r e s e n t s t h e s a m p l e m e a n . X i s r e a d a s " X - b a r " . T h e G r e e k s y m b o l i s r e a d a s " s i g m a " a n d i t m e a n s " t o s u m " .  X X . . . + X + + X = 1 2 n n X  . = n

  10. 3-2 The Sample Mean - Example 3-10 T h e a g e s i n w e e k s o f a r a n d o m s a m p l e o f s i x k i t t e n s a t a n a n i m a l s h e l t e r a r e 3 , 8 , 5 , 1 2 , 1 4 , a n d 1 2 . F i n d t h e a v e r a g e a g e o f t h i s s a m p l e . T h e s a m p l e m e a n i s  X 3 + 8 + 5 + 1 2 + 1 4 + 1 2 X = = n 6 5 4 = 9 w e e k s . = 6

  11. 3-2 The Population Mean 3-11 T h e G r e e k s y m b o l r e p r e s e n t s t h e p o p u l a t i o n m m e a n . T h e s y m b o l i s r e a d a s " m u " . m N i s t h e s i z e o f t h e f i n i t e p o p u l a t i o n . X X . . . + X + + = m 1 2 N N X  . = N

  12. A s m a l l c o m p a n y c o n s i s t s o f t h e o w n e r , t h e m a n a g e r , t h e s a l e s p e r s o n , a n d t w o t e c h n i c i a n s . T h e s a l a r i e s a r e l i s t e d a s $ 5 0 , , , , , a n d r e s p e c t i v e l y A s s u m e t h i s i s t h e p o p u l a t i o n T h e n t h e p o p u l a t i o n m e a n w i l l b e X N 3-2 The Population Mean-Example 3-12 0 0 0 2 0 0 0 0 1 2 0 0 0 , 9 , 0 0 0 9 , 0 0 0 . ( . )  = m 5 0 , 0 0 0 + 2 0 , 0 0 0 + 1 2 , 0 0 0 + 9 , 0 0 0 + 9 , 0 0 0 = 5 = $ 2 0 , 0 0 0 .

  13. 3-2 The Sample Mean for an Ungrouped Frequency Distribution 3-13 T h e m e a n f o r a n u n g r o u p e d f r e q u e n c y d i s t r i b u t u i o n i s g i v e n b y ( f X )  · X = . n H e r e f i s t h e f r e q u e n c y f o r t h e c o r r e s p o n d i n g v a l u e o f X , a n d n = f . 

  14. 3-2 The Sample Mean for an Ungrouped Frequency Distribution - Example 3-14 The scores for 25 students on a 4 – point quiz are given in the table. Find the mean score S c o r e , X F r e q u e n c y , f S c o r e , X F r e q u e n c y , f 0 2 0 2 1 4 1 4 2 1 2 2 1 2 3 4 3 4 4 3 4 3 5 5

  15. ? S c o r e , X F r e q u e n c y , f f X 0 2 0 1 4 4 2 1 2 2 4 3 4 1 2 3-2 The Sample Mean for an Ungrouped Frequency Distribution - Example 3-15 S c o r e , X F r e q u e n c y , f f X · 0 2 0 1 4 4 2 1 2 2 4 3 4 1 2 4 3 1 2 4 3 1 2 5 5 f X 5 2   X = = 2 . 0 8 . = n 2 5

  16. The mean for a grouped frequency distribution is given by × å ( f X ) X = . m n Here X is the correspond ing m class midpoint. 3-2 The Sample Mean for a Grouped Frequency Distribution 3-16

  17. C l a s s F r e q u e n c y , f 1 5 . 5 - 2 0 . 5 3 2 0 . 5 - 2 5 . 5 5 2 5 . 5 - 3 0 . 5 4 3 0 . 5 - 3 5 . 5 3 3 5 . 5 - 4 0 . 5 2 3-2 The Sample Mean for a Grouped Frequency Distribution -Example 3-17 C l a s s F r e q u e n c y , f 1 5 . 5 - 2 0 . 5 3 2 0 . 5 - 2 5 . 5 5 2 5 . 5 - 3 0 . 5 4 3 0 . 5 - 3 5 . 5 3 3 5 . 5 - 4 0 . 5 2 5 5

  18. ? C l a s s F r e q u e n c y , f X f X m m 1 5 . 5 - 2 0 . 5 3 1 8 5 4 2 0 . 5 - 2 5 . 5 5 2 3 1 1 5 2 5 . 5 - 3 0 . 5 4 2 8 3-2 The Sample Mean for a Grouped Frequency Distribution -Example 3-18 Table with class midpoints, Xm. C l a s s F r e q u e n c y , f X f X  m m 1 5 . 5 - 2 0 . 5 3 1 8 5 4 2 0 . 5 - 2 5 . 5 5 2 3 1 1 5 2 5 . 5 - 3 0 . 5 4 2 8 1 1 2 1 1 2 3 0 . 5 - 3 5 . 5 3 3 3 9 9 3 0 . 5 - 3 5 . 5 3 3 3 9 9 3 5 . 5 - 4 0 . 5 2 3 8 7 6 3 5 . 5 - 4 0 . 5 2 3 8 7 6 5 5

  19. 3-2 The Sample Mean for a Grouped Frequency Distribution -Example 3-19 f X 5 4 1 1 5 1 1 2 9 9 7 6 = + + + +   m = 4 5 6 a n d n = 1 7 . S o f X   X = m n 4 5 6 = 2 6 . 8 2 . = 1 7

  20. 3-2 The Median 3-20 • When a data set is ordered, it is called adata array. • Themedianis defined to be the midpoint of the data array. • The symbol used to denote the median isMD.

  21. 3-2 The Median - Example 3-21 • The weights (in pounds) of seven army recruits are 180, 201, 220, 191, 219, 209, and 186. Find the median. • Arrange the data in order and select the middle point.

  22. 3-2 The Median - Example 3-22 • Data array:180, 186, 191,201, 209, 219, 220. • The median,MD = 201.

  23. 3-2 The Median 3-23 • In the previous example, there was an odd number of values in the data set. In this case it is easy to select the middle number in the data array.

  24. 3-2 The Median 3-24 • When there is aneven numberof values in the data set, the median is obtained by taking theaverage of the two middle numbers.

  25. 3-2 The Median -Example 3-25 • Six customers purchased the following number of magazines: 1, 7, 3, 2, 3, 4. Find the median. • Arrange the data in order and compute the middle point. • Data array: 1, 2, 3, 3, 4, 7. • The median, MD = (3 + 3)/2 = 3.

  26. 3-2 The Median -Example 3-26 • The ages of 10 college students are: 18, 24, 20, 35, 19, 23, 26, 23, 19, 20. Find the median. • Arrange the data in order and compute the middle point.

  27. 3-2 The Median -Example 3-27 • Data array:18, 19, 19, 20,20,23, 23, 24, 26, 35. • The median,MD = (20 + 23)/2 = 21.5.

  28. 3-2 The Median-Ungrouped Frequency Distribution 3-28 • For an ungrouped frequency distribution, find the median by examining the cumulative frequencies to locate the middle value.

  29. 3-2 The Median-Ungrouped Frequency Distribution 3-29 • If n is the sample size, compute n/2. Locate the data point where n/2 values fall below and n/2 values fall above.

  30. 3-2 The Median-Ungrouped Frequency Distribution -Example 3-30 • LRJ Appliance recorded the number of VCRs sold per week over a one-year period. The data is given below. N o . S e t s S o l d F r e q u e n c y N o . S e t s S o l d F r e q u e n c y 1 4 1 4 2 9 2 9 3 6 3 6 4 2 4 2 5 3 5 3

  31. 3-2 The Median-Ungrouped Frequency Distribution -Example 3-31 • To locate the middle point, divide n by 2; 24/2 = 12. • Locate the point where 12 values would fall below and 12 values will fall above. • Consider the cumulative distribution. • The 12th and 13th values fall in class 2. Hence MD = 2.

  32. N o . S e t s S o l d F r e q u e n c y C u m u l a t i v e F r e q u e n c y 1 4 4 2 9 1 3 5 3 2 4 3-2 The Median-Ungrouped Frequency Distribution -Example 3-32 N o . S e t s S o l d F r e q u e n c y C u m u l a t i v e F r e q u e n c y 1 4 4 2 9 1 3 3 6 1 9 3 6 1 9 4 2 2 1 4 2 2 1 5 3 2 4 This class contains the 5th through the 13th values.

  33. The median can be computed from: - ( n 2 ) cf = + MD ( w ) L m f Where n = sum of the frequencies cf = cumulative frequency of the class immediately preceding the median class f = frequency of the median class w = width of the median class = the L lower boundary of median class m 3-2 The Median for a Grouped Frequency Distribution 3-33

  34. C l a s s F r e q u e n c y , f 1 5 . 5 - 2 0 . 5 3 2 0 . 5 - 2 5 . 5 5 2 5 . 5 - 3 0 . 5 4 3 0 . 5 - 3 5 . 5 3 3 5 . 5 - 4 0 . 5 2 3-2 The Median for a Grouped Frequency Distribution -Example 3-34 C l a s s F r e q u e n c y , f 1 5 . 5 - 2 0 . 5 3 2 0 . 5 - 2 5 . 5 5 2 5 . 5 - 3 0 . 5 4 3 0 . 5 - 3 5 . 5 3 3 5 . 5 - 4 0 . 5 2 5 5

  35. 3 2 0 . 5 - 2 5 . 5 5 8 2 5 . 5 - 3 0 . 5 4 1 2 3 0 . 5 - 3 5 . 5 3 1 5 3 5 . 5 - 4 0 . 5 2 1 7 3-2 The Median for a Grouped Frequency Distribution -Example 3-35 C l a s s F r e q u e n c y , f C u m u l a t i v e C l a s s F r e q u e n c y , f C u m u l a t i v e r e q u e n c y F F r e q u e n c y 1 5 . 5 - 2 0 . 5 3 1 5 . 5 - 2 0 . 5 3 3 2 0 . 5 - 2 5 . 5 5 8 2 5 . 5 - 3 0 . 5 4 1 2 3 0 . 5 - 3 5 . 5 3 1 5 3 5 . 5 - 4 0 . 5 2 1 7 5 5

  36. 3-2 The Median for a Grouped Frequency Distribution -Example 3-36 • To locate the halfway point, divide n by 2; 17/2 = 8.5  9. • Find the class that contains the 9th value. This will be the median class. • Consider the cumulative distribution. • The median class will then be 25.5 – 30.5.

  37. 3-2 The Median for a Grouped Frequency Distribution 3-37 n = 17 cf = 8 f = 4 w = 25.5 – 20.5 = 5 = L 25 . 5 m - ( n 2 ) cf (17 / 2) – 8 = + + MD ( w ) L = ( 5 ) 25 . 5 f 4 m = 26.125.

  38. 3-2 The Mode 3-38 • Themodeis defined to be the value that occurs most often in a data set. • A data set can have more than one mode. • A data set is said to have no mode if all values occur with equal frequency.

  39. 3-2 The Mode -Examples 3-39 • The following data represent the duration (in days) of U.S. space shuttle voyages for the years 1992-94. Find the mode. • Data set: 8, 9, 9, 14, 8, 8, 10, 7, 6, 9, 7, 8, 10, 14, 11, 8, 14, 11. • Ordered set: 6, 7, 7, 8, 8, 8, 8, 8, 9, 9, 9, 10, 10, 11, 11, 14, 14, 14.Mode = 8.

  40. 3-2 The Mode -Examples 3-40 • Six strains of bacteria were tested to see how long they could remain alive outside their normal environment. The time, in minutes, is given below. Find the mode. • Data set: 2, 3, 5, 7, 8, 10. • There isno modesince each data value occurs equally with a frequency of one.

  41. 3-2 The Mode -Examples 3-41 • Eleven different automobiles were tested at a speed of 15 mph for stopping distances. The distance, in feet, is given below. Find the mode. • Data set: 15, 18, 18, 18, 20, 22, 24, 24, 24, 26, 26. • There aretwo modes (bimodal). The values are 18 and 24. Why?

  42. 3-2 The Mode for an Ungrouped Frequency Distribution -Example 3-42 Given the table below, find the mode. V a l u e s F r e q u e n c y , f V a l u e s F r e q u e n c y , f 1 5 3 1 5 3 Mode 2 0 5 2 0 5 2 5 8 2 5 8 3 0 3 3 0 3 3 5 2 3 5 2 5 5

  43. 3-2 The Mode - Grouped Frequency Distribution 3-43 • Themode for grouped data is themodal class. • The modal class is the class with the largest frequency. • Sometimes the midpoint of the class is used rather than the boundaries.

  44. C l a s s F r e q u e n c y , f C l a s s F r e q u e n c y , f 1 5 . 5 - 2 0 . 5 3 1 5 . 5 - 2 0 . 5 3 2 0 . 5 - 2 5 . 5 5 2 0 . 5 - 2 5 . 5 5 2 5 . 5 - 3 0 . 5 7 2 5 . 5 - 3 0 . 5 7 3 0 . 5 - 3 5 . 5 3 3 0 . 5 - 3 5 . 5 3 3 5 . 5 - 4 0 . 5 2 5 3-2 The Mode for a Grouped Frequency Distribution -Example 3-44 Given the table below, find the mode. Modal Class 3 5 . 5 - 4 0 . 5 2 5

  45. 3-2 The Midrange 3-45 • The midrangeis found by adding the lowest and highest values in the data set and dividing by 2. • The midrange is a rough estimate of the middle value of the data. • The symbol that is used to represent the midrange isMR.

  46. 3-2 The Midrange -Example 3-46 • Last winter, the city of Brownsville, Minnesota, reported the following number of water-line breaks per month. The data is as follows: 2, 3, 6, 8, 4, 1. Find the midrange.MR = (1 + 8)/2 = 4.5. • Note:Extreme values influence the midrange and thus may not be a typical description of the middle.

  47. 3-2 The Weighted Mean 3-47 • Theweighted meanis used when the values in a data set are not all equally represented. • Theweighted meanof a variable Xisfound by multiplying each value by its corresponding weight and dividing the sum of the products by the sum of the weights.

  48. 3-2 The Weighted Mean 3-48 T h e w e i g h t e d m e a n w X w X . . . w X w X + + +  X = = 1 1 2 2 n n w w . . . w w + + +  1 2 n w h e r e w , w , . . . , w a r e t h e w e i g h t s 1 2 n f o r t h e v a l u e s X , X , . . . , X . 1 2 n

  49. Distribution Shapes 3-49 • Frequency distributions can assume many shapes. • The three most important shapes arepositively skewed, symmetrical, andnegatively skewed.

  50. Positively Skewed 3-50 Y P o s i t i v e l y S k e w e d X M o d e < M e d i a n < M e a n

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