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Data Description

Chapter 3. Data Description. Through this chapter you will learn Measure of Central tendency Measure of Dispersion Measure of Position. A statistic is a characteristic or measure obtained by using the data values from a sample.

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Data Description

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  1. Chapter 3 Data Description

  2. Through this chapter you will learn • Measure of Central tendency • Measure of Dispersion • Measure of Position

  3. A statistic is a characteristic or measure obtained by using the data values from a sample. • A parameter is a characteristic or measure obtained by using all the data values for a specific population.

  4. Population Arithmetic Mean X : Each value, N: Total number of values in the population

  5. Sample Arithmetic Mean X: Each value in the sample n: Total number of observations in the sample (sample size)

  6. Example 1 Find the mean of the following sample data 7 4 8 8 10 12 12 . X= 7+4+8+8+10+12+12 = 61

  7. Estimate the Mean of a Grouped Data into a Frequency Distribution f frequency of each class Xmclass midpoint of each class n Total number of frequencies

  8. Example 2 Given a frequency distribution Estimate the mean.

  9. Example 2 (Cont.) f Xm= 490 total n =f= 20

  10. Example 2 (Cont.)

  11. Median A median is the midpoint of the data array. Steps in finding the median of a data array: Step1: Arrange the data in order Step2: Select the midpoint of the array as the median.

  12. Example 3 2 3 6 7 7 8 9 9 9 10 10 We have 11 values. 8 is the exact middle value and hence it is the median. Find the median of the scores 7 2 3 7 6 9 10 8 9 9 10. Arrange the data in order to obtain

  13. Example 4 2 3 6 7 7 8 9 9 9 10 With these ten scores, no single score is at the exact middle. Instead, the two scores of 7 and 8 share the middle. We therefore find the mean of these two scores. Find the median of the scores 7 2 3 7 6 9 10 8 9 9 Arrange the data in order

  14. Example 4 (Cont.) the median is 7.5.

  15. The Estimate of Data Grouped into a Frequency Distribution

  16. LB Lower boundary of the median class n Total # of frequencies f frequency of the median class CF Cumulative frequency of the class preceding the median class. w class width

  17. Example 5 Given the frequency distribution as below. Estimate the median.

  18. Example 5 First find the cumulative frequency

  19. Example 5 w = 10, n = 50, and hence, n/2=25. The median falls in the class 60-69 ( 59.5-69.5)

  20. Example 6 Estimate the median for the frequency distribution below

  21. Mode • For grouped data into a frequency distribution, the estimate of mode can be the class midpoint of the modal class ( the class with the highest frequency) • It can also be found by the formula

  22. where • LB Lower boundary of the modal class • W class width • d1 difference between class frequency of the modal class and that of the class preceding it. • d2 difference between class frequency of the modal class and that of the class right after it.

  23. Example 7 Estimate the mode of the below distribution Modal class

  24. Example 7 (cont.) LB = 20.5 W = 5 d1= 5 - 3 =2 d2 = 5 – 4=1

  25. The Midrange

  26. Example 8 The midrange of this data set: 2, 3, 6, 8, 4, 1 is MR=(8+1)/2=4.5

  27. The Weighted Mean Xi : the values Wi : the weights

  28. Example 8 A student obtained 40, 50, 60, 80, and 45 marks in the subjects of Math, Statistics, Physics, Chemistry and Biology respectively. Assuming weights 5, 2, 4, 3, and 1 respectively for the above mentioned subjects. Find Weighted Arithmetic Mean per subject.

  29. Example 8 (cont.)

  30. Example 8 (cont.)

  31. Distribution Shapes

  32. Distribution Shapes (cont.)

  33. Distribution Shapes (cont.)

  34. Range The range is the highest value minus the lowest value. The symbol R is used for the range.

  35. Mean Deviation

  36. Example 9 The number of patients seen in the emergency room in a hospital for a sample of 5 days last year was: 103, 97, 101, 106, and 103. Determine the mean deviation and interpret.

  37. Example 9 First find the arithmetic mean

  38. Example 9 (Cont.)

  39. Example 9 (Cont.) Hence the mean deviation is 2.4 patients per day. The number of patients deviates, on average, by 2.4 patients from the mean of 102 patients per day.

  40. Example 10 The weight of a group of crates being shipped to Ireland is (in pounds) 95, 103, 105, 110, 104, 105, 112, and 90. a) What is the range of the weights? b) Compute the arithmetic mean weight. c) Compute the mean deviation of the weights. (answer: a) 22, b) 103, c) 5.25 pounds)

  41. Population Variance and Standard Deviation Remember: Standard deviation is the positive square root of variance.

  42. Example 11 Find the variance and standard deviation for the population data: 35, 45, 30, 35, 40, 25 Solution First find the arithmetic mean X= 35+ 45+ 30+ 35+40+25=210  = 210/6 = 35 then construct the table

  43. Example 11(cont.)

  44. Example 11(cont.) The population variance is The population standard deviation is

  45. Sample Variance and Standard Deviation Sample Variance (Conceptual formula) Sample Variance (Computational formula)

  46. Sample Variance and Standard Deviation (Cont.) • Sample Standard Deviation (Conceptual formula) • Sample Standard Deviation (Computational formula)

  47. Example 12 Find the sample variance and standard deviation for the amount of European auto sales for a sample of 6 years shown. The data are in millions of dollars. 11.2, 11.9, 12.0, 12.8, 13.4, 14.3

  48. Example 12 (Cont.) Method 1 Find the mean : 12.6 Total= 6.38

  49. Example 12 (Cont.) Method 1 The variance is defined by and hence, the standard deviation is

  50. Example 12 (Cont.) Method 2 We compute X= 11.2+11.9+12.0+12.8+13.4+14.3 =75.6 X2= 11.22 +11.92 +12.02 +12.82 +13.42 +14.32 =958.94 The variance is computed by Standard deviation is 1.13

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