CHAPTER 3 Data Description

1. CHAPTER 3Data Description

2. OUTLINE • 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 • 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 • Identify the position of a data value in a data set using various measures of position, such as percentiles, deciles and quartiles. • Use the techniques of exploratory data analysis, including stem and leaf plots, box plots, and five-number summaries to discover various aspects of data.

5. 3-1 Introduction • A statisticis a characteristic or measure obtained by using the data values from a sample. • A parameteris a characteristic or measure obtained by using the data values from a specific population.

6. 3-2 Measures of Central Tendency • Mean • Median • Mode • Mid-range

7. 3-2 The mean (arithmetric average) • The mean is defined to be the sum of the data values divided by the total number of values.

8. 3-2 The Sample Mean • The symbol X represents the sample mean. X is read as “X - bar”. The Greek symbol Σ is read as “sigma” and means “to sum”.

9. Example The following data represent the annual chocolate sales (in millions of RM) for a sample of seven states in Malaysia. Find the mean. RM2.0, 4.9, 6.5, 2.1, 5.1, 3.2, 16.6

10. Solution

11. 3-2 The Population Mean • The Greek symbol µ represents the population mean. The symbol µ is read as “mu”. N is the size of the finite population.

12. Example A small company consists of the owner, the manager, the salesperson, and two technicians. Their salaries are listed as RM 50,000, 20,000, 12,000, 9,000 and 9,000 respectively. Assume this is the population, find the mean.

13. Solution

14. Question In a random sample of 7 ponds, the number of fishes were recorded as the following, find the mean. 23 56 45 36 28 33 37

15. 3-2 The Sample Mean for an Ungrouped Frequency Distribution • The mean for an ungrouped frequency distribution is given by f = frequency of the corresponding value X n = f

16. Example The scores of 25 students on a 4-point quiz are given in the table. Find the mean score.

17. Solution

18. Question The number of balls in 17 bags were counted. Find the mean

19. 3-2 The Sample Mean for a Grouped Frequency Distribution • The mean for grouped frequency distribution is given by Xm= class midpoint = (UCL + LCL) / 2

20. Example The lengths of 40 bean pods were showed in the table. Find the mean.

21. Solution

23. Question Time (in minutes) that needed by a group of students to complete a game are shown as below. Find the mean.

24. Most common errors:

25. Example Mean = (f•X) / n = 23 / 12 = 1.92 Mean = (f • X) / n = (12 x 10) / 12 = 10

26. 3-2 The Median • When a data set is ordered, it is known as a data array. • The median is defined to be the midpoint of the data array. • The symbol used to denote the median is MD.

27. Example 1 The ages of seven preschool children are 1, 3, 4, 2, 3, 5, and 1. Find the median. 1. Arrange the data in order. 2. Select the middle point.

28. Data array: 1, 1, 2, 3, 3, 4, 5 Median The median (MD) age = 3 years.

29. 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. • When there is an even number of values in the data set, the median is obtained by taking the average of the two middle numbers.

30. Example 2 Six customers purchased these numbers of magazines: 1, 7, 3, 2, 3, 4. Find the median. 1. Arrange the data in order. 2. Select the middle point.

31. Data array: 1, 2, 3, 3, 4, 7 Median • The median (MD) = 3 + 3 • 2 • = 3

32. 3-2 The Median - Ungrouped Frequency Distribution • For an ungrouped frequency distribution, find the median by examining the cumulative frequencies to locate the middle value. • 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.

33. Example LRJ Appliance recorded the number of VCRs sold per week over a one-year period.

34. Solution • 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.

35. This class contains the 5th through the 13th values. Median

36. 3-2 The Median - Grouped Frequency Distribution • For grouped frequency distribution, find the median by using the formula as shown below: Median, MD = Lm + (W) n = sum of frequencies cf = cumulative frequency of the class immediately preceding the median class f = frequency of the median class w = class width of the median class Lm = Lower class boundary of the median class

37. Example Find the median by using the following data.

38. Solution

39. 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 26-30.

40. n = 17 cf = 8 f = 4 w = 30.5 – 25.5 = 5 = L 25.5 m - ( n 2 ) cf (17 / 2) – 8 = + + MD ( w ) L = ( 5 ) 25 . .5 f 4 m = 26.125.

41. Question Find the median by using the following data.

42. 3-2 The Mode • The mode is 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.

43. Example 1 Find the mode for the number of children per family for 10 selected families. Data set: 2, 3, 5, 2, 2, 1, 6, 4, 7, 3. Ordered set: 1, 2, 2, 2, 3, 3, 4, 5, 6, 7. Mode: 2.

44. Example 2 • 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 is no mode since each data value occurs equally with a frequency of one.

45. Example 3 • 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 18and 24.

46. 3-2 The Mode – Ungrouped Frequency Distribution • Example Find the mode by using the following data. Highest frequency Mode

47. The mode for grouped data is the modal class. • The modal class is the class with the largest frequency.

48. 3-2 The Mode – Grouped Frequency Distribution Example Find the mode by using the following data. Modal Class Highest frequency

49. 3-2 The Midrange • 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.

50. Example 1 • Last winter, the city of New York, 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.