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Chapter 3

3-1. Chapter 3. 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. 3-1 Chapter 3 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. Measure of central tendencyA single value which represents the distribution is called average(متوسط) ‌ . Since the averages tend to lie in the centre of a distribution they are called measure of central tendency. They are also called measure of location because they locate the centre of a distribution.

  7. Types of averages The most used averages are 1)The arithmetic mean 2)The geometric mean 3)The median and 4)The mode

  8. 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.

  9. 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.

  10. 3-2 The Mean (arithmetic average) 3-8 • The mean, in most cases, is not an actual data value.

  11. The Arithmetic Mean The Arithmetic mean is the most commonly used average. The arithmetic mean or simply the mean is defined as a value obtained by dividing the sum of the values by their number. Thus the mean of the values X1, X2 , X3 . . . Xn denoted by (read as X bar) or by “µ” is given by = When the data are grouped into a frequency distribution, we calculate arithmetic mean by using the following formula

  12. 3-2 The Population Mean 3-11

  13. 3-2 The Population Mean-Example 3-12

  14. 3-2 The Sample Mean 3-9

  15. 3-2 The Sample Mean - Example 3-10

  16. 3-2 The Sample Mean for an Ungrouped Frequency Distribution 3-13

  17. Example 2: Calculate Arithmetic Mean x f fx 1 5 5 2 13 26 3 4 12 22 43

  18. Find the mean from the following data(the score of students on 4 point quiz is given )

  19. 3-2 The Sample Mean for an Ungrouped Frequency Distribution - Example 3-14 Score, X Frequency, f Score, X Frequency, f 0 2 0 2 1 4 1 4 2 12 2 12 3 4 3 4 4 3 4 3 5 5

  20. × Score, X Frequency, f f X 0 2 0 1 4 4 2 12 24 3 4 12 4 3 12 3-2 The Sample Mean for an Ungrouped Frequency Distribution - Example 3-15 × Score, X Frequency, f f X 0 2 0 1 4 4 2 12 24 3 4 12 4 3 12 5 5

  21. The mean for a grouped frequency distributu ion 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

  22. 3-2 The Sample Mean for a Grouped Frequency Distribution -Example 3-17 Class Frequency, f Class Frequency, f 15.5 - 20.5 3 15.5 - 20.5 3 20.5 - 25.5 5 20.5 - 25.5 5 25.5 - 30.5 4 25.5 - 30.5 4 30.5 - 35.5 3 30.5 - 35.5 3 35.5 - 40.5 2 35.5 - 40.5 2 5 5

  23. × Class Frequency, f X f X m m 15.5 - 20.5 3 18 54 5 23 115 20.5 - 25.5 5 23 115 25.5 - 30.5 4 28 112 25.5 - 30.5 4 28 112 30.5 - 35.5 3 33 99 30.5 - 35.5 3 33 99 35.5 - 40.5 2 38 76 3-2 The Sample Mean for a Grouped Frequency Distribution -Example 3-18 × Class Frequency, f X f X m m 15.5 - 20.5 3 18 54 20.5 - 25.5 35.5 - 40.5 2 38 76 5 5

  24. Weighted Arithmetic mean Some times we want to find the average of certain values which are not of equal importance. When the values are not of equal importance , we assign them certain numerical values to express their relative importance. These numerical values are called weights. We calculate the weighted mean by using the following formula where w are weights.

  25. 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.

  26. 3-2 The Weighted Mean 3-48

  27. Example. A students final marks in Mathematics, Physics, English and statistics are respectively 82, 86, 90 and 70. If the respective credits received for these courses are 3,5,3 and 1. Determine an appropriate average marks.

  28. Example: In teacher selection we are checking the following points Experience, knowledge, behavior, Methodology ,personality and each one have different weight is shown in table.

  29. Example. A students final marks in Mathematics, Physics, English and statistics are respectively 82, 86, 90 and 70. If the respective credits received for these courses are 3,5,3 and 1. Determine an appropriate average marks. Example: Average Profit was $2.00 per order on 200 small orders and $4.60 per order on 50 large orders. Find average profit per order on all 250 orders.

  30. Total Value property of Arithmetic Mean ΣX = nX (To find ΣX multiply the mean by the number of data.) Example: Compute the total sales to 400 customers if the average sale per customer was $26.25. Solution: n=400 X=26.25 ΣX = nX = 400x 26.25= $10500

  31. The average and marginal addition Suppose we have produced 4 units at a cost of $220 so the average unit cost for the four is 220/4=$55 per unit. We wish to estimate what the additional cost for next unit, the fifth, should be if the average cost of the five is to be $50 per unit. Therefore total cost of the five units must be 5(50)= $250 as the cost of first 4 units is $220, the additional cost incurred by making the fifth must be therefore cost of 5 units – cost of 4 units= $250-$220=$30 The $30 is called the marginal cost of the fifth unit.

  32. Example: A student has an average of 61.25 on four tests. One test remaining and the course average will be the average of the five test grades. If 60 is the passing grade for the course, what is the lowest fifth test grade which will lead to a passing grade. Solution: Total grade of four tests= 4x61.25 = 245 Passing grade =60 Total passing grade of 5 tests = 5x60= 300 therefore the lowest fifth test grade = 300- 245= 55

  33. Q.4 A student has grades of 65, 70, 95 and 62 on four quizzes. If the student is to have an average of at least 75 after the next quiz. What is the lowest grade that can be received for next quiz. Q.5 Average cost when four units are produced in $10 per unit. What must be the marginal cost of the fifth unit if average cost is to decline to $9 per unit. Q.6 Compute the average number of new cars sold per day from the following table. Number of cars sold 0 1 2 3 4 Number of Days 3 19 48 15 5

  34. Q.7 Compute the average size of orders received from the following table. Size of order, dollars Relative frequency of orders 0.00 --------- 7.99 0.150 8.00 -------- 15.99 0.30 16.00 --------23.99 0.40 24.00 -------- 31.99 0.075 32.00 ---------39.99 0.075

  35. Q.8 A frequency distribution has class intervals of 3.00 and under 4.00, 4.00 and under 5.00, 5.00 and under 6.00 with respective frequencies of 4, 10,6. Estimate arithmetic mean for the distribution. Q.9 Last month an investor acquired 500 shares of a stock at an average price of $42 per share. This month the investor bought 500 shares of the stock at an average price of $36 per share. Find the average price per share paid during the two months.

  36. Q.3 A company has wage scale classes A,B and C. In period 1 there were 50 class A workers whose hourly wage rate was $4, 300 class B at $5 per hour and 150 class C at $6 per hour. In period 2, all hourly rates were increased 10 percent and the number of workers in classes A,B and C were respectively 160, 200 and 40. (a) Compute the average the hourly wage rate in period 1 (b) Compute the average the hourly wage rate in period 2 ( c) Explain why the answer of (a) and (b) are different.

  37. 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.

  38. 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.

  39. 3-2 The Median - Example 3-22 • Data array:180, 186, 191,201, 209, 219, 220. • The median,MD = 201.

  40. 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.

  41. 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.

  42. 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.

  43. 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.

  44. 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.

  45. 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.

  46. 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.

  47. 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. No. Sets Sold Frequency No. Sets Sold Frequency 1 4 1 4 2 9 2 9 3 6 3 6 4 2 4 2 5 3 5 3

  48. 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.

  49. 3-2 The Median-Ungrouped Frequency Distribution -Example 3-32 This class contains the 5th through the 13th values.

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