MODULE II MEASURES OF CENTRAL TENDENCIES 2.1 CENTRAL TENDENCY Central tendency is the point (or value) about which all other scores in the distribution or data set tend to cluster. For this reason, it is often referred to as averages. It is a single number which represents the general level of performance of a group. The three commonly measures of central tendencies are: The Mean The Median The Mode 2.2 THE MEAN The mean is defined as the sum of the values in the data set or distribution divided by the number of values. It is the most commonly used measure of central tendency. In ungrouped data, it is called arithmetic mean. When to use the mean 1. When the frequency distribution or set of data does not contain either open ended or extreme value. 2. When the measure of central tendency having the greatest stability is desired. 3. When other parameters or statistics like standard deviation, variance, coefficient of correlation, etc. are to be computed later. 4. When dealing with either interval or ratio scale of measurement. ISHRM 2014 Learning Material Statistics MT 312/MT 322/TM Module II Calculating the Arithmetic Mean of ungrouped data Formula : x = ๐ด๐ฅ Where: x = score N = number of scores Σ = summation ๐ Illustrative example: Last week, a fruit vendor earned the following income from selling fruits at the Baclaran: Tuesday - โฑ 650, Wednesday - โฑ 900, Thursday - โฑ 720, Friday - โฑ 610, Saturday - โฑ 780, and Sunday - โฑ 950. What is the arithmetic mean of his income? Solution: Given: Σx = โฑ 650 + โฑ 900 + โฑ 720 + โฑ 610 + โฑ 780 + โฑ 950 = โฑ 4610 N = 6 (number of days) X = ๐ด๐ฅ ๐ , x= โฑ 4,610 6 , x = 768.33 Calculating the Weighted Mean of ungrouped data. The weighted arithmetic mean is expressed as the sum of the product each value and its corresponding weight divided by the sum of the weights. Formula: X = ๐ดƒ๐ฅ ๐ดƒ or xw = ๐ด๐ค๐ฅ ๐ด๐ค ISHRM 2014 Learning Material Statistics MT 312/MT 322/TM Module II Illustrative example: The final grades of a student at the end of the semester are the following: Subject Accounting Human Resource Mgt. Economics English Finance Grades (x) 2.25 2.50 1.75 1.50 2.00 Units (ƒ) 6 3 3 3 3 What is the weighted mean grade of the student? Solution: Xw = 6(2.25)+ 3(2.50) + 3(1.75) + 3(1.50) + 3(2.00) 6+ 3+ 3+ 3+ 3 = 36.75 18 = 2. 04 Calculating the Mean of grouped data Formula: X = ๐ดƒ๐ฅ ๐ for sample mean µ= ๐ดƒ๐ฅ ISHRM 2014 Learning Material Statistics MT 312/MT 322/TM Module II ๐ for population mean Illustrative example: Given the following scores of 50 selected tourism students who took the test in Algebra: 47 29 31 37 34 41 36 36 36 42 36 48 30 42 39 31 34 37 44 39 43 37 35 33 34 31 30 43 38 35 31 31 35 38 45 40 38 35 32 25 40 38 35 32 22 41 38 35 33 23 n = 50 Step 1. Tabulate the data in a grouped frequency distribution and determine the Frequency (ƒ), and the Class Mark (X). (See table below) Step 2. Multiply the frequency and the class mark in every class and get the sum. Grouped Frequency Distribution of 50 selected Tourism students who took the test in College Algebra. 48 45 42 39 36 33 30 27 24 21 ci - 50 - 47 - 44 - 41 - 38 - 35 - 32 - 29 - 26 - 23 Tally ƒ 1 1 11 2 11111 5 111111 6 111111111111 12 11111111111 11 111111111 9 1 1 1 1 11 2 n = 50 X ƒX 49 49 46 92 43 215 40 240 37 444 34 374 31 279 28 28 25 25 22 44 ΣƒX = 1,790 ISHRM 2014 Learning Material Statistics MT 312/MT 322/TM Module II Step 3. Solve for the sample mean ( x ). X = ๐ดƒ๐ฅ ๐ = 1,790 50 = 35.84 Question: What does the sample mean of 35.84 tell you as regard to the performance of 50 TM students in College Algebra test? Another method for solving the sample mean of a grouped frequency distribution is by the use of Deviation or Coded formula X = AM + ๐ดƒ๐ ๐ (แผฐ) Where: AM = Assumed Mean (X with the highest ƒ) แผฐ = interval ƒ = frequency d = deviation from AM Study the table below Grouped Frequency Distribution of 50 selected Tourism students who took the test in College Algebra. 48 45 42 39 36 33 30 27 24 21 ci - 50 - 47 - 44 - 41 - 38 - 35 - 32 - 29 - 26 - 23 ƒ 1 1 11 2 11111 5 111111 6 111111111111 12 11111111111 11 111111111 9 1 1 1 1 11 2 n= 50 Tally X 49 46 43 40 37 34 31 28 25 22 d ƒd 4 3 2 1 0 -1 -2 -3 -4 -5 Σƒd = Solve for the sample mean using the deviation or coded formula. (Seatwork) ISHRM 2014 Learning Material Statistics MT 312/MT 322/TM Module II 2.3 THE MEDIAN The Median (Md) is a measure of central tendency that occupies the middle position in the frequency distribution or set of data. It is the value that divides the distribution or data set into two, such that 50% of the data lies above the median and the other 50% lies below it. When to use the Median 1. When the data set or distribution is open-ended or contains extreme values. 2. When the data is arranged in descending or ascending order. 3. When the exact midpoint of the distribution is desired. 4. When dealing with ordinal data. Calculating the Median of ungrouped data A. When N is even, the median value is obtained using the formula: Md = ๐๐1 + ๐๐2 2 Illustrative example: Ten middle-management employees of a certain company have the following ages: 52, 45, 49, 48, 53, 42, 50, 53, 55, 58. ISHRM 2014 Learning Material Statistics MT 312/MT 322/TM Module II Find the median age. Arrange the data in ascending order: 42, 45, 48, 49, 50, 52, 53, 53, 55, 58 n = 10 (even number) Solution: Md = ๐๐1 + ๐๐2 2 = 50 + 52 2 = 51 B. When N is odd, the median is the value exactly at the middle of the data set arranged either in descending or ascending order. Illustrative example: The daily rates of a sample of nine employees of RomAn firm are: โฑ 520, โฑ 450, โฑ 560, โฑ 500, โฑ 720, โฑ 660, โฑ 840, โฑ 490, โฑ 870 What is the median daily rate? Arrange the data set in descending order โฑ 870, โฑ 840, โฑ 720, โฑ 660, โฑ 560, โฑ 520, โฑ 500, โฑ 490, โฑ 450 n = 9 (odd number) Solution: The median daily rate is the middle value โฑ 560 . ISHRM 2014 Learning Material Statistics MT 312/MT 322/TM Module II Calculating the Median of grouped data Frequency distribution of 50 selected TM students who took the test in Algebra ci ๐ 48 - 50 1 45 - 47 2 ห c๐ 47.5 – 50.5 ห c๐ 50 44.5 - 47.5 49 3 41.5 - 44.5 47 8 38.5 - 41.5 42 14 35.5 - 38.5 36 26 32.5 - 35.5 24 37 29.5 - 32.5 13 46 26.5 - 29.5 4 47 23.5 - 26.5 3 48 20.5 - 23.5 2 50 X CB 1 49 ๐AMd class 46 AMd class 42 - 44 5 c๐p 43 39 - 41 6 40 36 - 38 12 37 33 - 35 11 n = 50 34 30 - 32 9 31 27 - 29 1 28 24 - 26 1 25 21 - 23 2 22 Formula: ๐ −๐๐๐ 2 Md = LB + ใ ๐(๐ด๐๐) ใแผฐ , Where: LB C๐p (AMd) แผฐ - Lower class boundary the ห c๐ preceding the ห c๐ of the AMd class frequency of the AMd class interval Solution: Step 1. Find the AMd class applying 1 2 (๐) = 1 2 (50) = 25. Counting from ห c๐ of the lowest class and moving upward (arrow), 25 falls on ห c๐ 36. Therefore, AMd class is ใ36 – 38ใ. ISHRM 2014 Learning Material Statistics MT 312/MT 322/TM Module II Step 2. Write the values of the following variables with respect to the AMd : ๐AMd class = 12 LB = 35.5 แผฐ = 3 C๐p = 24 Step 3. Substitute the formula and solve for the Md. Md = LB + ใ 1 (๐)−๐๐๐ 2 ๐๐ด๐๐ 1 ใ แผฐ = 35.5 + ใ 1 (50)−24 2 12 ใ3 = 35.5 +ใ0.08 ใ3 Md = 35.5 + ใ 12 ใ 3 = 35.5 + 0.24 Md = 35.74 For purposes of checking this solution, the ห c๐ is used in the formula: 1 Md = UB - ใ 2 (๐)−๐๐๐ ๐๐๐ ใแผฐ 2.4 THE MODE The Mode (Mo) is that single data or score which occurs most frequently in a data set. In grouped frequency distribution, the Mode (called crude mode) is the midpoint with the highest frequency. ISHRM 2014 Learning Material Statistics MT 312/MT 322/TM Module II When to use the Mode 1. When a quick and approximate measure of central tendency is desired 2. When the data is either quantitative or qualitative. 3. When the frequency distribution contains open-ended or extreme values. 4. When dealing with nominal data. Calculating the Mode of ungrouped data Data set may have no mode. If it contains one mode, it is called Unimodal. If there are two modes, it is called Bimodal, three modes or Trimodal. Illustrative examples: 1. The following data represent the total unit sales for MYRO 2000from a sample of 10 Gaming Centers for the month of October: 16, 18, 11, 13, 14, 11, 15, 11, 6, 7. Find the mode. Arrange the data in ascending or descending order. 6, 7, 11, 11, 11, 13, 14, 15, 16, and 18. The Mode is 11. Unimodal. ISHRM 2014 Learning Material Statistics MT 312/MT 322/TM Module II 2. An operation manager in charge of a company’s manufacturing keeps track of the number of manufactured LCD television in a day. If the number of LCD”s produced in the past 3 weeks were: 21, 19, 20, 26, 21, 22, 21, 26, 31, 30, 29, 30, 26, 26, 28, 27, 23, 21. What is the mode of these data? Arranging the data, 19, 20, 21, 21, 21, 21, 22, 23, 26, 26, 26, 26, 27, 28, 29, 30, 30, 31 There are two modes, 21 and 26. Bimodal 3. Nine middle-management of a certain company have the following ages: 47, 52, 59, 56, 55, 49, 60, 46, 54. What is the mode of this data set? None. ISHRM 2014 Learning Material Statistics MT 312/MT 322/TM Module II Calculating the Mode of grouped data Grouped Frequency Distribution of 50 selected Tourism students who took the test in College Algebra. 48 45 42 39 36 33 30 27 24 21 ci - f 1 2 5 6 12 11 9 1 1 2 50 47 44 41 38 35 32 29 26 23 CB 47.5 – 50.5 44.5 – 47.5 41.5 – 44.5 38.5 – 41.5 35.5 – 38.5 32.5 – 35.5 29.5 – 32.5 26.5 – 29.5 23.5 – 26.5 20.5 – 23.5 Find the Mode of this distribution using the formula: Mo = LB + ๐1 แผฐ, ๐1+๐2 Where: LB = Lower boundary of the modal class The modal class is the class with the highest frequency. d1 = Difference between the frequency of the modal class and the frequency of the class immediately below it. d2 = Difference between the frequency of the modal class and the frequency of the class above it. Given: Modal Class = (36 - 38) LB = 35.5 d1 = 12 – 11 = 1 d2 = 12 - 6 = 6 ISHRM 2014 Learning Material Statistics MT 312/MT 322/TM Module II Solution: ๐1 Mo = LB + ใ ๐1 + ใแผฐ ๐2 Mo = 35.5 + (0.14) 3 1 = 35.5 + ใ 1+6 ใ 3 = 35.5 + 0.42 Mo = 35.92 2.5 MEASURES OF LOCATION OR QUANTILES Quantiles are measures of position which divide the data set or frequency distribution into equal parts. They are the natural extension of the Median. While the median divides the distribution or data set into two parts, quantiles divide it into four, ten, and 100 equal parts. 2.6 THE QUARTILES Quartiles are quantiles that divide the data set or frequency distribution into four equal parts, such that 25 % of the data falls below the first quartile (Q1), 50 % of the data falls below second quartile (Q2), and 75 % of the data falls below the third quartile (Q3). Q1, Q2, and Q3 are the score – points which divides the set or distribution into four parts. Calculating the Quartiles of ungrouped data The general formula of Quartiles for ungrouped data Qk = ๐ (๐ต+๐) ๐ ISHRM 2014 Learning Material Statistics MT 312/MT 322/TM Module II Illustrative example: The following data give the average number of minutes required to complete an assembly job by the 14 workers of a manufacturing plant: 79, 87, 65, 56, 64, 80, 82, 50, 65, 81, 71, 57, 65, 80 1. Solve for the values of Q1, Q2, and Q3. 2. If Michelle can finish an assembly job in 65 minutes, where does she lie in relation to the three quartiles? Solutions: 1. Rank the given data in ascending order and calculate the Q1, Q2 (or Median), and Q3. 50 56 57 64 65 65 65 (even number) Q1 = 64 Q2 = 65+71 2 71 79 80 80 81 82 87 Q3 = 80 Q2 = 68 Check the answers using the formula. a. Q1 = 1 (14 +1) 4 = 15 4 = 3.75th. or the 4th. value which is 64 Q1 = 64 b. Q2 = 2(14 +1) 4 = 30 4 = 7.5th. value or 68 Q2 = 68 ISHRM 2014 Learning Material Statistics MT 312/MT 322/TM Module II c. Q3 = 3 ( 14 +1) 4 = 45 4 = 11.25th. or the 11th. value, which is Q3 = 80 2. Michelle’s minute falls within the first 50% below the second quartile (Q2) (or Md) but above the first quartile (Q1) 2.7 THE DECILES Deciles are quantiles that divide the data set or distribution into ten equal parts. There are nine score – points, denoted by D1, D2, D3. . . , D9. Calculating the Deciles of ungrouped data The general formula is Dk = ๐ (๐ต+๐) ๐๐ Illustrative example: A newly open supermarket conducted a survey to determine the age of customers that patronize the store on a given Sunday. A random sample of 24 customers was interviewed and the following ages were obtained: 47, 28, 19, 16, 41, 36, 25, 32, 18, 19, 35, 35, 21, 39, 26, 45, 22, 31, 42, 27, 40, 26, 9, 12. Find D3, D6, and D8 ISHRM 2014 Learning Material Statistics MT 312/MT 322/TM Module II Solution: Arrange the data in ascending order. 9, 12, 16, 18, 19, 19, 21, 22, 25, 26, 26, 27, 28, 31, 32, 35, 35, 36, 39, 40, 41, 42, 45, 47 D3 = 21.5 D5 = 27.50 D6= 32 D8 = 40 3 (24 +1) 1. D3 = 10 = 75 10 = 7.5th. value = 21.5 It means that 30% of the values in the data set falls below 21.5. 2. D6 = 6 (24+1) 10 = 150 10 = 15th. value = 32 60% of the values is less than 32. 3. D8 = 8 (24+1) 10 = 200 10 = 20th. value = 40 Supply your own description. 2.8 THE PERCENTILES Percentiles are score – points that divide the set or distribution into one hundred equal parts. It is used to assess performance by determining the position of one score relative to others. Percentile Rank (PR) is a percentage. It tells what percent of the cases got below the rank position. It is computed when the percentile is given. Percentile Point (Pn) is the score or value that corresponds to the given percentile rank. ISHRM 2014 Learning Material Statistics MT 312/MT 322/TM Module II Calculating the percentiles of ungrouped data The formula is: Pk = ๐(๐+1) 100 Illustrative example: A class of 20 BSBA students were given 30-item quiz in Statistics. The scores obtained are the following: 26, 29, 30, 5, 19, 23, 15, 17, 24, 9, 12, 11, 10, 7, 10, 14, 9, 10, 8, 5. 1. Solve for the P25, P50, and P75. 2. If Eunice Milca got a score of 26 and is ranked 3rd. in the class, how many students are ranked below her? 3. What percent of the students ranked below her? 4. What are the percentile rank and percentile point of Eunice? 5. What percent of the students ranked above her? Solutions: Arranging the data in ascending order. 3, 5, 7, 8, 9, 9, 10, 10, 10, 11, 12, 14, 15, 17, 19, 23, 24, 26, 29,30. 1. Solving for P25, P50, P75. P25 = 25(20+1) 100 = 525 100 = 5.25th. value = 9 This means that 25% of the scores falls below 9. It can also means that 25% of the students got a score below 9. ISHRM 2014 Learning Material Statistics MT 312/MT 322/TM Module II P50 = 50( 20+1) 100 = 1,050 100 = 10.5th. value = 11.5 Provide your own description here. P75 = 75(20+1) 100 = 1,575 100 = 15.75th. value = 22 Provide your own description here. 2. The number of students ranked below Eunice Milca: 20 – 3 = 17 (17 students got scores below her) 3. The percent of students ranked below her: %= ๐๐.๐๐ ๐ ๐ก๐ข๐๐๐๐ก๐ ๐ค๐๐กโ ๐ ๐๐๐๐ ๐๐๐๐๐ค 26 ๐๐.๐๐ ๐ ๐ก๐ข๐๐๐๐ก๐ ๐๐ ๐กโ๐ ๐๐๐๐ ๐ 4. Percentile rank = 17+1 20 x 100% = 90 % Percentile point = 26 5. Percent of students ranked above Eunice % = 2 20 x 100% = 10% Calculating the Quartiles of grouped data Formula: Qk = LB + ใ ๐ (๐ 4 ) −๐๐๐ ๐๐ด๐1 17 x 100% = 20 x 100% = 85 % ใแผฐ ISHRM 2014 Learning Material Statistics MT 312/MT 322/TM Module II The procedure for solving problems on quantiles is similar to the median (Md) because, as mentioned earlier, quantiles are natural extensions of the median. Illustrative example: The following is a distribution of the number of employees in 50 companies belonging to a certain service business: Number of Employees ci 69 - 76 61 - 68 53 - 60 45 - 52 37 - 44 29 - 36 21 - 28 No. of Companies ๐ 6 8 10 13 7 4 2 n = 50 CB 69.5 - 76.5 60.5 - 68.5 52.5 - 60.5 44.5 - 52.5 36.5 - 44.5 28.5 - 36.5 20.5 - 28.5 Solving for the first quartile (Q1) Solution: Step 1. Assumed Q1 class: 1 4 1 (๐) = 4 (50) = 12.5. Step 2. AQ1 class = (37 - 44) c๐p = 6 ๐AQ1 class = 7 LB = 36.5 แผฐ = 8 ISHRM 2014 Learning Material Statistics MT 312/MT 322/TM Module II ห c๐ 50 44 36 26 13 6 2 ห ๐c 6 14 24 37 44 48 50 Step 3. Substitution: 1 Q1 = LB +ใ 4 Q1 = 36.5 + ใ (๐) −๐๐๐ ๐๐ด๐1 12.5 − 6 7 1 ใแผฐ ใ8 (50) −13 = 36.5 + ใ 4 = 36.5 + ใ 6.5 13 ใ8 7 ใ8 = 36.5 + 7.43 Q1 = 43.95 ใ It falls within the ci = (37 – 44)ใ This means that 25 % of the data in the distribution falls below 43.95. Solving for the third quartile (Q3) Solution: Assumed Q3 class: 3 4 (๐) = 3 4 (50) = 37.5 (61 – 68) c๐p = 36 LB = 60.5 ๐AQ3 = 8 แผฐ= 8 3 Q3 = LB + ใ4 Q3 = 60.5 + ใ (๐) − ๐๐๐ ๐๐ด๐3 37.5 −36 8 3 ใแผฐ = 60.5 + ใ 4 ใ8 = 60.5 + ใ (50) −36 1.5 8 8 ใ8 Q3 = 62.00 (Give your own description.) ISHRM 2014 Learning Material Statistics MT 312/MT 322/TM Module II ใ8 = 60.5 + 1.5 Calculating the Deciles of grouped data Formula: ๐ Dk = LB + ใ 10 ( ๐) −๐๐๐ ๐๐ด๐ท๐ ใแผฐ A class of 60 BSHRM students was given a test in Accounting and the following scores are arranged in the distribution, as shown below: Class Interval Frequency (c i) (๐) 52 47 42 37 32 27 22 17 12 - 56 51 46 41 36 31 26 21 16 3 5 8 10 13 9 6 4 2 N = 60 Class Boundary (CB) 51.5 46.5 41.5 36.5 31.5 26.5 21.5 16.5 11.5 - 56.5 51.5 46.5 41.5 36.5 31.5 26.5 21.5 16.5 Cumulative Freq. ห c๐ 60 57 52 44 34 21 12 6 2 Determine the : (1) Fourth Decile (D4) (2) Seventh Decile (D7). ISHRM 2014 Learning Material Statistics MT 312/MT 322/TM Module II Cumulative Freq. ห c๐ 3 8 16 26 39 48 54 58 60 Solutions: (1) Fourth Decile (D4) 4 Assumed D4 class: 10 (๐) = 4 10 (60) = 24 ใ32 - 36 ใ c๐p = 21 ๐AD4 = 13 แผฐ = 5 LB = 31.5 D4 = LB + ใ 4 10 (๐) −๐๐๐ ๐๐ด๐ท4 ใแผฐ 24 − 21 D4 = 31.5 + ใ 13 = 31.5 + ใ 4 (60) −21 10 ใ5 = 13 3 ใ5 = 31.5 + ใ ใ 5 13 = 31.5 + 1.15 D4 = 32. 65 Means that 40 % of the scores or data in the distribution is less than 32.65 (2) Seventh Decile (D7) 7 Assume D7 Class: 10 (๐) = 7 10 (60) = 42 (37 - 41) c๐p = 34 ๐AD7 = 10 LB = 36.5 แผฐ = 5 7 D7 = LB + ใ 10 D7 = 36.5 + ใ (๐) − ๐๐๐ ๐๐ด๐ท7 42 − 34 10 7 ใแผฐ = 36.5 + ใ 10 ใ5 = 36.5 + ใ 10 ใ 5 8 D7 = 40.50 (Make your own description.) ISHRM 2014 Learning Material Statistics MT 312/MT 322/TM Module II (60) −34 10 ใ 5 = 36.5 + 4 Calculating the Percentiles of grouped data Formula: ๐ − ๐๐๐ Pk = LB + ใ100๐๐ด๐๐ ใ แผฐ Frequency distribution of scores of 60 BSHRM students in Statistics test. Class Interval Frequency (c i) (๐) 52 47 42 37 32 27 22 17 12 - 56 51 46 41 36 31 26 21 16 3 5 8 10 13 9 6 4 2 N = 60 Class Boundary (CB) 51.5 46.5 41.5 36.5 31.5 26.5 21.5 16.5 11.5 - Cumulative Freq. ห c๐ 60 57 52 44 34 21 12 6 2 56.5 51.5 46.5 41.5 36.5 31.5 26.5 21.5 16.5 Cumulative Freq. ห c๐ 3 8 16 26 39 48 54 58 60 Solve for the: ( 1 ) Thirtieth Percentile (P30) ( 2 ) Seventieth Percentile (P70) Solutions: (1) Thirtieth Percentile (P30) 30 Assumed P30 class: 100 (๐) = 30 100 (60) = 18 ใ27 - 31ใ ISHRM 2014 Learning Material Statistics MT 312/MT 322/TM Module II Cfp = 12 ƒAP30 = 9 LB = 26.5 แผฐ = 5 P30 = LB + [ 30 (12)−๐๐๐ 100 ใแผฐ ๐๐ด๐30 P30 = 26.5 + ใ 18 − 12 9 = 26.5 + ใ 30 ( 60) − 100 6 = 26.5 + ใ 9 ใ 5 ใ 5 12 9 ใ5 = 26.5 + 3.33 P30 = 29.83 Write your description here. (2) Seventieth Percentile (P70) 70 Assumed P70 class: 100 (๐) = 70 100 (60) = 42 (37 – 41) cƒp = 34 ƒAP70 = 10 LB = 36.5 แผฐ = 5 ๐ (๐) −๐๐๐ P70 = LB + [ 100๐๐ด๐70 P70 = 36.5 + ใ 42 − 34 10 70 ]แผฐ ใ5 = 36.5 + ใ 100 8 = 36.5 + ใ 10 ใ 5 P70 = 40.50 Write your description here. ISHRM 2014 Learning Material Statistics MT 312/MT 322/TM Module II (60) − 34 10 ใ5 = 36.5 + 4.0 WORK PROJECT / LEARNING ACTIVITY 1, MODULE 2 Name: _____________________________ Yr/Sec: ____________ Date: __________________ A. IDENTIFICATION. _______________________ 1. It is the point (or value) about which all other scores in the distribution or data set tend to cluster. _______________________ 2. It is defined as the sum of the values in the data set or distribution divided by the number of values. ________________________ 3. It is expressed as the sum of the product of each value and its corresponding weight divided by the sum of the weights. ________________________ 4. This refers to the measure of central tendency that occupies the middle position in the frequency distribution or set of data. ________________________ 5. It is that single data or score which occurs most frequently in a data set. ________________________ 6. These are measures of position which divide the data set or frequency distribution into equal parts. ________________________ 7. These quantiles divide the data set or frequency distribution into four equal parts. ________________________ 8. The quantiles that divide the data set or distribution into ten equal parts and consists of nine score – points. ________________________ 9. These are score – points that divide the set or distribution into one hundred equal parts. ________________________ 10. It tells what percent of the cases got below the rank position. It is computed when the percentile is given. ISHRM 2014 Learning Material Statistics MT 312/MT 322/TM Module II WORK PROJECT / LEARNING ACTIVITY 2, MODULE 2 Name: _____________________________ Yr/Sec: ____________ Date: __________________ B. MULTIPLE CHOICE. __________ 1. The score or value that corresponds to the given percentile rank is termed a. Percentile c. Percentile Point b. Percentile Rank d. None __________ 2. Which of the following is a percentage? a. Percentile Point c. Percentile b. Percentile Rank d. None __________ 3. The following are all measures of central tendencies, except a. Mode c. Median b. Mean d. Quartile __________ 4. The measures of central tendencies are also called a. Quantiles c. Centiles b. Measures of position d. Averages __________ 5. The following are all measures of position, except a. Decile c. Median b. Quartile d. Percentile ISHRM 2014 Learning Material Statistics MT 312/MT 322/TM Module II __________ 6. Which central tendency measures the level of performance of a group? a. Quartile c. Decile b. Percentile d. Mean __________ 7. The most commonly used measure of central tendency is the a. Mode c. Percentile b. Mean d. Median __________ 8. The measure of central tendency which divides a data set or frequency distribution into two parts, such that 50% of the data lies below it and the other 50% lies above it, is referred to as a. Weighted Mean c. Arithmetic Mean b. Sample Mean d. None __________ 9. The midpoint with the highest frequency in the data set or frequency distribution is referred to as a. Quartile c. Mode b. Percentile d. Median __________ 10. The following are all natural extensions of the Median, except a. Quatiles c. Deciles b. Percentiles d. None ISHRM 2014 Learning Material Statistics MT 312/MT 322/TM Module II __________ 11. These are score-points that divide a data set or frequency distribution into 25 %, 50 %, and 75 %. a. Quatiles c. Deciles b. Percentiles d. None __________ 12. Which of the following measures of position is used to assess performance by determining the position of one score relative to the others? a. Percentile b. Decile c. Quartile d. None __________ 13. It is used when other parameters or statistics like standard deviation, variance, coefficient of correlation, etc. are to be computed later. a. Quartile c. Percentile b. Median d. Mean __________ 14. Which of the following is affected by open ended or extreme value on either or both ends of the distribution? a. Mean c. Median b. Quartile d. Mode ___________ 15. It is used when a quick and approximate measure of central tendency is desired. a. Mean c. Quartile b. Median d. Mode ISHRM 2014 Learning Material Statistics MT 312/MT 322/TM Module II WORK PROJECT / LEARNING ACTIVITY 3, MODULE 2 Name: _____________________________ Yr/Sec: ____________ Date: __________________ C. PROBLEM SOLVING. 1. A researcher wished to study the trend in the attendance of the employees at a certain five-star hotel. 60 employees were selected at random and a record was consulted to determine the number of absences these employees incurred for a certain month. The following is a frequency distribution on the number of absences of the employees. No. of Absences (c แผฐ) 0–2 3–5 6–8 9 – 11 12 – 14 15 - 17 18 – 20 21 – 13 24 – 26 No. of Employees (ƒ) 3 5 8 11 14 17 20 23 26 n = 60 Complete the table with the required variables and solve for the following: a. Mean a.1 If Michelle, an employee, has 7 absences and is rank 8, what percent of the employees incurred absences more than her? a.2 What is Michelle’s level of performance in terms of the attendance record of the hotel? ISHRM 2014 Learning Material Statistics MT 312/MT 322/TM Module II b. Median c. Mode 2. The service time in minutes for 15 customers at a reservation counter is given below. 2, 1, 7, 8, 3, 5, 10, 15, 12, 13, 9, 5, 5, 9, 2 Rearrange the data set in ascending order and solve for the following: a. Arithmetic Mean b. Median c. Mode 3. The average family income (in thousand pesos) in 2013 for each of the 13 regions in the Philippines was 59, 24, 25, 28, 29, 30, 22, 21, 25, 39, 28, 19, 32 Rank the data set in ascending order and solve for the following: a. Q1, Q2, and Q3 and rank the data set according to these three quartiles. b. D2.5, D5, and D7.5 and rank the set according to these three deciles. c. P25, P50, and P75 and rank the set according to these three percentiles. d. What do you observed? Explain your observation. ISHRM 2014 Learning Material Statistics MT 312/MT 322/TM Module II 4. A BSHRM class of 50 students was given a test in Accounting, and the data obtained were arranged in a frequency distribution, as shown below Score (c แผฐ ) 37 - 41 32 - 36 27 - 31 22 - 26 17 - 21 12 - 16 7 - 11 No. of students (ƒ) 5 7 9 10 8 7 4 N = 50 ห c๐ 5 12 21 31 39 46 50 Determine the following: a. Q1, Q2, and Q3 b. D4 and D8 c. P20 and P70 c.1 If Lourdes got a score of 33 and is ranked 9, what percent of the students got a score below 33? What percent of students got a score above 33 c.2 What is the percentile rank of Lourdes? ISHRM 2014 Learning Material Statistics MT 312/MT 322/TM Module II