Measures of Location

Transcription

1 Chapter 7 Measures of Location Definition of Measures of Location (page 219) A measure of location provides information on the percentage of observations in the collection whose values are less than or equal to it. It indicates the relative position of an observation in the array. A measure of location is also referred to as a fractile, quantile, or measure of position. 1

2 Definition of Percentile (page 220) Definition 7.1 The percentiles divide the ordered observations into 100 equal parts. There are 99 percentiles, denoted by P 1, P 2,,P 99. The k th percentile, denoted by P k, is a value such that at least k% of the observations are less than or equal to it and at least (100-k)% are greater than or equal to it, where k = 1, 2, 3,, 99. Example: P 1 or the 1 st percentile is a value such that at least 1% of the observations are less than or equal to it and at least 99% of the observations are greater than or equal to it. P 50 or the 50 th percentile is a value such that at least 50% of the observations are less than or equal to it and at least 50% of the observations are greater than or equal to it. In other Chapter words, 7. Measures P 50 is the of Location median. Percentile Score vs Percentage Score (page 220) Percentage score is (total score/total number of points)100%. Example: Total no. of points=120 Score of Juan= 90 Percentage score of Juan= (90/120) x 100% = 75% (100) Percentile score indicates that at least (100%) of all scores in the collection are less than or equal to the individual s score while at least (1- )(100%) are greater than or equal to the individual s score, 0 < < 1. Example: Juan s percentile score for his section is This means that the scores of at least 95.5% of all students in his section are less than or equal to Juan s score and the scores of at least 4.5% are greater than or equal to Juan s score. Juan s percentile score for his school is 40. This means that the scores of at least 40% of all students who took the same test in his school are less than or equal to Juan s score and the scores of at least 60% are greater than or equal to Juan s score. 2

3 More About Percentile Scores Juan s percentile score for their section is This means that the scores of at least 95.5% of all students in his section are less than or equal to Juan s score and the scores of at least 4.5% are greater than or equal to Juan s score. Saying that at least 95.5% of all students in his section are less than or equal to Juan s score is equivalent to saying that at most 4.5% are greater than his score. So if there are 50 students in Juan s class, the number of students whose scores are greater than Juan s score will not exceed (50)(.045)=2.25 or 2. Saying that at least 4.5% are greater than or equal to Juan s score is equivalent to saying that at most 95.5% are less than his score. So if there are 50 students in Juan s class, the number of students whose scores are less than Juan s score will not exceed (50)(.955)=47.75 or 47. Additional Note on the Interpretation of P k (page 224) P k will be an interpolated value if nk/100 is an integer. If the values used in the interpolation are not tied values then P k will not be one of the observations. In such a case, the interpretation of P k will simplify as follows: k% of observations are less than P k. Likewise, (100-k)% are greater than P k. That is, nk/100 observations are less than P k so that the remaining n-(nk/100)=n(1-k/100) are greater than P k. 3

4 Some Methods for Interpolating P k (page 224) empirical distribution number with averaging weighted average estimate observation numbered closed to nk/100 empirical distribution number estimate Tukey s method Note: Before using any software to determine the percentile, make sure you know the method used by the software. Example: Excel uses an unusual method. The position of P k is determined using the formula (n-1)(k/100) +1 then it uses linear interpolation to compute for P k. Determining the k th Percentile, P k, using Empirical Distribution Number with Averaging (page 221) Step 1: Step 2: Step 3: Arrange the n observations in the collection in an array. Denote the i th ordered observation by X (i). Compute for nk/100. Determine P k using the given formula, nk Case (i). If 100 is an integer, X X nk /100 (( nk ) /1001) Pk 2 nk Case (ii). If 100 is not an integer, Pk X nk /

5 Example 7.1a (page 221) The following are the total receipts of seven mining companies (in million pesos): 4.6, 1.3, 7.3, 6.6, 10.5, 50.7, and Determine the 75th percentile. Solution: Arrange the data in an array (lowest to highest). Array: Notation: X (1) X (2) X (3) X (4) X (5) X (6) X (7) Find P 75. Thus k=75 nk/100= (7)(75)/100=(7)(.75)=5.25 is not an integer. P X X X 12.6 (5 1) (6) How many observations are less than or equal to 12.6? greater than or equal to 12.6? Example 7.1b (page 222) The following are the number of years of operation of 20 mining companies: 4, 5, 6, 6, 7, 8, 10, 10, 11, 16, 17, 17, 18, 19, 20, 20, 21, 23, 25, and 30. Determine the 90 th percentile. Solution Arrange the data in an array (lowest to highest) X (1) X (2) X (3) X (4) X (5) X (6) X (7) X (8) X (9) X (10) X (11) X (12) X (13) X (14) X (15) X (16) X (17) X (18) X (19) X (20) Find P 90. Thus k=90 nk/100 = (20)(90)/100 = (20)(0.9)=18 is an integer. X X nk nk 1 X (18) X X X (181) (18) (19) 2325 P Is 24 one of the observed values? Since no observation is equal to 24, How many observations are less than 24? greater than 24? 5

6 Determining the k th Percentile, P k, using Weighted Average Estimate (page 223) Step 1: Step 2: Arrange the n observations in the collection in an array. Compute for (n+1)(k/100) = j + g, where j=integer part and g=fractional part. Step 3: Determine P k by linear interpolation, that is, use the formula, P (1 g) X gx X g( X X ) k ( j) ( j1) ( j) ( j1) ( j) Example 7.3a (page 223) Use the following ordered data on total receipts of the seven mining companies in Example 7.1a to determine the 75 th percentile. Array: Notation: X (1) X (2) X (3) X (4) X (5) X (6) X (7) Solution: Compute for (n+1)k/100 = (8)(75)/100 = 6. The integer part of 6 is j=6 and since there is no fractional part then g=0. Using the formula, we get: P 75 = (1 0)X (6) + 0X (7) = X (6) = 12.6 We notice that whenever ( n 1) k is an integer then the k th percentile is the observation 100 in the array occupying that position. In other words, Pk X ( n 1) k

7 Example 7.3b (page 223) Use the following ordered data on the number of years of operation of 20 mining companies in Example 7.1b to determine the 90th percentile X(1) X(2) X(3) X (4) X (5) X(6) X (7) X (8) X (9) X(10) X (11) X (12) X (13) X (14) X (15) X (16) X (17) X (18) X (19) X (20) Solution: Compute for (n+1)k/100 = (21)(90)/100 = The integer part is j=18 and the fractional part is g=0.9. Using the formula, we get: P 90=(1 0.9)X (18) + 0.9X (19) = 0.1X (18) + 0.9X (19)= (0.1)(23)+(0.9)(25)=24.8. Since nk/100=18 is an integer and the interpolated percentile is not one of the data values then the interpretation of this percentile becomes simple. Once again, we can just say that 90% of the companies have been operating for less than 24.8 years. Exercise 2 (page 226) The employees of a drug company donated the following amount in pesos to the typhoon victims: 1500, 4200, 2600, 550, 2300, 1000, 2400, 5100, 3200, 1300, 550, 1600, 2400, 5100, 1300, 3100, 550, 2700, 4800, and Calculate the 80 th percentile and interpret. EMPIRICAL DISTRIBUTION NUMBER WITH AVERAGING Array: 550, 550, 550, 1000, 1300, 1300, 1500, 1600, 1600, 2300, 2400, 2400, 2600, 2700, 3100, 3200, 4200, 4800, 5100, What is k? What is n? WEIGHTED AVERAGE ESTIMATE nk/100=(20)(80)/100=16 P 80 = (X (16) +X (17) )/2 = ( )/2=3700 (n+1)(k/100)=(21)(.8)=16.8 P 80 = X (16) + (0.8)(X (17) X (16) ) = (0.8)( ) =

8 Exercise 2 modified with 4 additional observations (page 226) The employees of a drug company donated the following amount in pesos to the typhoon victims: 1500, 4200, 2600, 550, 2300, 1000, 2400, 5100, 3200, 1300, 550, 1600, 2400, 5100, 1300, 3100, 550, 2700, 4800, 1600, 5000, 1800, 1000, and Calculate the 80 th percentile and interpret. EMPIRICAL DISTRIBUTION NUMBER WITH AVERAGING nk/100=(24)(80)/100=19.2 P 80 = X (19+1) = X (20) = 4200 Array: 550, 550, 550, 1000, 1000, 1200, 1300, 1300, 1500, 1600, 1600, 1800, 2300, 2400, 2400, 2600, 2700, 3100, 3200, 4200, 4800, 5000, 5100, What is k? What is n? WEIGHTED AVERAGE ESTIMATE (n+1)(k/100)=(25)(.8)=20 P 80 = X (20) + (0)(X (21) X (20) ) = 4200 Approximating P k from FDT (page 225) Step 1. Compute for nk/100. Step 2. Construct the less than cumulative frequency distribution (<CFD). Step 3. Locate the k th percentile class (P k th class). The P k th class is the class interval where the less than cumulative frequency is greater than or equal to nk/100 for the first time, starting from the top. Step 4. Use the following formula to approximate P k. P =LCB k Pk (nk/100) - CF Pk-1 C f Pk 8

9 Example 7.5 (page 225). The table below gives us the weight in pounds of a sample of 75 pieces of luggage. Find the 75 th percentile. Weight Class Boundaries No. of Children < CF (in pounds) LCB - UCB f i Solution: Compute for nk/100 = 75(75)/100 = Thus, the 90 th percentile belongs in the class interval 71.5 to P Definition of Quartiles (page 227) Definition 7.2. The quartiles divide the ordered observations into 4 equal parts. There are 3 quartiles, denoted by Q 1, Q 2, and Q 3.. Q 1 or the 1 st quartile is a value such that at least 25% of the observations are less than or equal to it and at least 75% of the observations are greater than or equal to it. Q 2 or the 2 nd quartile is a value such that at least 50% of the observations are less than or equal to it and at least 50% of the observations are greater than or equal to it. In other words, Q 2 is the median. Q 3 or the 3 rd quartile is a value such that at least 75% of the observations are less than or equal to it and at least 25% of the observations are greater than or equal to it. 9

10 Relationship of Quartiles, Percentiles and Median (page 228) Value: Smallest Q 1 Q 2 Q 3 Largest P 25 P 50 P 75 Median Position: First (.25(n+1)) th (.5(n+1)) th (.75(n+1)) th Last To determine the quartile, compute for the corresponding percentile. Definition of Deciles (page 229) Definition 7.3. The deciles divide the ordered observations into 10 equal parts. There are 9 deciles, denoted by D 1, D 2,, D 9. D 1 or the 1 st decile is a value such that at least 10% of the observations are less than or equal to it and at least 90% of the observations are greater than or equal to it. D 5 or the 5 th decile is a value such that at least 50% of the observations are less than or equal to it and at least 50% of the observations are greater than or equal to it. In other words, D 5 is the median. D 9 or the 9 th decile is a value such that at least 90% of the observations are less than or equal to it and at least 10% of the observations are greater than or equal to it. 10

11 Relationship of Deciles and Percentiles (page 230) Deciles D 1 D 2 D 3 D 4 D 5 D 6 D 7 D 8 D 9 Percentiles P 10 P 20 P 30 P 40 P 50 P 60 P 70 P 80 P 90 To determine the deciles, compute for the corresponding percentile. Assignment 1. Page 227. Exercise no Page 229. Use data in Exercise no. 2. Determine the following using both empirical distribution number with averaging and weighted average estimate: a) First quartile b) 50 th percentile c) 2 nd decile 3. Page 229. Exercise no. 3 11