We formally define the maximum as a measure of location that pinpoints the highest value in the data distribution while the minimum locates the lowest value.
There are other measures of location that are becoming common because of its constant use in reporting rank in distribution of scores as the percentile rank in college entrance examination. These measures are referred to as percentiles, deciles, and quartiles.
Percentile is a measure that pinpoints a location that divides distribution into 100 equal parts. It is usually represented by Pj, that value which separates the bottom j%
of the distribution from the top (100-j)%. For example, P30 is the value that separates the bottom 30% of the distribution to the top 70%. Thus we say 30% of the total
number of observations in the data set are said to be less than or equal to P30 while the remaining 70% have values greater than P30.
Lifted from the workbook cited as reference at the end of this Teachers Guide, are the steps in finding the jth percentile (Pj)
Step 1: Arrange the data values in ascending order of magnitude.
Step 2: Find the location of Pj in the arranged list by computing j L=!$100"%×N
& ' , where N is the total number of observations in the data set.
Step 3:
a. If L is a whole number, then Pj is the mean or average of the values in the Lth and (L+1)th positions.
b. If L is not a whole number, then Pj is the value of the next higher position.
To illustrate we use the data on long test scores of 150 Grade 11 students of nearby Senior High School. An additional column on less than cumulative frequency was
To find P30 we note that j = 30. Since the observations are tabulated in increasing order, we could proceed to Step 2 which ask us to compute L as ! = !""! ×! =
!"
!"" ×150 = 45. The computed L which is equal to 45 is a whole number and thus
we follow the first rule in Step 3 which states that Pj is the average or mean of the values found in the Lth and (L+1)th positions. Thus, we take the average of the 45th and 46th observations which are both equal to 25. We then say that the bottom 30%
of the scores are said to be less than or equal to 25 while the top 70% of the observations (which is around 105) are greater than 25.
Deciles and quartiles are then defined in relation to percentile. If the percentile divides the distribution into 100 equal parts, deciles divide the distribution into 10 equal parts while quartiles divide the distribution into 4 equal parts. Thus, we say that 10th Percentile is the same as the 1st Decile, 20th Percentile same as 2nd Decile, 25th Percentile same as 1st Quartile, 50th Percentile same as 5th Decile or 2nd Quartile and so forth. Note also that by definition of the median in previous lesson, we could say that the median value is equal to the 50th Percentile or 5th Decile or 2nd Quartile.
Because of this relationship, the computation of the quartile and decile could be coursed through the computation of the percentile.
To illustrate, if we want to compute the 3rd Decile or D3 then we compute 30th Percentile or P30. In other words, D3 = P30 = 25 based on our earlier computation.
The 3rd Quartile or Q3 is equal to P75. To compute L as ! = !""! ×! = !""!" ×150 = 112.5. The computed L which is equal to 112.5 is not a whole number and thus we follow the second rule in Step 3 which states that Pj is the value found in the next higher position, specifically, in 113th position, the next higher position after 112.5.
Thus, we take the 113th observation which is equal to 38 as the value of P75. We then say that 75% of the class of 150 students or around 113 students correctly answered at most 38 out of the 50 items.
The median which is equal to P50 is computed as the mean or average of the 75th and 76th observations which are both equal to 33. Hence, we did get the same value as the one we obtained using the definition we had in the previous lesson.
KEY POINTS
• There are other measures of location that could further describe the distribution of the data set.
• The maximum and minimum values are measures of location that pinpoints the extreme values which are the highest and lowest values, respectively.
• Percentiles, quartiles and deciles are measures of locations that divide the distribution into 100, 4 and 10 equal parts, respectively.
ASSESSMENT
Note: Answers are provided inside the parentheses and in bold face.
1. A businesswoman is planning to have a restaurant in the university belt. She wants to study the weekly food allowance of the students in order to plan her pricing strategy for the different menus she is going to offer. She asked 213 students and gathered the following data:
Weekly we say that 60% of the students have at most 700 pesos as their weekly food allowance.)
b. What percentage of the students have a weekly food allowance that is at most 170 pesos?
(Here we are looking for the value of j. It is given that Pj = 170 is the 15th observation in the array of 213 values. Thus, 15 is the value of L and using this we compute the value of j as ! =!!×100 = !"#!" ×100 ≅ 7. Therefore we say that 7% of the students have a weekly food allowance of at most 170 pesos.)
c. If the business woman wanted to have at least 50% of the students could afford to eat in her restaurant, what should be the minimum total cost of the meals that the student could have in a week?
(The statistic we wanted is the median or P50. To compute L as ! = !""! ×
! = !""!" ×213 = 106.5 ≅ 107. Then we take the 107th observation which is equal to 600. Thus we say that at least 50% of the students could afford to eat in the restaurant if the minimum total cost of the meals that the student could have in a week is 600 pesos.)