Other Measures of Location

In the previous lesson we discussed a measure of location known as the measure of central tendency. There are other measures of location which are useful in describing the distribution of the data set. These measures of location include the maximum, minimum, percentiles, deciles and quartiles. How to compute and interpret these measures are also discussed in this lesson.

1. Other Measures of location

2. Learning outcomes Calculate measures of location other than the measure of central tendency, and Provide a sound interpretation of these summary measures

3. What is the highest score? Lowest score? Answer: Highest score is 50 while the lowest is 10. What is the most frequent score? Answer : Most frequent score is 38 which is the score of 28 students. What is the median score? Answer: The median score is 33 which implies that 50% of the students or around 75 students have score at most 33. What is the average or mean score? Answer: On the average, the students got 32.04667 or 32 (rounded off) out of 50 items correctly .

4. Measures of location Maximum, Minimum, Percentiles, Deciles and QuaNtiles

5. Maximum and minimum 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.

6. Percentile Percentile is a measure that pinpoints a location that divides distribution into 100 equal parts. It is usually represented by , that value which separates the bottom j% of the distribution from the top (100-j)%. For example 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 while the remaining 70% have values greater than  

7. STEPS in finding the percentile   Arrange the data values in ascending order of magnitude Find the location of in the arranged list by computing , where is the total number of observation in the data set. If is a whole number, then is the mean or average of the values in the and positions. If is not a whole number, then is the value of the next higher position.  

8. Find  

9. Step 1: To find we note that . Since the observations are tabulated in increasing order, we could proceed. Step 2: Compute for , . The computed which is equal to 45 is a whole number and thus we follow the first rule. Step 3: If is a whole number, then is the mean or average of the values in the and positions. Thus, we take and 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.  

10. DECILES AND QUARTILES 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 10 th Percentile is the same as the 1 st Decile, 20 th Percentile same as 2 nd Decile, 25 th Percentile same as 1 st Quartile, 50 th Percentile same as 5 th Decile or 2 nd 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 50 th Percentile or 5 th Decile or 2 nd Quartile. Because of this relationship, the computation of the quartile and decile could be coursed through the computation of the percentile.

11. If we want to compute the 3 rd Decile or then we compute 30 th Percentile or . In other words, based on our earlier computation. The 3 rd Quartile or is equal to . To compute , as The computed which is equal to 112.5 is not a whole number and thus we follow the second rule in Step 3 which states that is the value found in the next higher position, specifically, in 113 th position, the next higher position after 112.5. Thus, we take the 113 th observation which is equal to 38 as the value of . 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.  

12. 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 into100 , 4 and 10 equal parts, respectively.

13. quiz ¼ Sheet of paper

14. Determine the weekly food allowance where 60% of the students have at most. What percentage of the students have a weekly food allowance that is at most 170 pesos ? 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?

15. answers

16. . 60 % of the students have at most 700 pesos as their weekly food allowance . Here we are looking for . It is given that is the 15 th observation in the array of 213 values. Therefore we say that 7% of the students have a weekly food allowance of at most 170 pesos. or the median. 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.