8 Data Analysis 2
8.1 Percentiles and Quartiles
Learning Objectives
By the end of this section it is expected that you will be able to:
- Describe the measures of location: percentile and quartile
- Find the percentile represented by a given data value
- Determine the first, second and third quartiles for a set of data
Measures of Central Tendency
The mean, median and mode, as measures of central tendency, provide us with a point of comparison. As an example, consider Company ABC where the average (mean) salary is $55,000/year. An employee earning $38,000/year might feel unjustly treated or at the very least the employee might explore the reasons for the substantial difference. If in the process the employee learns that the median salary at his workplace is $26,000/year the employee would learn that relative to everyone else this employee’s salary is in the upper half of the employee group.
To provide additional comparison the employee could consider other measures of position or location. Two such measures are percentiles and quartiles.
Percentiles
Percentiles are useful for comparing values. If a data item is in the 75th percentile then three-quarters of the values are less than this value. This is not to be confused with a score of 75%, which is something very different. A student could score 35% on an exam but be in the 75th percentile. This means that relative to the rest of the class the student had a score that was higher than 75% of the students.
Percentiles
Percentiles divide ordered data into hundredths. A data item is said to be in the kth percentile of a data set if k% of the data items are less than the item.
The notation Pk can be used to represent the kth percentile. A data set can be divided into one hundred equal parts by ninety-nine percentiles P1 , P2 , P3 , … P99 . The 60th percentile would be denoted P60 . If an item is in the 60th percentile, then 60 percent of the data items are less than this item.
Consider a set of math exam scores. A student scoring in the 60th percentile achieved a score equal to or higher than 60 percent of the other students. This does not mean that the student scored 60% on the exam. Perhaps the student’s score was 78%, which would mean that 60 percent of the other students in the class had exam scores less than (or equal to) 78%.
It is important to note that since percentiles divide a data set into one hundred equal parts, percentiles are best used with large data sets. Percentiles are mostly used with very large populations. For a specified percentile Pk if you were to say that k percent of the data values are less (and not the same or less) than a specified data value, it would be acceptable because removing one particular data value is not significant.
Refer again to the employee earning $38,000/year at Company ABC. If the employee learns that their salary is in the 90th percentile then 90 percent of the other employees at Company ABC have a salary less than (or possibly equal to) this salary. In