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.

The notation  P_k can be used to represent the k^th percentile. A data set can be divided into one hundred equal parts by ninety-nine percentiles P₁ , P₂ , P₃ , … P₉₉ . The 60^th 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 P_k 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 relation to the other employees this salary ranks among the upper portion of the employee group.

Percentiles are useful for comparing values. For this reason, universities and colleges use percentiles on entrance exams. Rather than set one value as an acceptance score, a university may set a percentile target. Perhaps all students scoring  in the the 80th percentile or above will receive an acceptance letter. Every year there is likely to be a different acceptance score. Students will be accepted based on their score relative to all other applicants.

Quartiles

Quartiles divide ordered data into quarters. Quartiles are special percentiles. The first quartile, Q₁, is the same as the 25^th percentile, and the third quartile, Q₃, is the same as the 75^th percentile. The median is a number that separates ordered data into halves. Half the values are the same as or smaller than the median, and half the values are the same as or larger than the median. The median can be called both the second quartile Q₂ and the 50^th percentile.

Determining Quartiles

We will consider two methods for determining quartiles. As with percentiles, the data values must first be ordered from smallest to largest. The first method involves dividing the data set into four equal parts. The second method involves the use of formulas.

Determining Quartiles: Method 1

Step 1: Order the data from smallest to largest.

Step 2: Determine the number of data values n.

Step 3: Determine the median (Q₂) of the data set. This will divide the data set into two equal parts.

Step 4: Determine Q₁. This will divide the first half of the data set into two equal parts.

Step 5: Determine Q₃. This will divide the second half of the data set into two equal parts.

Note: The median and the quartiles may not be actual observations from the data set.

Method 1

Consider the following data set:

15       4       20       8      3     12      14      11      7     2     6     23     16

Step 1: To determine the  quartiles, order the data values from smallest to largest:

2      3       4     6     7       8      11     12      14      15    16     20     23

Step 2: The number of data values is 13.

Step 3: Determine the median, which measures the “centre” of the data. It is the number that separates ordered data into halves. Half the observations are the same number or smaller than the median, and half the observations are the same number or larger.

 

Since there are 13 observations, the median will be in the seventhh position. The median, and therefore the 2nd quartile Q₂ , is eleven. The median is often referred to as  the “middle observation,” but it is important to note that it does not actually have to be one of the observed values.

Step 4: The first quartile, Q₁, is the middle value of the lower half of the data.

To determine the first quartile, Q₁, consider the lower half of the data observations:

2      3       4      6      7       8

Since there are six observations, the middle observation will be the average of the third and fourth data values  or  (4 + 6)/2 = 5  therefore   Q₁  is 5

Step 5: The third quartile, Q₃, is the middle value of the upper half of the data.

To determine the third quartile, Q₃, consider the upper half of the data observations:

12      14      15     16    20    23

Since there are six observations, the middle observation will be the average of 15 and 16 , or 15.5 therefore    Q₃  is 15.5.

Figure 2 illustrates the three quartiles, which divide the data set into four equal parts.

The number 4.5 is the first quartile, Q₁. One-fourth of the entire set of observations lie below 4.5 and  three-fourths of the data observations lie above 4.5.

The third quartile, Q₃, is 15.5. Three-fourths (75%) of the ordered data set lie below 15.5. One-fourth (25%) of the ordered data set lie above 15.5.

It is important to note that a quartile may not be a data observation. Sometimes there may be a need to average or weight the data values when determining the quartiles.

Method 2:

Consider the following data set:

15       4       20       8      3     12      14      11      7     2     6     23     16

Step 1: To determine the  quartiles, order the data values from smallest to largest:

2      3       4     6     7       8      11     12      14      15    16     20     23

Step 2: The number of data values is 13.

Step 3: Use the formula to determine the position for the median (Q₂) of the data set.

Count from left to right to determine the corresponding data value in the 7th position. The corresponding value is 11.

Step 4: Use the formula to determine the position for the first quartile (Q₁) of the data set.

Since 3.5 is a fraction, the first quartile will be the average of the two data values that are in the 3rd and 4th positions. Count from left to right to determine the corresponding data values. The data value 4 is in the 3rd position and the data value 6 is in the 4th position so these will be averaged (4 + 6)/2 = 5. The first quartile will be 5.

Step 5: Use the formula to determine the position for the third quartile (Q₃) of the data set.

Since 10.5 is a fraction, the third quartile will be the average of the two data values that are in the 10th and 11th positions. Count from left to right to determine the corresponding data values. The data value 15 is in the 10th position and the data value 16  is in the 11th position so these will be averaged (15 + 16)/2 = 15.5. The third quartile will be 15.5.

Figure 3 illustrates the three quartiles, which divide the data set into four equal parts.

 

EXAMPLE 2

A shoe store wanted to determine the popularity of different shoe sizes for women’s tennis shoes. It planned to place its next order using this information. In  a five day period it sold nine pairs of women’s tennis shoes in the following sizes:    7,  8, 11,  10,  7,   6,   9, 10,  7

Solution

Method 1:

To determine the quartiles:

1. Order the shoe sizes from smallest to largest:  6,  7,   7,   7,  8,   9,  10,   10,   11
2. Count the number of values: n = 9
3. Determine Q₂, the median, which is the middle observation. Since there are nine data observations (shoe sizes) the median, or second quartile, will be the 5th data value. The 5th data value is 8. 

4. Determine the first quartile Q_1. It will be the middle observation of the lower half of data values. This will be the average of the 2nd and 3rd data values  so (7 +7)/2 = 7.

5. Determine the third quartile Q₃. This will be the middle observation of the upper half. This will be the average of the 7th and 8th data values  so (10+10)/2 = 10 

 

Method 2:

The formulas can be used to determine the quartiles.

1. Order the shoe sizes from smallest to largest:  6,  7,   7,   7,  8,   9,  10,   10,   11 .
2. Determine the number of data values, n.     n = 9
3. Use the formula to determine the median. The median , or second quartile,  can be determined as follows:

Counting from left to right, the 5^th data value is 8. The median, or 2nd quartile Q₂, is 8.

4 & 5.  The first and third quartiles can be determined as follows:

The first quartile is the 2.5th data value. To determine the 2.5^th data value we must take the average of the 2nd and 3rd data values. The 2nd data value is 7 and the 3rd data value is 7 so  (7+7)/2 = 7.

The first quartile, Q₁ = 7

The third quartile is the 7.5th data value. This will be the average of the 7th and 8th data values. The 7th data value is 10 and the 8th data value is also 10  so   (10+10)/2 = 10.

The third quartile, Q₃ = 10

We can see that  Q₂ = 8  splits the data set into two halves. Q₁= 7  is the middle value of the lower half of the data set and Q₃ = 10 is the middle value of the upper half of the data set.   

In Example 2 the number of data items was odd. When n  is odd the median or Q₂ will be one of the data observations. When n is odd the formula for finding quartiles is straight forward.

It is important to note that a quartile may not be a data observation. When the number of data values n  is even the median or Q₂ will not be one of the actual data observations. As a result, when n is even an adjustment must be made to the value of n that is to be used in the formula to determine the first and third quartiles.

Method 1:

Consider the following data set:

1;  11.5;  6;  7.2;  4;  8;  9;  10;  6.8;   8.3;   2;   2;   10;  1

Step 1: To determine the  quartiles, order the data values from smallest to largest:

1   1   2   2   4   6   6.8   7.2   8    8.3   9    10   10   11.5

Step 2:  The number of data values is 14

Step 3: Determine the median, which measures the “centre” of the data. It is the number that separates ordered data into halves. Half the observations are the same number or smaller than the median, and half the observations are the same number or larger.

Since there are 14 observations, the median lies between the seventh observation, 6.8, and the eighth observation, 7.2. To find the median, add the two values together and divide by two.   Median = (6.8 + 7.2)/2 = 7

The median, and therefore the 2nd quartile Q₂ , is seven. It is important to note that the median is not actually one of the observed data values.

Step 4: The first quartile, Q₁, is the middle value of the lower half of the data.

To determine the first quartile, Q₁, consider the lower half of the data observations:

1     1     2     2    4    6    6.8.

Since there are seven observations, the middle observation will be the 4th item. The middle or 4^th item of these data observations  is 2.

Step 5: The third quartile, Q₃, is the middle value of the upper half of the data.

To determine the third quartile, Q₃, consider the upper half of the data observations:

7.2     8     8.3    9    10    10     11.5.

Since there are seven observations, the middle observation will be the 4th item in the upper half. The middle item of these data observations is 9.

Figure 4 illustrates the three quartiles, which divide the data set into four equal parts.

The number 2 is the first quartile, Q₁. One-fourth of the entire set of observations lie below 2 and  three-fourths of the data observations lie above 2.

The third quartile, Q₃, is 9. Three-fourths (75%) of the ordered data set lie below 9. One-fourth (25%) of the ordered data set lie above 9.

Method 2:

Consider the following data set:

1;  11.5;  6;  7.2;  4;  8;  9;  10;  6.8;   8.3;   2;   2;   10;  1

Step 1: To determine the  quartiles, order the data values from smallest to largest:

1   1   2   2   4   6   6.8   7.2   8    8.3   9    10   10   11.5

Step 2:  The number of data values is 14 so n is an even number.

Step 3: Use the formula to determine the position of  Q₂, the median. The position will be (14 + 1)/2 = 7.5. This means that the median,  or Q₂, will be in the 7.5th observation or halfway between the 7th and 8th position. The observation 6.8 is in the 7th position and the observation 7.2 is in the 8th position therefore the average of these (6.8 + 7.2)/ 2 is the median or Q₂.

Note that the median is not an actual observation in the data set. If we use the formula to find Q₁ and Q₃ then we must adjust “n” to include this additional item so in effect “n” will be 15. This is done only when determining the positions of Q₁ and Q₃ (and not for determining the position of Q₂)

Step 4: Use the formula to determine the position of  Q₁, the first quartile. Remember than n will now be 15, not 14. The position will be (15 + 1)/4 = 4 th. This means that  Q₁ will be in the 4th position. Counting from the left, the data value 2 is in the 4th position so Q₁= 2.

Step 5: Use the formula to determine the position of  Q₃, the third quartile. Remember than n will now be 15, not 14. The position will be 3(15 + 1)/4 = 12th. This means that  Q₃ will be in the 12th position. Refer to Figure 5. Counting from the left, we include the median value of  7,  to determine that the data value in the 12th position. This value is 9  so  Q₃ will be 9.

It is also important to recognize that the median of 7 is not an actual data value in this set. It was included in Figure 5 to illustrate that its position must be counted when determing the position of the third quartile. It is not actually part of the data set. The actual data set is illustrated in Figure 6  (and Figure 4).

Consider Figure 7 where the data set that has an even number of data values:   1   2    4    5 

In this data set  Q₁ = 1.5,  Q₂ = 3,  and Q₃ = 4.5   This illustrates that quartile values need not be actual values in the data set. The second quartile Q₂ is 3 which  is the average of the data values 2 and 4. Similarily the first quartile of 1.5 is the average of two data values 1 and 2 and the third quartile of 4.5 is the average of the two data values 4 and 5. Determining the quartile values can become complex as it may require different weightings of the data values but this is beyond the scope of this textbook.

Example 3 illustrates two techniques for determining quartiles when the number of data observations is even.

We have seen that either of Method 1 or Method 2 will produce the same quartile values although the formula method can be less intuitive when n is even.

Key Concepts

• A data set can be divided into one hundred equal parts by ninety-nine percentiles P₁ , P₂ , P₃ , … P₉₉ . Percentiles are best used with large sets of data.
• Quartiles divide the data set into four equal parts. The first quartile, Q₁, is the same as the 25^th percentile, and the third quartile, Q₃, is the same as the 75^th percentile. The median can be called both the second quartile, Q₂ , and the 50^th percentile.
• Quartiles may or may not be actual observations within a set of data.

Glossary

8.1 Exercise Set

1. Your instructor announces to the class that anyone with a midterm exam score of 63% scored in the 80th percentile.  If you received a score of 63% how did you do in relation to your classmates?
2. A test consists of 40 marks. Fifty students wrote the test and their scores are in the table below. Determine the percentiles that are associated with scores of:
   1. 1. 15
      2. 25
      3. 37