Lesson 2.2 Quartiles and Percentiles

Quartiles

Quartiles divide an ordered set (smallest to largest) of data into quarters. Consider the following ordered set of 17 data values:

{2, 2, 3, 3.5, 4, 4, 4, 6, 7.5, 8, 8, 10, 10, 11.5, 12, 12, 12}

The value that divides the set in halves is called the second quartile (Q2). The second quartile, Q2, is equal to 7.5. The second quartile is also called the median and the 50th percentile.

The lower half of the data is

2, 2, 3, 3.5, 4, 4, 4, 6

The value that divides the lower half into halves is called the first quartile (Q1). The first quartile, Q1, is between the two middle values 3.5 and the first 4.

Q1 = (3.5 + 4)/2 = 3.75

Notice that 3.75 is not part of the data.

The upper half of the data is:

8, 8, 10, 10, 11.5, 12, 12, 12

The value that divides the upper half into halves is called the third quartile (Q3). The third quartile, Q3, is between the two middle values 10 and 11.5.

Q3 = (10 + 11.5)/2 = 10.75

Notice that 10.75 is not part of the data.

• The data that falls below Q1= 3.75 is (2, 2, 3, 3.5) and is 25% of the data. We say that 25% of the data falls below Q1 = 3.75.
  
• The data that is more than Q1 = 3.75 but less than Q2 = 7.5 is (4, 4, 4, 6) and is 25% of the data. We say that 25% of the data falls between Q1 =3.75 and Q2 = 7.5.
  
• The data that is more than Q2 = 7.5 but less than Q3 = 10.75 (8, 8, 10, 10) is 25% of the data. We say that 25% of the data falls between Q2 = 7.5 and Q3 = 10.75.
  
• The data that falls above Q3 = 10.75 (11.5,12, 12, 12) is 25% of the data. We say that 25% of the data falls above Q3 = 10.75.

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Percentiles

Percentiles divide an ordered set (smallest to largest) of data into hundredths. Consider the ordered set of the 100 numbers 1, 2, 3, 4, 5, ..., 99, 100. Ten percent of 100 numbers is 10 numbers. The 10 numbers 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 fall below the 10th percentile. This means that the 10th percentile is between 10 and 11. The 10th percentile (10th %ile) is equal to 10.5. Similarly, the 90th percentile (90th %ile) is equal to 90.5.

An easy way to find percentiles is to look at a frequency table that has a Cum. RF column (see Frequency Tables in Lesson 1.6).

The following table has, as data, the number of DVDs 20 students watched at home during a recent one week holiday.

• 50% of the data include the two 0s, the four 1s, and four of the 2s (10 values) since 50% of 20 is 10. The 50th %ile is between the 10th and 11th values and is equal to 2. If you look at the Cum. RF column, you can see that all the 0s and 1s are 30% of the data. All the 0s, 1s, and 2s are 70% of the data. You can see that the 50th %ile has to fall within the 2s. Notice that the 50th %ile is the same as Q2 and the median.
  
• The 25th %ile falls within the 1s because the two 0s are 10% of the data and the two 0s together with the four 1s are 30% of the data. The 25th%ile is 1. Notice that the 25th%ile = Q1.
  
• 70% of the data include all the 0s, 1s, and 2s (14 values). Since, by definition, 70% of the data fall below the 70th %ile, the 70th %ile is between the last 2 (14th value) and the first 3 (15th value). The 70th %ile = 2.5.

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Think About It

A measure that is often used to find the range of the middle 50% of the data is called the Interquartile Range (IQR). The Interquartile Range is defined as the difference between the third and first quartiles.

IQR = Q3 - Q1

For the Percentile DVD example above, find the IQR if Q3 = 3 and Q1 = 1.

The IQR can help to determine outliers. A value is suspected to be an outlier if it is more than (1.5)(IQR) below Q1 or (1.5)(IQR) above Q3.

On the TI-83, the quartiles are calculated at the same time the mean is calculated. The use of TI-83 is a great example of how easy it is to calculate the quartiles using technology.

Please continue to the next section of this lesson.

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Up » 2.1 Graph » 2.2 Quartiles and Percentiles » 2.3 Mean, Median and Mode » 2.4 Variance and Standard Deviation