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The third quartile , Q 3, is nine. Three-fourths (75%) of the ordered data set are less than nine. One-fourth (25%) of the ordered data set are greater than nine. The third quartile is part of the data set in this example.
The interquartile range is a number that indicates the spread of the middle half or the middle 50% of the data. It is the difference between the third quartile ( Q 3 ) and the first quartile ( Q 1 ).
IQR = Q 3 – Q 1
The IQR can help to determine potential outliers . A value is suspected to be a potential outlier if it is less than (1.5)( IQR ) below the first quartile or more than (1.5)( IQR ) above the third quartile . Potential outliers always require further investigation.
A potential outlier is a data point that is significantly different from the other data points. These special data points may be errors or some kind of abnormality or they may be a key to understanding the data.
For the following 13 real estate prices, calculate the
IQR and determine if any prices are potential outliers. Prices are in dollars.
389,950; 230,500; 158,000; 479,000; 639,000; 114,950; 5,500,000; 387,000; 659,000; 529,000; 575,000; 488,800; 1,095,000
Order the data from smallest to largest.
114,950; 158,000; 230,500; 387,000; 389,950; 479,000; 488,800; 529,000; 575,000; 639,000; 659,000; 1,095,000; 5,500,000
M = 488,800
Q 1 = = 308,750
Q 3 = = 649,000
IQR = 649,000 – 308,750 = 340,250
(1.5)( IQR ) = (1.5)(340,250) = 510,375
Q 1 – (1.5)( IQR ) = 308,750 – 510,375 = –201,625
Q 3 + (1.5)( IQR ) = 649,000 + 510,375 = 1,159,375
No house price is less than –201,625. However, 5,500,000 is more than 1,159,375. Therefore, 5,500,000 is a potential outlier .
For the following 11 salaries, calculate the IQR and determine if any salaries are outliers. The salaries are in dollars.
Order the data from smallest to largest.
Median = $54,000
Q 1 = $40,500
Q 3 = $69,000
IQR = $69,000 – $40,500 = $28,500
(1.5)( IQR ) = (1.5)($28,500) = $42,750
Q 1 – (1.5)( IQR ) = $40,500 – $42,750 = –$2,250
Q 3 + (1.5)( IQR ) = $69,000 + $42,750 = $111,750
No salary is less than –$2,250. However, $120,000 is more than $11,750, so $120,000 is a potential outlier.
For the two data sets in the test scores example , find the following:
The five number summary for the day and night classes is
| Minimum | Q 1 | Median | Q 3 | Maximum | |
|---|---|---|---|---|---|
| Day | 32 | 56 | 74.5 | 82.5 | 99 |
| Night | 25.5 | 78 | 81 | 89 | 98 |
The IQR for the night group is Q 3 – Q 1 = 89 – 78 = 11
The interquartile range (the spread or variability) for the day class is larger than the night class IQR . This suggests more variation will be found in the day class’s class test scores.
Since the minimum and maximum values for the day class are greater than 16.25 and less than 122.25, there are no outliers.
Night class outliers are calculated as:
For this class, any test score less than 61.5 is an outlier. Therefore, the scores of 45 and 25.5 are outliers. Since no test score is greater than 105.5, there is no upper end outlier.
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