Range and Interquartile Range

Range: the difference between the highest and lowest values in a set of data
Example- Set A: 1, 2, 3, 4, 5, 6, 7, 13
Range: 13-1=12
Interquartile Range: the range of the middle 50% of the data. Since this process excludes outliers, the interquartile range is a more accurate representation of the "spread" of the data than range. The quartiles divide the data between the 25th, 50th, and 75th percentiles. The median of the values between the 25th and 50th percentiles is called the lower quartile, and the median of the values between the 50th and 75th percentiles is called the upper quartile.

Note: "percentile" refers to the data value for which the stated percentage of the data points lie below that value. For example, if the the 10th percentile is 86.2, then 10% of the data are less than or equal to 86.2 .

Example- Set B: 1, 13, 6, 21, 19, 2, 137

  1. Put the data in numerical order: 1, 2, 6, 13, 19, 21, 137
  2. Find the median of the data: 13
  3. Divide the data into four QUARTILES by finding the median of all the numbers below the median of the full set and then find the median of all the numbers above the median of the full set.*
  4. Subtract the lower quartile from the upper quartile: 21-2=19. This is the Interquartile range, or IQR.
*How to find the quartiles:
  1. To find the lower quartile, take all of the numbers below the median: 1, 2, 6
  2. Find the median of these numbers
    1. Take the first and last number in the subset and add their POSITIONS (not values) and divide by two. This will give you the position of your median: 1+3=4/2=2, so the median of the subset is the second position, which is two.
    2. If there is an even number of values, then the position of the median will be in between two numbers. In that case, take the average of the two numbers that the median is between. Example: 1, 3, 7, 12. Median is 1+4=5/2=2.5th position, so it is the average of the second and third positions, which is 3+7=10/2=5. This median separates the first and second quartiles.
3. Repeat with numbers above the median of the full set: 19, 21, 137. Median is 1+3=4/2=2nd position, which is 21. This median separates the third and fourth quartiles.

A boxplot (or box and whisker plot) separates the quartiles of the data. All outliers are displayed as regular points on the graph. The vertical line in the box indicates the location of the median of the data. The box starts at the lower quartile and ends at the upper quartile, so the difference, or length of the boxplot, is the IQR.box_plot_hi_billy.png
On this boxplot, the IQR is about 300, because Q1 starts at about 300 and Q3 ends at 600, and 600-300=300.
In a boxplot, if the median (Q2 vertical line) is in the center of the box, the distribution is symmetrical. If the median is to the left of the data (such as in the graph above), then the distribution is considered to be skewed right, because there is more data on the right side of the median. Similarly, if the median is on the right side of the box, the distribution is skewed left, because there is more data on the left side. The range of this data is 1700 (biggest outlier)- -500 (smallest outlier)=2200. If you wanted to leave out the outliers for a more accurate reading, you would subtract the values at the ends of both "whiskers:" 1000-0=1000. To calculate whether something is truly an outlier or not you use the formula 1.5(IQR). Once you get that number, the range that includes numbers that are not outliers is [Q1-1.5IQR, Q3+1.5IQR]. Anything lying outside those numbers are true outliers.

Practice question 1- 13 students receive these scores on their tests,
60, 78, 90, 67, 88, 92, 81, 100, 95, 83, 83, 86, 74
what is the interquartile range for the data?


Source: McGraw-Hill's SAT Subject Test: Math Level 2, Second Edition. John J. Diehl

Practice question 2- Consider the boxplot below.

Which of the following statements are true?
I. The distribution is skewed right.
II. The interquartile range is about 8.
III. The median is about 10.
(A) I only
(B) II only
(C) III only
(D) I and III
(E) II and III
The correct answer is (B). Most of the observations are on the high end of the scale, so the distribution is skewed left. The IQR is indicated by the length of the box, which is 18 minus 10 or 8. And the median is indicated by the vertical line running through the middle of the box, which is roughly centered over 15. So the median is about 15. Source: StatTrek. AP Statistics: Boxplots.