1. Interquartile Range Lecture 21 Sec. 5.3.1 – 5.3.3 Mon, Feb 23, 2004

2. Measuring Variation or Spread Static view – Given a sample or a population, how spread out is the distribution? Static view – Given a sample or a population, how spread out is the distribution? Dynamic view – If we are taking measurements on units in the sample or population, how much will our measurements vary from one to the next? Dynamic view – If we are taking measurements on units in the sample or population, how much will our measurements vary from one to the next?

3. Measures of Variation or Spread These are two aspects of the same phenomenon. These are two aspects of the same phenomenon. The more variability there is in a population, the more difficult it is to estimate its parameters. The more variability there is in a population, the more difficult it is to estimate its parameters. For example, it is easier to estimate the average size of a crow than the average size of a dog (thanks to selective breeding). For example, it is easier to estimate the average size of a crow than the average size of a dog (thanks to selective breeding).

4. The Range By far, the simplest measure of spread is the range. By far, the simplest measure of spread is the range. Range – The difference between the largest value and the smallest value of a sample or population. Range – The difference between the largest value and the smallest value of a sample or population. How would you expect the range of a sample compare to the range of the population? How would you expect the range of a sample compare to the range of the population?

5. The Range Is the sample range a good estimator of the population range? Is the sample range a good estimator of the population range? Would you expect it to systematically overestimate or underestimate the population range? Why? Would you expect it to systematically overestimate or underestimate the population range? Why? In general, the range is a poor measure of variability since it does not take into account how the values are distributed in between the maximum and the minimum. In general, the range is a poor measure of variability since it does not take into account how the values are distributed in between the maximum and the minimum.

6. Percentiles The p th percentile – A value that separates the lower p% of a sample or population from the upper The p th percentile – A value that separates the lower p% of a sample or population from the upper (100 – p)%. The median is the 50 th percentile; it separates the lower 50% from the upper 50%. The median is the 50 th percentile; it separates the lower 50% from the upper 50%. The 25 th percentile separates the lower 25% from the upper 75%. The 25 th percentile separates the lower 25% from the upper 75%.

7. Finding percentiles To find the p th percentile, compute the value To find the p th percentile, compute the value r = (p/100)  (n + 1). This gives the position (r = rank) of the p th percentile. This gives the position (r = rank) of the p th percentile. Round r to the nearest whole number. Round r to the nearest whole number. We will use the number in that position as the p th percentile. We will use the number in that position as the p th percentile.

8. Finding percentiles Special case: If r is a “half-integer,” for example 10.5, then take the average of the numbers in positions r and r + 1, just as we did for the median when n was even. Special case: If r is a “half-integer,” for example 10.5, then take the average of the numbers in positions r and r + 1, just as we did for the median when n was even. Note: The “official” procedure says to interpolate when r is not a whole number. Note: The “official” procedure says to interpolate when r is not a whole number. Therefore, by rounding, our answers may differ from the official answer. Therefore, by rounding, our answers may differ from the official answer.

9. Example Find the 30 th percentile of Find the 30 th percentile of 5, 6, 8, 10, 15, 30. p = 30 and n = 6. p = 30 and n = 6. Compute r = (30/100)(7) = 2.1  2. Compute r = (30/100)(7) = 2.1  2. The 30 th percentile is 6. The 30 th percentile is 6. The “official” answer is 6.2. The “official” answer is 6.2. Find the 35 th percentile. Find the 35 th percentile.

10. Quartiles The first quartile is the 25 th percentile. The first quartile is the 25 th percentile. The second quartile is the 50 th percentile, which is also the median. The second quartile is the 50 th percentile, which is also the median. The third quartile is the 75 th percentile. The third quartile is the 75 th percentile. The first quartile is denoted Q1. The first quartile is denoted Q1. The third quartile is denoted Q3. The third quartile is denoted Q3. There are also quintiles and deciles. There are also quintiles and deciles.

11. The Interquartile Range The interquartile range (IQR) is the difference between Q3 and Q1. The interquartile range (IQR) is the difference between Q3 and Q1. The IQR is a commonly used measure of spread. The IQR is a commonly used measure of spread. Like the median, it is not affected by extreme outliers. Like the median, it is not affected by extreme outliers.

12. Example See Example 5.4, p. 281. See Example 5.4, p. 281. n = 20. n = 20. For Q1, r = (0.25)(21) = 5.25  5. For Q1, r = (0.25)(21) = 5.25  5. Q1 = 41. Q1 = 41. For Q3, r = (0.75)(21) = 15.75  16. For Q3, r = (0.75)(21) = 15.75  16. Q3 = 47. Q3 = 47. Therefore, IQR = 47 – 41 = 6. Therefore, IQR = 47 – 41 = 6.

13. Computing Quartiles on the TI-83 Follow the procedure used to find the mean and the median. Follow the procedure used to find the mean and the median. Scroll down the display to find Q1 and Q3. Scroll down the display to find Q1 and Q3.

14. Computing Quartiles on the TI-83 In our last example, the TI-83 says that In our last example, the TI-83 says that Q1 = 41 Q1 = 41 Q3 = 46.5 Q3 = 46.5 Other software might compute Q3 = 46.75. Other software might compute Q3 = 46.75.