Measuring Variation Lecture 16 Sec. 5.3.1 – 5.3.3 Mon, Oct 4, 2004

Measuring Variation or Spread 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?

Measures of Variation or Spread These are two aspects of the same phenomenon. The more variability or spread there is in a population, the more difficult it is to estimate its parameters. Examples: Hurricane history Hurricane computer models

The Range Range – The difference between the largest value and the smallest value of a sample or population.

Questions about the Range How would you expect the range of a sample compare to the range of the population? Would you expect it to systematically overestimate or underestimate the population range? Why? Is the sample range a good estimator of the population range?

Percentiles The pth percentile – A value that separates the lower p% of a sample or population from the upper (100 – p)%. p% of the values fall at or below the pth percentile. (100 – p)% of the values fall at or above the pth percentile.

Excel’s Percentile Formula To find position, or rank, of the pth percentile, compute the value

Excel’s Percentile Formula This gives the position (r = rank) of the pth percentile. Round r to the nearest whole number. The number in that position is the pth percentile.

Excel’s Percentile Formula 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. M icrosoft Excel will interpolate whenever r is not a whole number. Therefore, by rounding, our answers may differ from Excel.

Example Find the 30th percentile of 5, 6, 8, 10, 15, 30. p = 30 and n = 6. Compute r = 1 + (30/100)(6 – 1) = 2.5. The 30th percentile is the average of the 2nd and 3rd numbers, i.e., 7. Find the 35th percentile. Excel spreadsheet Percentiles.xls.

Excel’s Percentile Formula The formula may be reversed to find the percentile percentage of a number, given its position, or rank, in the sample. The formula is

Example In the sample 5, 6, 8, 10, 15, 30 what percentile percentage is associated with 15? n = 6 and r = 5. Compute p = 100(5 – 1)/(6 – 1) = 80. Therefore, 15 is the 80th percentile. What is the percentile percentage of 8?

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

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

Example Example 5.4, p. 281. Note the discrepancies in the answers. For Q1, r = 1 + (0.25)(19) = 5.75  6. Q1 = 41. For Q3, r = 1 + (0.75)(19) = 15.25  15. Q3 = 46. Therefore, IQR = 46 – 41 = 5. Excel spreadsheet Ages.xls. Note the discrepancies in the answers.

TI-83 – Finding Quartiles Follow the procedure used to find the mean and the median. Scroll down the display to find Q1 and Q3.

Example Use the TI-83 to find Q1 and Q3 for the age data.

Let’s Do It! Collect data from the class on the number of miles from HSC to home. Find Q1 and Q3. Find the 40th percentile. Find the 90th percentile.

Homework For the age data (p. 184), use the formula, with rounding, to find The 10th percentile. The 43rd percentile. The 69th percentile. The 95th percentile. Use the formula to find the percentile percentages of the following ages. 41, 47, 50, 39.