Lecture 16 Sec – Wed, Feb 15, 2006 Measuring Variation 1

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? These are two aspects of the same phenomenon. These are two aspects of the same phenomenon. Measuring Variation or Spread

Measures of Variation or Spread The more variability or spread there is in a population, the more difficult it is to estimate its parameters. The more variability or spread there is in a population, the more difficult it is to estimate its parameters. That is, when variability is low, a small sample is likely to be representative, but when variability is high, a small sample is not likely to be representative. That is, when variability is low, a small sample is likely to be representative, but when variability is high, a small sample is not likely to be representative.

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.

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

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

Percentiles – Textbook’s Definition Find the 25 th percentile of the following sample: Find the 25 th percentile of the following sample: 20, 35, 40, 50, 80. Consider each number: Consider each number: 20 has 20% at or below and 100% at or above. 20 has 20% at or below and 100% at or above. 35 has 40% at or below and 80% at or above. 35 has 40% at or below and 80% at or above. 40 has 60% at or below and 60% at or above. 40 has 60% at or below and 60% at or above. 50 has 80% at or below and 40% at or above. 50 has 80% at or below and 40% at or above. 80 has 100% at or below and 20% at or above. 80 has 100% at or below and 20% at or above. Therefore, 35 is the 25 th percentile. Therefore, 35 is the 25 th percentile.

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

Excel’s Percentile Formula 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. The number in that position is the p th percentile. The number in that position is the p th 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. 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. Microsoft Excel will interpolate whenever r is not a whole number. Microsoft Excel will interpolate whenever r is not a whole number. Therefore, by rounding, our answers may differ from Excel. Therefore, by rounding, our answers may differ from Excel.

Example Find the 25 th percentile of Find the 25 th percentile of 20, 35, 40, 50, 80. p = 25 and n = 5. p = 25 and n = 5. Compute r = 1 + (25/100)(5 – 1) = 2. Compute r = 1 + (25/100)(5 – 1) = 2. The 25 th percentile is the 2 nd number, i.e., 35. The 25 th percentile is the 2 nd number, i.e., 35. Excel spreadsheet: HSC Prof Tenures 2.xls. Excel spreadsheet: HSC Prof Tenures 2.xls.HSC Prof Tenures 2.xlsHSC Prof Tenures 2.xls

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

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

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.

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.

Example Example 5.4, p Example 5.4, p n = 20. n = 20. For Q1, r = 1 + (0.25)(19) = 5.75  6. For Q1, r = 1 + (0.25)(19) = 5.75  6. Q1 = 41. Q1 = 41. For Q3, r = 1 + (0.75)(19) =  15. For Q3, r = 1 + (0.75)(19) =  15. Q3 = 46. Q3 = 46. Therefore, IQR = 46 – 41 = 5. Therefore, IQR = 46 – 41 = 5. Excel spreadsheet HSC Prof Tenures 2.xls. Excel spreadsheet HSC Prof Tenures 2.xls.HSC Prof Tenures 2.xlsHSC Prof Tenures 2.xls Note the discrepancies in the answers. Note the discrepancies in the answers.

TI-83 – Finding Quartiles 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.

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

Example Use the spreadsheet HSC Prof Tenures 2.xls. Use the spreadsheet HSC Prof Tenures 2.xls.HSC Prof Tenures 2.xlsHSC Prof Tenures 2.xls Find Q1 and Q3. Find Q1 and Q3. Find the IQR. Find the IQR.

Homework (2 problems – 10 points each) For the % on-time-arrival data (p. 252), use the formula, with rounding, to find For the % on-time-arrival data (p. 252), use the formula, with rounding, to find The 10 th percentile. The 10 th percentile. The 43 rd percentile. The 43 rd percentile. The 69 th percentile. The 69 th percentile. The 95 th percentile. The 95 th percentile. Use the formula to find the percentile percentages, with rounding, of the following % on-time arrivals. Use the formula to find the percentile percentages, with rounding, of the following % on-time arrivals. 76.0, 81.1, 85.8, , 81.1, 85.8, 90.3.