Interquartile RangeInterquartile Range

Lecture 21Lecture 21

Sec. 5.3.1 – 5.3.3Sec. 5.3.1 – 5.3.3

Mon, Feb 23, 2004Mon, Feb 23, 2004

Measuring Variation or Measuring Variation or SpreadSpread Static view – Given a sample or a Static view – Given a sample or a

population, how population, how spread outspread out is the is the distribution?distribution?

Dynamic view – If we are taking Dynamic view – If we are taking measurements on units in the measurements on units in the sample or population, how much sample or population, how much will our measurements will our measurements varyvary from from one to the next?one to the next?

Measures of Variation Measures of Variation or Spreador Spread These are two aspects of the same These are two aspects of the same

phenomenon.phenomenon. The more variability there is in a The more variability there is in a

population, the more difficult it is population, the more difficult it is to estimate its parameters.to estimate its parameters.

For example, it is easier to For example, it is easier to estimate the average size of a estimate the average size of a crow than the average size of a crow than the average size of a dog (thanks to selective breeding).dog (thanks to selective breeding).

The RangeThe Range

By far, the simplest measure of By far, the simplest measure of spread is the range.spread is the range.

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

How would you expect the range of How would you expect the range of a sample compare to the range of a sample compare to the range of the population?the population?

The RangeThe Range

Is the sample range a good estimator of Is the sample range a good estimator of the population range?the population range?

Would you expect it to systematically Would you expect it to systematically overestimate or underestimate the overestimate or underestimate the population range? Why?population range? Why?

In general, the range is a poor measure of In general, the range is a poor measure of variability since it does not take into variability since it does not take into account how the values are distributed in account how the values are distributed in between the maximum and the minimum.between the maximum and the minimum.

PercentilesPercentiles

The The ppthth percentile percentile – A value that – A value that separates the lower separates the lower pp% of a sample % of a sample or population from the upper or population from the upper (100 – (100 – pp)%.)%.

The median is the 50The median is the 50thth percentile; it percentile; it separates the lower 50% from the separates the lower 50% from the upper 50%.upper 50%.

The 25The 25thth percentile separates the percentile separates the lower 25% from the upper 75%.lower 25% from the upper 75%.

Finding percentilesFinding percentiles

To find the To find the ppthth percentile, compute percentile, compute the valuethe value

rr = ( = (pp/100) /100) ( (nn + 1). + 1). This gives the position (This gives the position (rr = rank) of = rank) of

the the ppthth percentile. percentile. Round Round rr to the nearest whole number. to the nearest whole number. We will use the number in that We will use the number in that

position as the position as the ppthth percentile. percentile.

Finding percentilesFinding percentiles

Special case: If Special case: If rr is a “half-integer,” for is a “half-integer,” for example 10.5, then take the average example 10.5, then take the average of the numbers in positions of the numbers in positions rr and and rr + 1, + 1, just as we did for the median when just as we did for the median when nn was even.was even.

Note: The “official” procedure says to Note: The “official” procedure says to interpolate when interpolate when r r is not a whole is not a whole number.number.

Therefore, by rounding, our answers Therefore, by rounding, our answers may differ from the official answer.may differ from the official answer.

ExampleExample

Find the 30Find the 30thth percentile of percentile of

5, 6, 8, 10, 15, 30.5, 6, 8, 10, 15, 30. pp = 30 and = 30 and nn = 6. = 6. Compute Compute rr = (30/100)(7) = 2.1 = (30/100)(7) = 2.1 2. 2. The 30The 30thth percentile is 6. percentile is 6.

The “official” answer is 6.2.The “official” answer is 6.2. Find the 35Find the 35thth percentile. percentile.

QuartilesQuartiles

The The first quartilefirst quartile is the 25 is the 25thth percentile. percentile. The The second quartilesecond quartile is the 50 is the 50thth

percentile, which is also the median.percentile, which is also the median. The The third quartilethird quartile is the 75 is the 75thth percentile. 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 There are also quintilesquintiles and and decilesdeciles..

The Interquartile The Interquartile RangeRange The The interquartile rangeinterquartile range (IQR) is (IQR) is

the difference between Q3 and the difference between Q3 and Q1.Q1.

The IQR is a commonly used The IQR is a commonly used measure of spread.measure of spread.

Like the median, it is not affected Like the median, it is not affected by extreme outliers.by extreme outliers.

ExampleExample

See Example 5.4, p. 281.See Example 5.4, p. 281. nn = 20. = 20. For Q1, For Q1, rr = (0.25)(21) = 5.25 = (0.25)(21) = 5.25 5. 5.

Q1 = 41.Q1 = 41. For Q3, For Q3, rr = (0.75)(21) = 15.75 = (0.75)(21) = 15.75

16.16. Q3 = 47.Q3 = 47.

Therefore, IQR = 47 – 41 = 6.Therefore, IQR = 47 – 41 = 6.

Computing Quartiles Computing Quartiles on the TI-83on the TI-83 Follow the procedure used to find Follow the procedure used to find

the mean and the median.the mean and the median. Scroll down the display to find Q1 Scroll down the display to find Q1

and Q3.and Q3.

Computing Quartiles Computing Quartiles on the TI-83on the TI-83 In our last example, the TI-83 In our last example, the TI-83

says thatsays that Q1 = 41Q1 = 41 Q3 = 46.5Q3 = 46.5

Other software might compute Q3 Other software might compute Q3 = 46.75.= 46.75.