Measures of Position

Where does a certain data value fit in relative to the other data values?

Nth Place

• The highest and the lowest• 2nd highest, 3rd highest, etc.• “If I made $60,000, I would be 6th richest.”

Another view: “How does my compare to the mean?”

• “Am I in the middle of the pack?”• “Am I above or below the middle?”• “Am I extremely high or extremely low?”

• Score is the measuring stick

Score: is how many standard deviations away from the mean?

If you know the x value• Population:

• Sample

To work backward from z to x• Population

• Sample

score is also called “Standard Score”

• No matter what is measured in or how large or small the values are….

• The score of the mean will be 0– Because numerator turns out to be 0.

• If is above the mean, its is positive.– Because numerator turns out to be positive

• If is below the mean, its is negative.– Because numerator turns out to be negative

score basics, continued

• Typically round to two decimal places.– Don’t say “0.2589”, say “0.26”

• If not two decimal places, pad– Don’t say “2”, say “2.00”– Don’t say “-1.1”, say “-1.10”

• scores are almost always in the interval . Be very suspicious if you calculate a score that’s not a small number.

Practice computing z scores

• What are the scores for the salary values ?• What are the salaries corresponding to the

scores ?• Helpful necessary information:

Two parallel axes (scales), and

scores can compare unlike values

• Textbook’s example on next slide – they compare test scores on two different tests to ascertain “Which score was the more outstanding of the two?”

• Be careful if the scores turn out to be negative. Which is the better performance? or ?

Example 3-29: Test ScoresA student scored 65 on a calculus test that had a mean of 50 and a standard deviation of 10; she scored 30 on a history test with a mean of 25 and a standard deviation of 5. Compare her relative positions on the two tests.

10Bluman, Chapter 3

She has a higher relative position in the Calculus class.

65 50 1.5 Calculus10

X Xzs

30 25 1.0 History5

X Xzs

Percentiles

• “What percent of the values are lower than my value?”– 90th percentile is pretty high– 50th percentile is right in the middle– 10th percentile is pretty low

• If you scored in the 99th percentile on your SAT, I hope you got a scholarship.

Given value , what’s its percentile?

• With these salary values again

• What’s thepercentile for a salary of $59,000 ?

• You can see it’s going to be higher than 50th.

Example: Finding the percentile

• Count = how many values below $59,000• Formula for percentile

• 78th percentile

Excel will find the percentile

• Excel will compute it but slightly differently.• PERCENTRANK.EXC(cells, value)• For $59,000

Excel gives 0.74• It does some fancy

“interpolation”to come up withits results

Given Percentile, what’s value?

• Formula: position from bottom – Again, how many data values in the set– and the percentile rank that’s given.– If there’s a decimal remainder, drop it.– If it’s integer, take average of th and th.

• 33rd percentile: • So we look 6 positions from the bottom

Given percentile, find (continued)

• 33rd percentile: • So we look 6 positions from the bottom• $43,546

• Excel: =PERCENTILE.EXC(cells,0.33)=$44,411

Quartiles Q1, Q2, Q3

• Data values are arranged from low to high.• The Quartiles divide the data into four groups.• Q2 is just another name for the Median.• Q1 = Find the Median of Lowest to Q2 values

• Q3 = Find the Median of Q2 to Highest values

• It gets tricky, depending on how many values.

Quartiles example

• 0, 10, 20, 30, 40, 50, 60, 70, 80, 90, 100• Q2 = median = 50 in the middle. • Remove it and split into subsets left and right.• Q1 = median(0, 10, 20, 30, 40) = 20

• Q3 = median(60, 70, 80, 90, 100) = 80

Quartiles example

• 10, 20, 30, 40, 50, 60, 70, 80, 90, 100• Q2 = median =. (two middle #s)• 55 isn’t really there so you can’t remove it!• Leave the 50 and 60 in place• Q1 = median(10, 20, 30, 40, 50) = 30

• Q3 = median(60, 70, 80, 90, 100) = 80

Quartiles example

• 0, 10, 20, 30, 40, 50, 60, 70, 80, 90, 100, 110• Q2 = median = (two middle #s). • 55 isn’t really there so you can’t remove it!• Leave the 50 and 60 in place• Q1 = median(0, 10, 20, 30, 40, 50) = 25

• Q3 = median(60, 70, 80, 90, 100, 110) = 85• Two middle numbers happened again!

Quartiles with TI-84

• 0, 10, 20, 30, 40, 50, 60, 70, 80, 90, 100, 110• Put values into a TI-84 List• Use STAT, CALC, 1-Var Stats

Quartiles in Excel

• =QUARTILE.INC(cells, 1 or 2 or 3) seems to give the same results as the old QUARTILE function

• There’s new =QUARTILE.EXC(cells, 1 or 2 or 3)

• Excel does fancy interpolation stuff and may give different Q1 and Q3 answers compared to the TI-84 and our by-hand methods.

Quintiles and Deciles

• You might also encounter– Quintiles, dividing data set into 5 groups.– Deciles, dividing data set into 10 groups.

• Reconcile everything back with percentiles:– Quartiles correspond to percentiles 25, 50, 75– Deciles correspond to percentiles 10, 20, …, 90– Quintiles correspond to percentiles 20, 40, 60, 80

Interquartile Range and Outliers

• Concept: An OUTLIER is a wacky far-out abnormally small or large data value compared to the rest of the data set.

• We’d like something more precise.• Define: IQR = Interquartile Range = Q3 – Q1.• Define: If , is an Outlier.• Define: If , is an Outlier.• (Other books might make different definitions)

Outliers Example

• Here’s an quick elementary example:• Data values 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 20• Mean and

• Anything more than 9 units away from is abnormal. Outlier, Outlier, Pants on Fire.

• The 20 is an outlier.

No-Outliers Example

• Data values 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 10• Mean and

(coincidence that , insignificant)

• Anything more than 9 units away from is abnormal.

• This data set has No Outliers.

Outliers: Good or Bad?

• “I have an outlier in my data set. Should I be concerned?”– Could be bad data. A bad measurement. Somebody

not being honest with the pollster.– Could be legitimately remarkable data, genuine true

data that’s extraordinarily high or low.• “What should I do about it?”

– The presence of an outlier is shouting for attention. Evaluate it and make an executive decision.