1 / 27

Measures of Position

Measures of Position. Where does a certain data value fit in relative to the other data values?. N th Place. The highest and the lowest 2 nd highest, 3 rd highest, etc. “If I made $60,000, I would be 6 th richest.”. Another view: “How does my compare to the mean?”.

spiro
Download Presentation

Measures of Position

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript


  1. Measures of Position Where does a certain data value fit in relative to the other data values?

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

  3. 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

  4. Score: is how many standard deviations away from the mean? If you know the x value To work backward from z to x Population Sample • Population: • Sample

  5. 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

  6. 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.

  7. Practice computing z scores • What are the scores for the salary values ? • What are the salaries corresponding to the scores ? • Helpful necessary information:

  8. Two parallel axes (scales), and

  9. 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 ?

  10. Example 3-29: Test Scores A 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. She has a higher relative position in the Calculus class. Bluman, Chapter 3

  11. 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.

  12. 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.

  13. Example: Finding the percentile • Count = how many values below $59,000 • Formula for percentile • 78th percentile

  14. Excel will find the percentile • Excel will compute it but slightly differently. • PERCENTRANK.EXC(cells, value) • For $59,000Excel gives 0.74 • It does some fancy“interpolation”to come up withits results

  15. 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

  16. Given percentile, find (continued) • 33rd percentile: • So we look 6 positions from the bottom • $43,546 • Excel: =PERCENTILE.EXC(cells,0.33)=$44,411

  17. 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.

  18. 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

  19. 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

  20. 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!

  21. 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

  22. 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.

  23. 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

  24. 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)

  25. 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.

  26. 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.

  27. 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.

More Related