HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Section 3.3 Measures of Relative Position With some added content by D.R.S., University of Cordele

Measures of Relative Position “How do I compare with everybody else?” 1.nth place 2.Percentiles a.Given percentile P, find data value there. b.Given data value, what’s its percentile? 3.Quartiles 4.Five Number Summary and the Box Plot diagram 5.Standard Score (also known as z-score) 6.Outliers

N th Place The highest and the lowest 2 nd highest, 3 rd highest, etc. “Olin earned $41,246. He’s in ___ th place out of ___.” 3

Getting a handle on the idea of Percentiles If your test score were at this percentile, do you consider it to be high or low or middleish? 90 th percentile is _______________ (≥90% of the pop.) 70 th percentile is _______________ (≥70% of the pop.) 40 th percentile is _______________ (≥40% of the pop.) 10 th percentile is _______________ (≥10% of the pop.) “Olin’s $ salary is the same or higher than ____% of the population.” FRACTION: > or = how many? how many in population? and convert it to a percent: _____ % =

Two Kinds of Percentile Problems. The ______th Percentile The Data Value is _______ Percentile is given. You have to find the data value. Question is like this: “The salary at the 90 th percentile is $how much?” Data value is given. They ask for percentile. The question is like this: “A $50,000 salary puts you in the the ?th percentile?” Example 3.18 is this kind of problem Example 3.19 is this kind of problem

“What is the data value at the P th percentile?” This is like Example 3.18

HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. If you know the value, what’s its percentile? For this formula, always ROUND in the usual rounding way of rounding (5 or higher round up; 4 or lower chop down)

.. Avoid this common error: If your answer is “36%”, you are WRONG. The correct answer is “The 36 th Percentile”. Percents and Percentiles are related, sure. But good grammar and proper usage matter.

HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Quartiles Q 1 = First Quartile: 25% of the data are less than or equal to this value. Q 2 = Second Quartile: 50% of the data are less than or equal to this value. Q 3 = Third Quartile: 75% of the data are less than or equal to this value.

Quintiles and Deciles You might also encounter – Quintiles, dividing data set into 5 groups. – Deciles, dividing data set into 10 groups. These are done by the Percentile method: – Deciles correspond to percentiles 10, 20, …, 90 – Quintiles correspond to percentiles 20, 40, 60, 80 10

HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Five-Number Summary and Box Plots Interquartile Range (IQR) The interquartile range is the range of the middle 50% of the data, given by IQR = Q 3  Q 1 where Q 3 is the third quartile and Q 1 is the first quartile. How “wide” is the “middle half” of the data set?

HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved.

HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 3.23: Creating a Box Plot Draw a box plot to represent the five-number summary for a data set whose five-number summary was 12.1, 19.8, 23.6, 25.3, 35.9 _____, _____, _____, _____, _____ Step 1: Label the horizontal axis at even intervals.

HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 3.23: Creating a Box Plot (cont.) Step 2:Place a small line segment above each of the numbers in the five ‑ number summary.

HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 3.23: Creating a Box Plot (cont.) Step 3:Connect the line segment that represents Q 1 to the line segment that represents Q 3, forming a box with the median’s line segment in between.

HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 3.23: Creating a Box Plot (cont.) Step 4:Connect the “box” to the line segments representing the minimum and maximum to form the “whiskers.” TI-84 Boxplot information is at this link: 84.pdf

z Scores

The Literature test The mean score was 77 points. The standard deviation was 11 points Sue earned 91 points Find her z score for this test: The Biology test The mean score was 47 points The standard deviation was 6 points Sue earned 55 points Find her z score for this test: 18 On which test did she have the “better” performance?

Interquartile Range and Outliers Extra topic for awareness 19

Outliers Example 20

Outliers Example 21

No-Outliers Example 22

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. 23