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. Example 3.18: Finding Data Values Given the Percentiles A car manufacturer is studying the highway miles per gallon (mpg) for a wide range of makes and models of vehicles. See separate handout for the data. a. Find the value of the 10 th percentile. b. Find the value of the 20 th percentile.

HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 3.18: (a) Find the mpg value for the 10 th percentile a.There are ____ values in this data set, thus n = ___. We want the 10 th percentile, so P = ___. Compute Location Is it an exact integer? No. ALWAYS BUMP UP, so take the data value in position # ______, which is ______ mpg. Answer: “The 10 th percentile is _____ mpg.”

HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 3.18: (b) Find the mpg value for the 20 th percentile a.There are ____ values in this data set, thus n = ___. We want the 20 th percentile, so P = ___. Is it an exact integer? ________. so take the data values in position # ______ and #______, and average them. Answer: “The 20 th percentile is ___ mpg.”

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)

HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 3.19: Finding the Percentile of a Given Data Value In the data set from the previous example, the Nissan Xterra averaged 21.1 mpg. In what percentile is this value? Solution We begin by making sure that the data are in order from smallest to largest. We know from the previous example that they are, so we can proceed with the next step.

HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 3.19: Finding the Percentile of a Given Data Value (cont.) Your calculation here: And round to nearest whole number: ______th percentile This is your answer.

.. 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. Excel gives different answers Excel does some fancy interpolation 14

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.

HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 3.20: Finding the Quartiles of a Given Data Set – TWO DIFFERENT WAYS Using the mpg data from the previous examples, find the quartiles. a.Use the percentile method to find the quartiles. b. Use the approximation method to find the quartiles. c. How do these values compare? The Percentile method says to find the 25 th percentile and that’s Q1. And find the 50 th percentile and that’s Q2. And find the 75 th percentile and that’s Q3. The approximation method says to find the median and that’s Q2. If it landed on an actual value (odd # of data values), don’t include it in next steps. Q1 is the median of the values to the left of Q2. Q3 is the median of the values to the right of Q2.

HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 3.20: Finding the Quartiles of a Given Data Set with the Percentile Method Count up to 34 th position: “Q 1 is 19.8 mpg” Count up to 68 th position: “Q 2 is 23.6 mpg” “Median is 23.6 mpg” Count up to 102 nd position: “Q 3 is 25.3 mpg” If we hurry through these, it’s because most or all of the problems seem to be done with the Approximation Method instead.

HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Positions Pos.#____ Positions # #68 - ____ thru ____ _____ -135 Example 3.20: Finding the Quartiles with the Approximation Method b.Approximation Method (probably more common in this course, and also same as TI-84’s 1-Var Stats) First find the Median, that’s same as Q 2. Q1 = median of these Q3 = median of these Positions #1, 2, 3,…, 67 Position #______ Positions #69, 70, 71, …, 135 has data ______ mpg Positions # Pos.#____ Positions # 1 thru ____ _____mpg ___ thru 67 Q2 Q1 Q3

HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 3.20: Finding the Quartiles of a Given Data Set (cont.) c. If you put all 135 data values into a TI-84 list and did 1-Var Stats, the results look like this. Scroll down to the second page of results. n is __________________________________ minX is _______________________________ Q1 is _________________________________ Q2 is same as Med which is _______________ Q3 is_________________________________ maxX is _______________________________

Additional examples of finding Quartiles and the Five-Number Summary for a data set. #!stats/chapter3/section3/position/5number A difficult, challenging example is at this link: #!stats/chapter3/section3/position/other/03

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 21

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. For the vehicle mpg ratings example, IQR = _____ - _____ = _____ mpg 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. Example 3.23: Creating a Box Plot Draw a box plot to represent the five-number summary from the previous example. Recall that the five-number summary was 12.1, 19.8, 23.6, 25.3, Solution 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

HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Standard Scores Standard Score The standard score for a population value is given by where x is the value of interest from the population, μ is the population _____________ σ is the population ___________________.

HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Standard Scores The standard score for a sample value is given by where x is the value of interest from the sample, is the sample _____________ s is the sample ____________________.

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HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 3.25: Calculating a Standard Score If the mean score on the math section of the SAT test is 500 with a standard deviation of 150 points, what is the standard score for a student who scored a 630? Solution (note this formula is for a ______________) μ = 500 and σ = 150. The value of interest is x = 630, so we have the following.

Excel STANDARDIZE function to convert a data value (x) to a standard score (z)

If you know the x valueTo work backward from z to x 32 These formulas agree with the labeling of the axes you did in the Empirical Rule and Chebyshev’s Theorem problems. In those problems, the z values were always nice integers: -3, -2, -1, 0, 1, 2, 3.

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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: 34 On which test did she have the “better” performance?

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Interquartile Range and Outliers Extra topic for awareness 36

Outliers Example 37

Outliers Example 38

No-Outliers Example 39

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. 40