section 3.3

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Section 3.3

Measures of Relative Position

With some added contentby D.R.S., University of Cordele

Measures of Relative Position

“How do I compare with everybody else?”

1. nth place2. Percentiles

a. Given percentile P, find data value there.b. Given data value, what’s its percentile?

3. Quartiles4. Five Number Summary and the Box Plot diagram5. Standard Score (also known as z-score)6. Outliers

Nth Place

The highest and the lowest2nd highest, 3rd highest, etc.“Olin earned $41,246. He’s in ___th place out of ___.”

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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?90th percentile is _______________ (≥90% of the pop.)70th percentile is _______________ (≥70% of the pop.)40th percentile is _______________ (≥40% of the pop.)10th percentile is _______________ (≥10% of the pop.)

“Olin’s $41.246 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 90th percentile is $how much?”

Data value is given.They ask for percentile.The question is like this:“A $50,000 salary puts youin 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 Pth percentile?”This is like Example 3.18

Formula: Location (=number of data values)• Your data values are in order from lowest to highest.• Compute Location and then:

If happens to be an exact integer…

If is NOT an exact integer…

…take the average of the values in positions and .

…bump up to the next highest integer (“ceiling”) -- never round down, but always bump up -- and take value in that position.






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 10th percentile.

b. Find the value of the 20th percentile.






Example 3.18: (a) Find the mpg value for the 10th percentile

a. There are ____ values in this data set, thus n = ___. We want the 10th 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 10th percentile is _____ mpg.”

𝐿=𝑛 ∙𝑃100 =¿






Example 3.18: (b) Find the mpg value for the 20th percentile

a. There are ____ values in this data set, thus n = ___. We want the 20th percentile, so P = ___.

Is it an exact integer? ________.so take the data values in position # ______ and #______, and average them.

Answer: “The 20th percentile is ___ mpg.”

Location Calculate:






If you know the value, what’s its percentile?

Pth Percentile of a Data Value Figure out how many values < or = your data value.Formula:

where P is the percentile rounded to the nearest whole number, L is the number of values in the data set less than or equal to the given value, andn is the number of data values in the data set.

For this formula, always ROUND in the usual rounding way of rounding

(5 or higher round up; 4 or lower chop down)






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.






Example 3.19: Finding the Percentile of a Given Data Value (cont.)

The Xterra’s value of _____ mpg is repeated in the data set, in both the 48th and 49th positions, so we will pick the one with the largest location value, which is the ____th. Using a sample size of n = ____ and a location of L = ____, we can substitute these values into the formula for the percentile of a given data value, which gives us the following.

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 36th Percentile”.• Percents and Percentiles are related, sure.• But good grammar and proper usage matter.






Excel gives different answers

Excel does some fancy interpolation

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Quartiles

QuartilesQ1 = First Quartile: 25% of the data are less than or

equal to this value.Q2 = Second Quartile: 50% of the data are less than or

equal to this value.Q3 = Third Quartile: 75% of the data are less than or

equal to this value.






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 25th percentile and that’s Q1.And find the 50th percentile and that’s Q2.And find the 75th 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.






Example 3.20: Finding the Quartiles of a Given Data Set with the Percentile Method

a. Percentile Method First quartile is 25th percentilePosition Second quartile is 50th percentilePosition Third quartile is 75th percentilePosition

Count up to 34th position: “Q1 is 19.8 mpg”

Count up to 68th position: “Q2 is 23.6 mpg”“Median is 23.6 mpg”

Count up to 102nd position: “Q3 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.






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 Q2.

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






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 theFive-Number Summary for a data set.http://www.drscompany.com/edu/Examples/index.htm#!stats/chapter3/section3/position/5number

A difficult, challenging example is at this link:http://www.drscompany.com/edu/Examples/index.htm#!stats/chapter3/section3/position/other/03

http://www.drscompany.com/edu/Examples/index.htm#!stats/chapter3/section3/position/5number

http://www.drscompany.com/edu/Examples/index.htm#!stats/chapter3/section3/position/5number

http://www.drscompany.com/edu/Examples/index.htm#!stats/chapter3/section3/position/other/03

http://www.drscompany.com/edu/Examples/index.htm#!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

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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 = Q3 - Q1 where Q3 is the third quartile andQ1 is the first quartile.

For the vehicle mpg ratings example, IQR = _____ - _____ = _____ mpg

How “wide” is the “middle half”of the data set?






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, 35.9. Solution Step 1: Label the horizontal axis at even intervals.






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. ‑







Step 3: Connect the line segment that represents Q1 to the line segment that represents Q3, forming a

box with the median’s line segment in between.







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:http://www.drscompany.com/edu/QuickNotes/Statistics/DataDescription/Boxplot_TI-84.pdf

http://www.drscompany.com/edu/QuickNotes/Statistics/DataDescription/Boxplot_TI-84.pdf

http://www.drscompany.com/edu/QuickNotes/Statistics/DataDescription/Boxplot_TI-84.pdf






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 ___________________.

𝑧=𝑥−𝜇𝜎






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 ____________________.

x xz

s-

x

Standard Score answers the question“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 stickIf z= 0, then I’m ________________________.If z > 0,then I’m ________________________.If z < 0, then I’m ________________________.z is almost always between _____ and _____.

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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.

𝑧=𝑥−𝜇𝜎

𝑧=❑

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

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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

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.

score values

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.

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Example: Using scores to compare unlike items

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:

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

scores caution with negatives

Example: 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 ?Stop and think back to your basic number line and the meaning of “<“ and “>”

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Interquartile Range and Outliers Extra topic for awareness

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)

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Outliers Example

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

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Outliers Example

Data values 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 20We found IQR = 6 and the mean is 6.8One definition uses to define outliersHere, Anything more than 9 units away from is then considered to be abnormally small or large., nothing smaller than : the 20 is an outlier.

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No-Outliers Example

Data values 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 10Mean and (coincidence that , insignificant)

Anything more than 9 units away from is abnormal. This data set has No Outliers.

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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.

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section 3.3

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