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Chapter 2: Describing LocationIn a Distribution

Section 2.1

Measures of Relative Standing

And Density Curves

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Case Study

• Read page 113 in your textbook

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Where are we headed?

Analyzed a set of observations

graphically and numerically

Consider individual observations

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Consider this data set:

6 7

7 2334

7 5777899

8 00123334

8 569

9 03

How good is

this score

relative to the

others?

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Measuring Relative Standing: z-scores

• Standardizing: converting scores from the original values to standard deviation units

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Measuring Relative Standing:z-scores

A z-score tells us how many standard deviations away from the mean the original observation falls, and in

which direction.

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Practice: Let’s Do p. 118 #1

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Measuring Relative Standing:Percentiles

• Norman got a 72 on the test. Only 2 of the 25 test scores in the class are at or below his.

• His percentile is 2/25 = 0.08, or 8%. So he scores in the 8th percentile.

6 7

7 2334

7 5777899

8 00123334

8 569

9 03

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Density Curves

Histogram of the scores of all 947 seventh-grade students in Gary, Indiana.

The histogram is:

•Symmetric

•Both tails fall off smoothly from a single center peak

•There are no large gaps

•There are no obvious outliers

Mathematical ModelFor the

Distribution

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Density Curves

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Density Curves: Normal Curve

This curve is an example of a

NORMAL CURVE.

More to come later….

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Describing Density Curves

• Our measure of center and spread apply to density curves as well as to actual sets of observations.

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Proportions in a Density Curve

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Describing Density Curves

• MEDIAN OF A DENSITY CURVE:– The “equal-areas point”– The point with half the area under the curve to

its left and the remaining half of the area to its right

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Describing Density Curves

• MEAN OF A DENSITY CURVE:– The “balance point”– The point at which the curve would balance if

made of solid material

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Mean of a Density Curve

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Notation

• Use English letters for statistics– Measures on a data set– x = mean– s = standard deviation

• Use Greek letters for parameters– Measures on an idealized distribution– µ = mean– σ = standard deviation

Usually