Chapter 2: Describing Location In a Distribution Section 2.1 Measures of Relative Standing And Density Curves.

42510011 0010 1010 1101 0001 0100 1011

Chapter 2: Describing LocationIn a Distribution

Section 2.1

Measures of Relative Standing

And Density Curves

4251

0011 0010 1010 1101 0001 0100 1011

Case Study

• Read page 113 in your textbook

4251

0011 0010 1010 1101 0001 0100 1011

Where are we headed?

Analyzed a set of observations

graphically and numerically

Consider individual observations

4251

0011 0010 1010 1101 0001 0100 1011

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?

4251

0011 0010 1010 1101 0001 0100 1011

Measuring Relative Standing: z-scores

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

4251

0011 0010 1010 1101 0001 0100 1011

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.

4251

0011 0010 1010 1101 0001 0100 1011

Practice: Let’s Do p. 118 #1

4251

0011 0010 1010 1101 0001 0100 1011

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

4251

0011 0010 1010 1101 0001 0100 1011

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

4251

0011 0010 1010 1101 0001 0100 1011

Density Curves

4251

0011 0010 1010 1101 0001 0100 1011

Density Curves: Normal Curve

This curve is an example of a

NORMAL CURVE.

More to come later….

4251

0011 0010 1010 1101 0001 0100 1011

Describing Density Curves

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

4251

0011 0010 1010 1101 0001 0100 1011

Proportions in a Density Curve

4251

0011 0010 1010 1101 0001 0100 1011

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

4251

0011 0010 1010 1101 0001 0100 1011

4251

0011 0010 1010 1101 0001 0100 1011

Describing Density Curves

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

made of solid material

4251

0011 0010 1010 1101 0001 0100 1011

Mean of a Density Curve

4251

0011 0010 1010 1101 0001 0100 1011

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