2.1 Power Point

8/7/2019 2.1 Power Point

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Section 2.1:

Measures of Relative

Standing and Density Curves


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So Where Do I Stand?

Here are the scores of all 25 students

in a statistics class:

79, 81, 80, 77, 73, 83, 74, 93, 78, 80, 75,67, 73, 77, 83, 86, 90, 79, 85, 83, 89, 84,

82, 77, 72

Lets say that you scored the 86. How well

did you perform relative to the rest of the

class?


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First Graph!


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Then Calculate Your Statistics:


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B

ut How Much Above Average? We can safely say

that your score is

above averagesince it is greater

than both the mean

and median, but

exactly how much

above average is it?


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Standardizing One way to describe your position in the

distribution is to tell how many standard

deviations it is above or below the mean.

Since the mean is 80 and the SD is 6, your

score of 86 is 1 SD above the mean.

Converting scores like this from original

values to standard deviations is known as

standardizing.


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Getting Some Zzzzzzzzzs

A standardized value is usually called a

z-score.

If an observation, x, from a distribution

that has a known mean and SD, the

standardized value ofx is:

mean

standard deviation

x

x

xx

z

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The heights of adult men aged 20

to 29 are approximately normallydistributed with a mean of 69.3

inches and a standard deviation of

2.8 inches. Find the standardizedvalues (z-scores) of the following

heights:

1. 6 ft.

2. 6 ft. 6 in.

3. 5 ft. 5 in.


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Who did better?

Suppose Bubbas score on his history testwas 65. The mean and SD for his class

were 50 and 10 respectively. Bubbettesscore was an 88 on her English exam,where the mean and SD were 74 and 12.Both of the distributions of test scores were

approximately normal.

Who did better on their test compared to therest of their class?


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Another Measure of Relative

Standing We can also describe your test score of 86

using percentiles.

Remember the pth percentile of a distributionis the value with p percent of the

observations less than or equal to it.

Since your score was the 22nd highest out of

25 scores, what is your percentile?


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Scores on this national test

have a very regular distribution. The histogram is symmetric,

and both tails fall off smoothly

from a single center peak.

There are no large gaps orobvious outliers.

The smooth curve drawn over

the histogram is a good

description of the overall patternof the data.

The curve is a mathematical

model for the distribution.

This is thehistogram of

the scores of all 947 7th

graders in Gary, IN, onthe vocabulary part of

the Iowa Test of Basic

Skills.


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Mathematical Model A mathematical model is an idealized

description.

It gives a compact picture of the overall pattern ofthe data but ignores minor irregularities as well as

any outliers.

It is much easier to work with the smooth curve

than the histogram since the histogram depends

on your choice of intervals (classes), while we

can create a curve that doesnt depend on our

choices.


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The blue bars represent the

students with vocabularyscores of 6.0 or below.

Remember that the area of

the bars represents the

proportion of students with

scores of 6.0 or below.

287 students had such

scores, so 287/947 = 0.303

of all the students scored

6.0 or below. In other words, a score of

6.0 corresponds to about the

30th percentile.


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Now the blue represents thearea under the curve to the left

of 6.0. If we adjust the scale of the

graph so that the total areaunder the curve is 1 (100%),then the curve is a density

curve. Areas under the curve now

represent the proportions ofthe observations.

The blue area is 0.293 or about

the 29th

percentile. Our estimate is 0.010 away

from the histogram result. Wecan see that areas underdensity curves give very goodapproximations of histogramareas.


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Density Curve

* Of course, no set of real data is exactly described by a density

curve. The curve is an approximation that is easy to use and

accurate enough for practical use.


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Skewed Which Way?


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The median is the point with half of theobservations on each side. So in a densitycurve, the median is the line that splits the curveinto equal areas.

Median of a Density Curve


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Its All a Balancing Act

The mean of a density curve is the

balance point, the point at which the

curve would balance if made of solidmaterial.


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A Normal Density Curve


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Notation Again Mean computed from actual observations:

Standard Deviation computed from actual

observations:

Mean of a density curve:

Standard Deviation of a density curve:

x

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