AP Statistics
Chapter 2 Notes
Measures of Relative Standing

Percentiles
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The percent of data that lies at or below a
particular value.
e.g. standardized test score reports
baby weight/height/head size.
Standardized value (z-score)
x - mean x - x
z=
=
std dev
s
Z-score


Z-score tells you how many standard
deviations away from the mean a given
observation is.
Z-scores are more useful with symmetric
distributions.
Density Curve

An idealized mathematical model used to
represent a distribution.



Always on or above the horizontal axis.
Has an area of exactly 1 underneath it.
The area under the curve for any given interval is
equal to the proportion of all observations that fall
in that interval.


median: equal areas point
Mean: balance point
Notation used for density curves

Observed data


Idealized Data
(Sample)
Statistics

x

s
(population)
Parameters
MEAN
standard deviation
μ
σ
Normal Distribution


Always symmetric, but
the exact shape
depends upon μ and
σ.
Change in curvature
(point of inflection)
shows where 1
standard deviation
from the mean is
located.
Empirical Rule (68-95-99.7 Rule)
Example


IQ scores are Normally
distributed with a mean of
100 and a std dev of 15.
What % of people have IQ
scores…

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Between 70 and 130?
Less than 85?
Greater than 145?
Less than 115?
Between 55 and 70?
Probability Calculations

We define a Normal distribution by its mean and
standard deviation.

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
N(μ, σ)
If we standardize the distribution by calculating zscores, we create the distribution: N(0,1).
The z-table provides the percentiles associated
with various z-scores.
When performing a calculation, be sure to draw a
sketch of the region under the Normal curve that
you are working with, and answer the question in
context.
Examples

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IQ scores are Normally distributed with a mean of
100 and a std dev of 15.
What percent of people have IQ scores less than
82? Less than 121?
What percent of people have IQ scores greater
than 107?
What percent of people have IQ scores between
88 and 104?
A person is considered a genius if they are in the
top 2% in terms of IQ. What IQ score does a
person need to be considered a genius?
Using the calculator

To calculate the % of observations within a certain
interval, use the z-table or the graphing calculator.

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2nd Vars (Dist), choose option 2.
Normalcdf (min, max, μ, σ)
To calculate raw data scores from percentiles:

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2nd Vars (Dist), choose option 3.
invNorm(%, μ, σ)