Download Chapter 2 The Normal Distributions

Chapter 2
The Normal Distributions
Section 2.1
Density curves and the normal
distributions
Activity

Roll one die 20 times and record the results
 Graph
Roll two dice 20 times and record the sum of the two die
 Graph
Page 85 2.5 (using a calculator for a simulation)

http://www.shodor.org/interactivate/activities/PlopIt/


We now have a clear strategy for exploring data on a
single quantitative variable:
◦ Always plot your data (usually a histogram or a
stemplot)
◦ Look for the overall pattern (shape, center, spread) and
for striking deviations such as outliers
◦ Calculate a numerical summary to briefly describe the
center and spread
 We now add a new step:
◦ Sometimes the overall pattern of a large number of
observations is so regular that we can describe it by a
smooth curve

 Called a Density Curve
Density Curves

A density curve is a mathematical
model for the distribution
◦ A mathematical model is an idealized
description
Gives a compact picture of the overall
pattern of the data
 Ignores minor irregularities as well as any
outliers

Density Curves
Is always on or above the horizontal axis
 Has area exactly 1 underneath it
 Describes the overall pattern of a
distribution
 The area under the curve and between
any range of values is the proportion of
all observations that fall in that range

Standard Normal Distributions
Mean and median of a density curve





Median of a density curve is the equal-areas point
(hard to find with skewed curves)
In a symmetric distribution the mean and median
are in approximately the same place
Mean is the balance point of a density curve
In a skewed distribution the mean is pulled towards
the tail
Can basically locate the mean, median, and quartiles
of any density curve (not the standard deviation)
Normal Distributions
Normal curves describe normal distributions
 All normal distributions have the same overall shape
 Symmetric, single-peaked, bell-shaped
 The exact density curve for a particular normal distribution is
described by giving its mean μ and its standard deviation σ
 Changing μ without changing σ moves the normal curve along the
horizontal axis without changing its spread
 The standard deviation σ controls the spread of a normal curve
 Inflection points are the points at which the change of the curvature
takes place and are located at a distance σ on either side of the mean
◦ They change as σ changes

Points of Inflection
Empirical Rule: 68-95-99.7
68% within one Standard deviation
 95% within two Standard Deviations
 99.7% within three Standard Deviations

Normal Distribution

Notation N(μ, σ)
◦ Ex: distribution of women’s heights
 N(64.5, 2.5)

Give good descriptions for some distributions of
real data
◦ Ex: SAT scores

Give good approximations to the results of
many kinds of chance outcomes
◦ Ex: tossing a coin

Statistical inference procedures are based on
normal distributions
Percentiles





Used when we are most interested in seeing
where an individual observation stands relative
to the other individuals in the distribution
In practice the number of observations is quite
large so can talk about the distribution as a
density curve
Median score is the 50th percentile
First quartile is the 25th percentile
Third quartile if the 75th percentile
www.tc3.edu
Section 2.2
Standard Normal Calculations
Standard Normal Calculations


Normal distributions vary from one
another in two main ways:
◦ The mean μ may be located anywhere on the
x-axis
◦ The bell shape may be more or less spread
according to the size of the standard
deviation σ

We can standardize our distribution in
order to find proportions and compare to
other distributions
z-score
z

x

The standardized value, or z value, or z score tells
us the number of standard deviations the original
measurement is from the mean, and in what
direction
◦ observations larger than the mean are positive
when standardized
◦ observations smaller than the mean are negative
when standardized
Calculating the Z-score


This says that she is 1.4 standard deviations
above the mean.
Calculating the Z-score

How about a woman who is 5 feet (60 inches)
tall? What would be her standardized height?

This says that she is 1.8 standard deviations
below the mean.