DENSITY CURVES
and
NORMAL DISTRIBUTIONS
The histogram displays the Grade equivalent
vocabulary scores for 7th graders on the Iowa
Test of Basic Skills. The scores of students
on this national test have a regular
distribution.
This histogram is mostly symmetric. Both
tails fall of smoothly from the center peak.
There are no obvious gaps or outliers.
THE SMOOTH CURVE IS A GOOD
DESCRIPTION OF THE OVERALL PATTERN
To change from a histogram to a smooth curve




Use PROPORTIONS of the observations that
fall in each range rather than actual counts of
observations.
Each bar area will represent the proportion of
observations in that class
For the curve, the area under the curve
represents the proportions of the
observations.
Adjust the scale of the graph so that the total
area under the curve is equal to 1.
DENSITY CURVES
Describe the overall shape of
distributions
Idealized mathematical models for
distributions
Show patterns that are accurate enough
for practical purposes
Always on or above the horizontal axis
The total area under the curve is
exactly 1
Areas under the curve represent relative
frequencies of observations



The MEDIAN (M) is the point with half the
observations on either side. The QUARTILES
divide the area under the curve into quarters.
The median is the POINT OF EQUAL AREAS.
The MEAN (or arithmetic average) is the point at
which the curve would balance if made of a solid
material.
DENSITY CURVES can be symmetrical or skewed.
Remember that for symmetrical distributions, the
Median and Mean are equal.
Because a Density Curve is an
Idealized Description of the
distribution of data, we must
distinguish between:
The Mean x , and standard
deviation (s ) ; computed from the
actual observations
&
The mean (μ ) and standard
deviation (σ ) of the idealized
distribution.
EXAMPLE:
Consider the unusual
density curve:
Find the % of the data
in the following
intervals
0 < X < 0.6 ?
0.2 < X < 0.4 ?
0 < X < 0.8 ?
THE NORMAL DISTRIBUTION
Gauss used the Normal Distribution to analyze
astronomical data in 1809.
The normal curve is often called the Gaussian
Distribution or more normally, “The Bell Shaped
Curve”
The normal curve is the most used statistical
distribution because
1. Normality arises naturally in many biological,
physical, and social measurement situations.
2. It is a good approximation of many distribution
curves of chance occurrence (i.e. binomial)
3. Normality is important in statistical inference.
The Normal Distribution is characterized
by two parameters:
The Mean ( μ ) - A measure of center or
location. The mean can be any + value. The
mean is in the same location as the median.
The Standard Deviation ( σ ) – A measure of
spread. The standard deviation must be a
positive number.
Together, the Mean and the Standard Deviation
define a specific normal distribution.
The mean and standard deviation
determine the shape of the normal
curve
AND
The standard deviation can be
located visually by finding the
INFLECTION POINTS on either
side of the mean
AND
The INFLECTION POINTS of the
curve are the places where the
CONCAVITY changes.
INFLECTION POINTS ON THE
NORMAL CURVE
In the normal distribution with a mean (mu)
and standard deviation (sigma):
68% of all observations lie within one
standard deviation of the mean
95% of all observations lie within two
standard deviations of the mean
99.7% of all observations fall within three
standard deviations of the mean.
Because normal distributions are used so
frequently, a short notation is often used to
describe the parameters of mean and standard
deviation.
N ( μ ,σ )
For example: N ( 64.5, 2.5 ) indicates a
normal distribution with a mean = 64.5 and a
standard deviation of 2.5.
Percentile is a familiar term because it is so
frequently used in the reporting of
standardized test scores. Percentiles are
used when we are interested in seeing where
an individual observation stands in relation to
other observations in the distribution.
An observations PERCENTILE is the
percent of the distribution that lies to the
LEFT of the observation
HOMEWORK
Chapter 2
2.6 – 2.9
2.11 – 2.18