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Chapter 2
2.2 Density Curves and Normal Distributions
I will estimate the relative locations of the median and mean on a density
curve. I will use the 68-95-99.7 rule to estimate areas (proportions of values) in
a Normal distribution.
Density Curves
• So far we have worked only with jagged
histograms and stem plots to analyze data.
• As we begin to explore more fully the many
statistical calculations and analyses one can
perform on data it will become clear that
working with smooth curves is much easier than
jagged histograms.
• These smooth curves are called Density curves.
Characteristics of a density curve
• A density curve is a smooth curve that describes the overall pattern of a
distribution by showing what proportions of observations (not counts) fall
into a range of values.
• Areas under a density curve represent proportions of observations
• The scale of a density curve is adjusted in such a way that the total
area under the curve is always equal to 1
• The density curve is always on or above the horizontal
axis.
• The median of a density curve is the equal areas point.
• The mean of a density curve is the balance point (like a
see-saw) if the curve was made of solid material
• The mean and the median are equal in a symmetric density curve.
• The mean and the median are not equal in a skewed density
curve.
Check your knowledge!!!
• What percent of data is above the median?
• If my mean is below my median, than my distribution
is skewed _______________________.
• If my mean is above my median, than my distribution
is skewed _______________________.
Normal Distributions
A particularly important class of density curves are
normal distributions.
A density curve that is normally distributed has the
following characteristics:
•Symmetric
•Single peak
•Bell shape
• The mean of a density curve (including the normal curve) is
denoted by μ (the Greek lowercase letter mu) and the standard
deviation is denoted by σ (the Greek lowercase letter sigma)
• All Normal distributions have the same overall shape. Any
differences can be explained by μ and σ.
Calculus connection:
On the normal curve, at a distance of σ on either
side of the mean μ, are two points. These points are
called inflection points and they signify where the
curve changes concavity. The normal curve
changes from concave up to concave down at
these points. To find the inflection points, you would
need to find the second derivative of the function.
An unbelievable property that all
normal curves have is that they
follow the 68-95-99.7 rule
In a normal distribution with mean
μ and standard deviation σ :
• 68% of the observations fall
within 1 standard deviation of
the mean.
• 95% of the observations fall
within 2 standard deviations of
the mean.
• 99.7% of the observations fall
within 3 standard deviations of
the mean.
The Standard Normal Distribution
• The standard Normal distribution is the Normal distribution N(0,1)
with mean 0 and standard deviation 1.
Table A
• The table entry for each value z is the area
under the curve to the left of z.
• How do we find the area to the right of z?
– What is the area under the normal curve?
Example
• What are the proportions of
observations from the
standard Normal distribution
that are less than .81?
– Locate .8 in the left-hand
column of Table A
– Then locate the remaining digit
1 as .01 in the top row.
– 0.7910
• What about finding areas to
the right?
• Let’s find the proportion of
observations from the
standard Normal table that
are greater than -1.78
– Locate -1.7 on the left-hand
column.
– Locate .08 in the top row
– 0.0375 (but this is the area TO
THE LEFT)
– So, we do 1-0.0375=0.9625
From z-scores to Areas
(and vice versa)
• Finding Areas on Your Calculator
– 2nd  VARS (Distr)
– 2: normalcdf(
– Lower:
– Upper:
– 𝜇:
– 𝜎:
– If you are finding the area to the LEFT, the lower bound is −∞, which we
denote as -1E99.
– If you are finding the area to the RIGHT, the upper bound is ∞, which we
denote as 1E99
– To get the E in the calculator: 2nd  , (comma)
We can work backwards too!
• So far, we have used Table A
to find areas from z-scores
• What if we want to find a
specific z-score from an
area?
• For example, suppose you
want to score within the top
10% of SAT scores. What score
will you need to earn?
• Use Table A – look for the
value that is closest to 90%
• Use algebra to figure out
what score you need!
From Areas to z-scores
• Finding z-scores on Your Calculator
– 2nd  VARS (Distr)
– 3: invNorm(
– area:
– 𝜇:
– 𝜎:
– Always ensure your area is represented as a decimal.
Now Let’s Take a Second and
Breathe.
*Remember*
• What is a z-score?
Now, What happens if your score does not fall
exactly 1, 2, or 3 standard deviation(s) from the
mean? What if you scored a 550?
Now, What happens if your score does not fall
exactly 1, 2, or 3 standard deviation(s) from the
mean? What if you scored a 550?
Let’s try to finish that problem now.
Finding the area between 2 variables
• What is the percentage of students who scored between
425 and 550?
• Z-scores!
Practice!
1. Find the area under the standard normal curve to the left of z = 1.15
2. Find the proportion of observations from the standard normal curve
that are less than 1.15 standard deviations below the mean.
3. Find the probability of randomly selecting an individual from a
normal population whose z-score is 1.15 or less.
4. What percent of z-scores are less than or equal to 1.15?
Homework!
Pg. 129-130
#41, 43, 47, 53, 55.
Chapter 2
2.2 Density Curves and Normal Distributions
PRACTICE
The distribution of heights of adult American men is approximately
Normal with mean 69 inches and standard deviation 2.5 inches.
Draw an accurate sketch of the distribution of men’s heights. Be
sure to label the 68-95-99.7 Rule.
a. Between what heights do the middle 95% of men fall?
b. What percent of men are taller than 74 inches?
c. What percent of men are between 64 and 66.5 inches tall?
d. A height of 71.5 inches corresponds to what percentile of adult male
American heights?
Assessing Normality
• Method 1: Construct a histogram, stem and leaf plot or
box plot to determine if the shape is approximately bell
shaped with symmetry about the mean.
– *Use calculator. This makes it easier.
Assessing Normality
• Method 1: Construct a histogram, stem and leaf plot or
box plot to determine if the shape is approximately bell
shaped with symmetry about the mean.
– *Use calculator. This makes it easier.
• Method 2: Check the normal probability plot (on TI-83).
This is an easy and quick way to check for normality.
You want it to be one with a linear trend to it.
Assessing Normality
• Method 1: Construct a histogram, stem and leaf plot or box plot
to determine if the shape is approximately bell shaped with
symmetry about the mean.
– *Use calculator. This makes it easier.
• Method 2: Check the normal probability plot (on TI-83). This is an
easy and quick way to check for normality. You want it to be
one with a linear trend to it.
• Method 3: You can improve upon the accuracy of methods 1
and 2 by checking to see if the 68-95-99.7 rule applies
(approximately) to the data. Find the mean, and standard
deviation of the data. Find out if approximately 68% of the data
points are within 1 standard deviation of the mean, 95% are
within 2 standard deviations, and approximately 99.7% are within
3 standard deviations.
– *May only want to use it as a back-up plan.
Example
• The following are the heights of 50 of my former male
students, randomly selected from my classes. Are male
student heights at DHS normally distributed?
68, 68, 73, 74, 75, 68, 68, 66, 70, 72, 69, 63, 68, 69, 68, 65,
68, 67, 69, 70, 71, 68, 66, 72, 69, 69, 70, 67, 64, 69, 70, 71,
68, 68, 67, 67, 69, 65, 68, 70, 69, 67, 66, 61, 68, 69, 69, 71,
72, 70