Download Density Curves

Density Curves and
Normal Distributions
Section 2.1
Sometimes the overall pattern of a distribution can be
described by a smooth curve. This histogram shows
the distribution of vocab scores. We could use it to
see how many students scored at each value, or what
percent of students got 4’s, above 10, etc.
Density Curves
A
density curve is an idealized
mathematical model for a set of
data.

It ignores minor irregularities and
outliers
Density Curves

Page 79-80
Density Curves
0.303
Page 79-80
0.293
Density Curve
 Always
on or above the
horizontal axis
 Has
an area of exactly 1
underneath it
Types of Density Curves
 Normal
curves
 Uniform density curves

Later we’ll see important density curves
that are skewed left/right and other
curves related to the normal curve
Density Curve
 Area = 1, corresponds to 100% of the data

What would the results look like if we
rolled a fair die 100 times?






Press STAT ENTER
Choose a list: highlight the name and press
ENTER.
Type: MATH  PRB 5:randInt(1,6,100)
ENTER
Look at a histogram of the results:
2ND Y= ENTER
Press WINDOW and change your settings
Press GRAPH. Use TRACE button to see
heights.
What would the results look like if we
rolled a fair die 100 times?
30% or 0.3
Relative Frequency

20% or 0.2
10% or 0.1
1
2
3
4
Outcomes
5
6
In a perfect world…

The different outcomes when you roll a die
are equally likely, so the ideal distribution
would look something like this:
An example of a
uniform density
curve.
Other Density Curves

What percent of observations are between 0
and 2? (area between 0 and 2)
Area of rectangle: 2(.2) = .4
Area of triangle: ½ (2)(.2) = .2
Total Area = .4 + .2 = .6 = 60%
Other Density Curves

What percent of observations are between 3
and 4?
Area: (1)(.2) = .2 = 20%
Normal curve

Density Curves: Skewed
M
Median: the equal-areas point
of the curve
Half of the area on each side
Density Curves: Skewed
Mean: the balance point of the curve
(if it was made of solid material)
Mean and Median
Of Density Curves
Just remember:
 Symmetrical distribution

Mean and median are in the center
 Skewed

distribution
Mean gets pulled towards the skew
and away from the median.
Notation
Since density curves are idealized, the mean
and standard deviation of a density curve will
be slightly different from the actual mean and
standard deviation of the distribution
(histogram) that we’re approximating, and we
want a way to distinguish them
Notation


For actual observations (our sample):
use x and s.
For idealized (theoretical): use μ (mu)
for mean and σ (sigma) for the standard
deviation.
Normal Curves are always:

Described in terms of their mean (µ) and
standard deviation (σ)

Symmetric

One peak and two tails
Normal Curves
Concave
down
Inflection point
Concave up
σ
µ
Inflection points – points at which this change
of curvature takes place.
Normal Curves
The Empirical Rule

The 68-95-99.7 Rule
-3
-2
-1
0
1
2
3
The Empirical Rule

68% of the observations fall within σ of the mean µ.
68 % of data
-3
-2
-1
0
1
2
3
The Empirical Rule

95% of the observations fall with 2σ of µ.
95% of data
-3
-2
-1
0
1
2
3
The Empirical Rule

99.7% of the observations fall within 3σ of µ.
99.7% of data
-3
-2
-1
0
1
2
3
Heights of Young Women

The distribution of heights of young women aged 18 to 24 is approximately
normal with mean µ = 64.5 inches and standard deviation σ = 2.5 inches.
64.5 – 2.5 = 62
64.5 + 2.5 = 67
62
64.5
67
Height (in inches)
Heights of Young Women

The distribution of heights of young women aged 18 to 24 is approximately
normal with mean µ = 64.5 inches and standard deviation σ = 2.5 inches.
5
59.5
62
64.5
67
Height (in inches)
69.5
Heights of Young Women

The distribution of heights of young women aged 18 to 24 is approximately
normal with mean µ = 64.5 inches and standard deviation σ = 2.5 inches.
99.7% of data
59.5
62
64.5
67
Height (in inches)
69.5
Shorthand with Normal Dist.

N(µ,σ)
Ex: The distribution of young women’s heights
is N(64.5, 2.5).
What this means:
Normal Distribution centered at µ = 64.5 with a
standard deviation σ = 2.5.
Heights of Young Women

What percentile of young women are 64.5 inches or shorter?
50%
99.7% of data
57
59.5
62
64.5
67
Height (in inches)
69.5
72
Heights of Young Women

What percentile of young women are 59.5 inches or shorter?
2.5%
99.7% of data
57
59.5
62
64.5
67
Height (in inches)
69.5
72
Heights of Young Women

What percentile of young women are between 59.5 inches and 64.5 inches?
64.5 or less = 50%
59.5 or less = 2.5%
50% – 2.5% = 47.5%
99.7% of data
57
59.5
62
64.5
67
Height (in inches)
69.5
72
Practice

For homework:


2.1, 2.3, 2.4 p. 83
2.6, 2.7, 2.8 p. 89