§ 5.1
Introduction to Normal
Distributions and the Standard
Distribution
Properties of Normal Distributions
A continuous random variable has an infinite number of
possible values that can be represented by an interval on
the number line.
Hours spent studying in a day
0
3
6
9
12
15
18
21
24
The time spent
studying can be any
number between 0
and 24.
The probability distribution of a continuous random
variable is called a continuous probability distribution.
Density Functions
•A Density Function is a function that describes the
relative likelihood for this random variable to have
a given value. For a given x-value, the probability of
x, P(x), is called the Density, or Probability Density.
•Density Functions can take any shape.
•The Sum of all Probabilities, P(x), of a Probability
Density Function, that is, Densities, is always 1.
•The Sum of probabilities in an interval is the area
under the Density Function of the interval.
Density Function
•Consider the function f(x) below defined on the interval
[-1, 1]. Which of the following statements is true?
1
-1
1
a) f(x) could not be a continuous probability density function
2
7
b) P(−1 ≤ 𝑥 ≤ .5) =
c) P(−1 ≤ 𝑥 ≤ .5) =
d) P(−1 ≤ 𝑥 ≤ .5) =
3
11
12
e) P(−1 ≤ 𝑥 ≤ .5)
8
3
=
4
Uniform Distribution
• Let 𝑋~𝑈𝑛𝑖𝑓𝑜𝑟𝑚 0,25 , state the piecewise function
that defines this uniform distribution.
1
𝑓 𝑥 = 25
0
0 < 𝑥 < 25
𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒
a. Using your uniform distribution, sketch a graph of the
probability function.
Complete 5.1 Handout
Uniform Density Function
Practice
Properties of Normal Distributions
The most important probability distribution in
statistics is the normal distribution.
Normal curve
x
A normal distribution is a continuous probability
distribution for a random variable, x. The graph of a
normal distribution is called the normal curve.
Notation Used is 𝑵~(𝝁, 𝝈)
Means and Standard Deviations
A normal distribution can have any mean and
any positive standard deviation.
The mean gives
the location of
the line of
symmetry.
Inflection
points
1 2 3 4 5 6
Inflection
points
x
Mean: μ = 3.5
Standard
deviation: σ  1.3
1 2
3 4
5
6 7
8
9 10 11
Mean: μ = 6
Standard
deviation: σ  1.9
The standard deviation describes the spread of the data.
x
Means and Standard Deviations
Example:
1. Which curve has the greater mean?
2. Which curve has the greater standard deviation?
B
A
x
1
3
5
7
9
11
13
The line of symmetry of curve A occurs at x = 5. The line of symmetry of curve B
occurs at x = 9. Curve B has the greater mean.
Curve B is more spread out than curve A, so curve B has the greater standard
deviation.
Interpreting Graphs
Example:
The heights of fully grown magnolia bushes are normally
distributed. The curve represents the distribution. What
is the mean height of a fully grown magnolia bush?
Estimate the standard deviation.
μ=8
6
The inflection points are one standard
deviation away from the mean.
σ  0.7
7
8
9
Height (in feet)
10
x
The heights of the magnolia bushes are normally distributed with a mean height of
about 8 feet and a standard deviation of about 0.7 feet.
Complete 5.1 Handout
Normal Distribution
Practice