Name: __________________________________________ Per _____ Normal Probability Distribution
Assignment: Section 11.3 #1-5, 7-8,12
Many continuous variables follow a distribution that is symmetric and mounded in a “bell” shape.
Graphs of distributions for large populations, such as heights, clothing sizes, and test scores often have this shape. The bell-shaped distribution is so common that it is called the NORMAL DISTRIBUTION and its graph is called the NORMAL CURVE.
The formula you’ve learned for the SAMPLE MEAN and SAMPLE STANDARD DEVIATION are both
statistics because they are numbers that describe a sample. We use these two statistics as estimates for the values in the population. Write the formula for both of these statistics if the sample size is n=3.
Sample Mean 𝑥̅ =____________________________________
Sample Standard Deviation s = ____________________________________
If you find the mean and standard deviation for an entire population, they are called population
parameters. We use Greek Letters to distinguish between population parameter and sample statistics.
Population Mean is pronounced “mew” and written with the Greek Letter,

Population Standard deviation is pronounced “sigma” and is written with the Greek letter,

.
(1) The parent graph of the Normal Probability Distribution is called the STANDARD NORMAL
Distribution and has a mean equal to

=0 and standard deviation equal to

=1.
The equation is 𝑦 =
1 𝜎√2𝜋
(√𝑒)
−𝑥
2 or just 𝑦 =
1
√2𝜋
(√𝑒)
−𝑥
2
(2) If you want to graph a Normal Distribution with a different mean value (keeping

=1) you shift
the graph right by the mean value.
The resulting equation is 𝑦 =
1 𝜎√2𝜋
(√𝑒)
−(𝑥−𝜇)
2
or just 𝑦 =
1
√2𝜋
(√𝑒)
−(𝑥−𝜇)
2
(3) If you want to graph a Normal Distribution with a different standard deviation value (keeping
 =0) you need to dilate the graph horizontally by the new standard deviation,  .
The resulting equation is 𝑦 =
1 𝜎√2𝜋
(√𝑒)
−( 𝑥 𝜎
)
2
If you want to graph a Normal Distribution with a BOTH a different standard deviation value
2
AND a different mean value the resulting equation is 𝑦 =
1 𝜎√2𝜋
(√𝑒)
−(
(𝑥−µ) 𝜎
)

GRAPH
(1) Standard Normal

=0
 = 1
𝑦 =
1
√2𝜋
(√𝑒)
−𝑥
2
You should notice that there is a line of symmetry at the population mean: x =  = 0
(2) Change ONLY the Mean

=3

= 1


Shift right by

𝑦 =
1
√2𝜋
(√𝑒)
−(𝑥−3)
2
You should notice that the graph has the same shape as the standard normal, but this graph, that has a larger mean, so it is shifted to the right by 3.
(3) Change ONLY the
Standard Deviation

=0

= 0.5

Vertical Dilation by

𝑦 =
1
0.5√2𝜋
(√𝑒)
−( 𝑥
0.5
)
2
You should notice that the graph is in the SAME location as standard normal but it has a different shape. The graph with the smaller standard deviation is taller and skinnier (because smaller standard deviations mean that the graph is less spread out).
(4) Change BOTH the
mean and standard
deviation.

=3

= 0.5
Shift right by

Vertical Dilation by

𝑦 =
1
0.5√2𝜋
(√𝑒)
−(
(𝑥−3)
0.5
)
2
Compare the graphs. You should notice that all Normal Distributions are “Bell-Shaped”. Changing the mean only changes the LOCATION of the graph (shifted right or shifted left). Changing the standard deviation changes the SHAPE of the “bell” (taller and skinner if the standard deviation is smaller and shorter and fatter if the standard deviation is larger).

Write down what you have noticed about the graphs of Normal Distributions.
(a) How does changing the mean affect the graph?
(b) How does changing the standard deviation affect the graph?
Calculating Probabilities using the Normal Probability Distribution (aka, The Bell-Curve)
In the Sections 11-2 we were able to calculate probabilities using the area under the probability distribution curves (see examples 2 & 3 from yesterday’s notes). Recall that we used formulas from geometry to find the areas we needed. Since there are no formulas from geometry that we can use to calculate the area under the curve for the Bell-Curve, we use the 68-95-99.7% Rule
Demonstrate on the calculator how to find these and other area under the curve. 68% of the data will fall within ______ standard deviation away from the mean.
95% of the data will fall within ______ standard deviations away from the mean.
99.7% will fall within ______ standard deviations away from the mean.
Example #1: From the equation of the Normal Distribution, estimate the mean and standard deviation.
𝑦 =
1
3√2𝜋
(√𝑒)
−(
(𝑥−10)
3
)
2
𝑦 =
0.4
(0.60653)
+(
(𝑥−100)
15
)
2
15
Example #2: From the graph, estimate the mean and standard deviation.
Example #3: Sketch these on the back of this paper since there is no room to show work here.
Assume the mean height of an adult male gorilla is 6.7 feet with a standard deviation of 0.6 feet.
(a) Sketch the graph of the normal distribution of gorilla heights.
(b) Give the range of heights for 65% of the gorillas
(c) Give the range of heights for 95% of the gorillas.
(d) Give the range of heights for 99.7% of the gorillas.