MI-4: Normal Distributions
Learning Objectives:
Students will learn to evaluate normal distributions using Microsoft Excel and Wolfram Mathematica.
More specifically, students will learn:
1. What a normal distribution is
2. What standard deviation and mean are
3. How standard deviation and mean affect the graph of a normal distribution using Wolfram
Mathematica
4. How to calculate probability under a normal distribution using Microsoft Excel
5. What implications normal distributions hold in daily life
What is a normal distribution?
A normal distribution is a symmetric curve that has significantly more data points near the mean than at
the extremities of the graph, which results in a bell shaped curve.
The equation for a normal distribution is described by:
𝑓(𝑥) =
1
√2𝜋𝜎 2
−(𝑥−𝜇)2
𝑒 2𝜎2
Normal Distributions: Josh Lam and Nate Suek
Here is an example of a normal distribution graph:
1
What is Mean?
Mean is the average of all of the data points. The mean can be calculated by:
𝑆𝑢𝑚 𝑜𝑓𝑑𝑎𝑡𝑎 𝑝𝑜𝑖𝑛𝑡𝑠
𝑁𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑑𝑎𝑡𝑎 𝑝𝑜𝑖𝑛𝑡𝑠
Calculate the mean of these data sets:
1. {1,5,20,32,47}
2. {a,2,5,g,z}
Standard deviation is a description of how far, or how dispersed, data points are from the mean. A low
standard deviation would tell us that the data points are very close to the mean whereas a high stand
deviation would tell us that the data points are further spread out from the mean. Standard deviation is
usually denoted by the Greek symbol, σ (sigma).
The equation for sample standard deviation is:
Where:
x= the data points
μ= the means
n= number of data points
𝜎=√
∑(𝑥−𝜇)2
𝑛−1
Normal Distributions: Josh Lam and Nate Suek
What is Standard Deviation?
2
Calculate the standard deviation of the following sets:
1. {1,5,20,32,47}
1 3 4 1
6
2. {2 , 5 , 3 , 7 , 11}
How are mean, standard deviation, and normal distribution connected?
Open up Wolfram Mathematica and copy the following command into the box:
Manipulate[Plot[1/( Sqrt[2 Pi]) E^(-(x-)^2/(2
^2)),{x,-10,10},PlotRange {0,1}],{{,1},0.1, 10},
{{,0},-10, 10}]
This application allows you to manipulate both the mean and standard deviation
so you can see how each factor affects the graph. Mean is represented by μ and
standard deviation is represented by σ in case you’ve forgotten.
Normal Distributions: Josh Lam and Nate Suek
As shown earlier, both mean and standard deviation are represented as variables in the equation for a
normal distribution graph. The highest point of the graph is centered on the mean and it is where the
most data points are clustered. Normal distribution graphs are generally divided up by standard
deviations that help define the regions in which data points are located.
3
What happens to the graph as you change the mean?
What happens to the graph as you change the standard deviation?
How can you calculate the probability under a normal distribution graph?
Within the first standard of deviation both ways [-σ, +σ], approximately 68% of points are present;
within the first 2 standards of deviation [-2σ, +2σ], approximately 95%; within the first 3 standards of
deviation [-3σ, +3σ], approximately 99.7%.
Normal Distributions: Josh Lam and Nate Suek
Fill in the blanks below for percentages of points:
4
Calculate the probability of a point being:
1. Between μ-σ and μ+σ
2. Between μ-σ and μ+2σ
3. Between μ-2σ and
μ+σ
2
Open Microsoft Excel and input into the cell:
=NORM.DIST or =NORMDIST depending on which version of Excel you have
Excel will ask you to input x, mean, standard_dev, cumulative.
Input your given value for x, the mean, the standard deviation, and “TRUE” for
cumulative, each separated by a comma.
This function in Excel is extremely useful for calculating the probability of a point being in a certain
specified range of numbers. Calculating this by hand requires a lot of computation and the use of a zscore table, which can all be eliminated by the use of Excel.
Here are some practical applications to solve. Use Excel to calculate:
2. The average SAT score for any given subject is 450 and the standard deviation is 50 points. What
is the probability that a test taker will score higher than a 630?
3. If the average IQ is 100 points and the standard deviation is 10 points, what is the probability
that one’s IQ would be lower than 127 points, but higher than 83 points?
Normal Distributions: Josh Lam and Nate Suek
1. In a math test, the average score was 80 points out of a hundred and the standard deviation was
5 points. What is the probability that a student would score lower than 76 points?
5