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Welcome to Tutorial #3 of Introduction to Statistics
Normal Distributions
Normal Distribution is also called Gaussian Distribution
Normal Distribution is very popular.
The key question is: "Why Normal Distribution is so popular?"
Normal Distribution is compact
All data is distributed three standard deviations to the left and to the right of the mean.
Almost all data is within 6 standard deviations.
six standard deviation from left to right
i would suggest to you know
Please, if the terminology of this tutorial is not easy for you, review Tutorial 1 and Tutorial 2.
this is the formula discovered by Gauss.
The formula may be
scary
but imagine that I can justify these two simplifications.
Standard deviation is equal to 1.
And the expression (x-mu)/sigma), well, I can rename this expression to Z-value...
Z-value is known as standardized score.
Z is equal to the deviation from the mean
divided by the standard deviation. I can always call the ratio Z.
And you already know that Z is very important.
so
Normal Distribution in which all values of the variable are recalculated as Z-value is called
Standard Normal Distribution.
Any Standard Normal Distribution has standard deviation 1, and the Z-value of the mean is always zero.
All x-scores are standardized.
Z = (x - mu)/sigma
this is how the formula of Standard Normal Distribution looks like.
Next, I can use this formula in Excel
B7 is the value of Z recorded in a cell with address B7, for example.
The constant e is approximately 2.72 - this is very popular constant in math, statistics and
natural science
Normal Distribution
um...
is always
symetrical
The normal distribution depends on the values of the
random variable x
also, normal distribution depends on the mean of the population and the standard deviation.
and the value of the distribution functions changes as
x-value changes.
Normal Distribution is symmetrical
with respect to the mean
Normal Distribution always has the mode, median and mean in the same location
in the middle
Normal Curve is simple and elegant.
Normal Distribution can be represented as a Standard Normal Distribution by re-calculating all x-values as corresponding Z-values.
Let us consider these x-values: mu - 3*sigma to mu + 3*sigma.
We would like to recalculate these x-values as
Z-values.
For this re-calculation, we need to subtract from each x-value the mean.
The difference between the x-value and the mean is called deviation from the mean.
For the calculation of deviations from the mean, we can cancel mu in each x-value.
The results will be from -3*sigma to +3*sigma.
mu - 3*sigma - mu is
exactly
-3*sigma
mu - 2*sigma - mu = -2*sigma
Negative 3sigma divided by sigma will be exactly -3.
and so on -- we will have that the deviations from the mean are from -3*sigma to +3*sigma.
Let's see how to do these calculations
with the excel
I have hear one example of distribution
with mean = 200
and standard deviation = 50
to calculate Z-values that is 3 standard deviations
from the mean,
we have to subtract 3*50 from
the mean, which is 200
200 - 150 is
50
So, tree standard deviation to the left of the mean, the x-value is
fifty
I will copy this formula for all
values of x.
three hundred and fifty
So, if I have x-values, in this case, from 50 through 350.
I would like to re-calculate x-into Z-values.
Starting with 50 minus the mean, which is 200 = -150.
Negative 150, divided by the
standard deviation is (-150)/50 = -3/
Z-values are
(x - mu)/sigma
mean and standard deviation
you see what is happening
the z-values show how many standard deviation from the mean is the x-values.
So, the Z-value is the number of standard deviations from the mean.
The Z-value is the deviation measured in sigma.
Now, we will use the formula for Standard Normal Distribution
to calculate the functional values.
This is the formula
i will show you
called too
how to calculate this formula
This is the formula
and the way we type the formula in Excel
Don't forget to use equal sign
= one divided by quantity of sqrt (square root)
of 2 *3.14 (two pi)
close the parenthesis.
close the denominate.
time
EXP which is exponential notation with base e=2.72
"e" is the Neper or Euler number.
0.5, negative, times the address of the cell with Z-value
in our case
this is B3
squared
or to the power of 2 (^2)
And I will close the parenthesis and press Enter,
This is the value of the function.
This is the
probability
of this Z-value
in the interval a little bit more or less of exact value of Z.
i will copy this formula for all Z-values from -3 to positive 3
because if you remember, normal distribution is always captured in 3 standard deviations
to the left and right of the
mean
For sum reason I cannot move this
OK
I will copy the values of the function
without formula.
I will try to simplify the Z=values by using only
integers
It will be *** the figure if we keep all Z-values.
Control + C = and I will copy all functional values.
So, these are the Z-values from -3 to +3.
and the probabilities corresponding to these values.
So, you can see the figure
I will delete the figure to show you how to visualize
Let me show you how to do this figure.
First, select all Z and functional values.
Click Insert
Select the Line
Let's grab this figure and move it to the top together with the formula
to make sense
and this is the standard normal distribution from -3 to positive 3.
you can see that this is a beautiful
What else is important for understanding Normal Distribution.
I would like to remind you some important characteristic of Standard Normal Distribution
Back to PowerPoint
from andy field deposit defeat gets it all the data
also about 68% of all data is between negative one to positive 1.
from negative one to positive one all almost sixty eight percent of full data
is sketch it from negative one
to positive one
and almost
ninety five percent of all data is two standard deviations to the left and to the right of the mean.
And this is 1-2-3 rule that you should remember.
but don't think that all calculation will be done with 1-2-3 rule.
the probabilities are not always
calculated with 1-2-3 rule.
This is one memorable feature of standard normal distribution.
So, why normal distribution is so popular?
It is easy to calculate
Normal Distribution is symmetrical
compact
and has many applications.
This is all for today
thank you and
shell see you online.