Welcome to lesson on the Normal Distribution.
If you were to stand at the door of your math class and watched the students coming in, think about how the students would enter.
A few would arrive early then more and more students come in. Finally, the number of students entering decreases with a few students arriving late.
So the distribution of the number of students arriving, might take on this shape here, which is called a bell shaped curve.
Though the bell shaped curve, can take on various shapes.
If we were to measure height or shoe size of males and females,
in most cases, the distribution of the data would fit a similar distribution.
In fact this distribution is so common, it is called the normal distribution.
However, it is also called the Gaussian distribution.
Other data that would resemble a normal distribution, include,
IQ scores ,SAT scores, the life span of light bulbs,
and one of my favorite, kernels of corn popped when popping microwave popcorn.
If you've ever popped microwave popcorn you know
it starts popping slowly, then pops quickly,
and then slows down again . Before we discuss all the characteristics
of the have the normal distribution, let's look a few normal distribution curves.
All the curves illustrated here,
demonstrate the normal distribution. The mean and standard deviation,
effect the shape for the bell curve. Though the distribution will still have the
same characteristics,
of the normal distribution. So now let's talk about the characteristics,
Of the normal distribution. Number 1, the mean,
median and mode are all equal and located at the center of the distribution,
highlighted here. Next the distribution follows the empirical rule,
also called the 68,95,99.7 rule.
Which means, approximately 68% of the data,
falls plus or minus one standard deviation from the mean.
Approximately 95% of the data,
falls plus or minus two standard deviations from the mean.
and approximately 99.7% of the data,
falls, plus or minus 3 standar deviations from the mean.
So our text uses these percentages,
though some text, may use more accurate percentages,
given here. Let's talk more about the empirical rule
or the 68,95,99.7 rule.
We use mu for the mean, and sigma
for standard deviation. So 68% of the data,
falls between the mean and plus or minus one standard deviation.
So because of the symmetry, we could say that 34% falls
between the mean in one standard deviation above the mean,
and 34 percent falls between the mean
and one standard deviation below the mean. And because 95% the data
falls between the mean and plus or minus two standard deviations.
Notice half of 95 % would be 47.5,
so 47.5% would be between the mean.
and 2 standard deviations above the mean here.
Notice how if we subtract the 34 percent,
that tells us that 13.5%
falls between one and two standard deviations above the mean,
as well as 13.5%,
between one and two standard deviations below the mean.
So representing the normal distribution this way can also be helpful.
And then, finally because 99.7% of the data,
falls between the mean and plus or minus 3 standard deviations.
If we divide a 99.7 in half, that would give us 49.85%,
Which means 49.85% falls
between the mean and 3 standard deviations above the mean here.
So if we take 49.85%, subtract 34 %,
subtract 13.5%, that will leave us with 2.35 % of the data.
between two and three standard deviations above the mean.
As well as 2.35 percent between two and 3 standard deviations
below to mean. So again, it can be helpful to represent the normal distribution
this way as well. And this is all because of the symmetry
of the normal distribution. Let's take a look at some examples.
In a call center, the distribution of the number of phone calls
answered each day by each of the 12 receptionists
is bell-shaped and has a mean of 63
and a standard deviation of 3. So we know mu
will equal 63 and Sigma equals
3. Using the empirical rule,
what is the approximate percentage of daily phone calls
numbering between 60 and 66?
Well notice 60 is one standard deviation below the mean
and 66 is one standard deviation above the mean.
So more specifically we can say that 60,
is equal to the mean of 63
minus 1, times the standard deviation.
Which would be mu minus one sigma or 1 standard deviation.
And 66 is equal to 63 plus
1, times the standard deviation.
Which would be mu plus one standard deviation.
And therefore, we have mu plus or minus
one standard deviation. Which by the empirical rule we know
represents 68 percent,
of this case, the number of daily phone calls.
Which would be the percent of data in this region here.
Next, the scores of a midterm are normally distributed
with a mean have 85 percent and a standard deviation
6 percent. So again, we know that mu
equals 85 percent
and Sigma equals 6 percent.
Find the percentage of the class scores above and below
the given score. Again, using the 68,95
99.7 rule from the text.
So notice that the first score is 91%. Notice that 91
percent
is one standard deviation above the mean.
So 91 percent is equal to the mean of 85 percent
plus one times the standard deviation.
So we have mu plus one standard deviation.
So looking at the normal distribution
91 percent would be here.
And we know that 34 percent of the data falls between
the mean one standard deviation above the mean.
And then half the data is below the mean. So 50 percent plus 34 percent
means 84 percent of the data
for the midterm scores
are below 91 percent. And therefore, one hundred percent
minus 84 percent or 16 percent of the midterm scores
are above 91 percent.
Next we have a score of 73 percent. Notice that 73 percent
would be 2 standard deviations below 85 percent.
So 73 percent
is equal to the mean of 85 percent
minus 2 times the standard deviation
of 6 percent. So here we have mu
minus 2 standard deviations.
Which means a score of 73 percent would be here.
We need to be careful here. We can not say
that 2.35 percent of the test scores would be below
73 percent because this is not represent
100 percent of the test scores. But we do know that 50 percent
of the test scores are above the mean which would be the percent of test scores
over here on the right. So this should be 50 percent.
And then because we know that 95 percent
of the test scores fall between plus or minus two standard deviations.
And therefore, half of 95 percent or 47.5 percent would fall between
the mean
and two standard deviations below the mean. Meaning 47.5 percent of the
test scores
would be in this region here.
And therefore 50 percent plus 47.5 percent
or 97.5 percent of the test scores
would be above 73 percent. And therefore,
100 percent minus 97.5percent
or 2.5 percent of the midterm scores
are below 73 percent.
Ok, I think we'll stop here for this introductory video.
I hope you found this helpful.