### Video Transcript

In this video, we will learn how to
calculate the standard deviation of discrete random variables.

Standard deviation of a random
variable is a measure of spread of the probability distribution. Given a random variable 𝑋, the
standard deviation is denoted 𝜎 or 𝜎 sub 𝑋. Its square, which is called the
variance, or var of 𝑋, is defined as follows: 𝜎 squared, or the variance of 𝑋, is
equal to the 𝐸 of 𝑋 minus 𝐸 of 𝑋 all squared, where 𝐸 of 𝑋 denotes the
expected value of the random variable 𝑋. The standard deviation 𝜎 is
therefore obtained by taking the positive square root of the variance.

Looking at this formula a little
closer, we see that the variance of 𝑋 is the average value of the square distance
of data points from the expected value. In short, the standard deviation
represents how far, on average, outcomes of the random variable are from the
expected value. This can be demonstrated
graphically. In the picture shown, the
probability distribution of the random variable 𝑋 is given where 𝐸 of 𝑋 denotes
the expected value or mean and 𝜎 denotes the standard deviation. This formula is cumbersome to use
in practice, so we introduce a variant of this formula, which we will see
shortly. Since this alternative formula is
simpler to use, we will use it instead as the definition.

Given a random variable 𝑋, the
variance of 𝑋 is given by var of 𝑋 is equal to 𝐸 of 𝑋 squared minus 𝐸 of 𝑋 all
squared where 𝐸 of 𝑋 is the expected value or mean. The standard deviation 𝜎 or 𝜎 sub
𝑋 can therefore be calculated by square rooting this variance of 𝑋. The process of computing the
standard deviation of a discrete random variable can therefore be summarized in four
steps.

Step one is to compute 𝐸 of
𝑋. For any discrete random variable
𝑋, taking values 𝑥 sub one, 𝑥 sub two, and so on up to 𝑥 sub 𝑛, the expected
value is equal to 𝑥 sub one multiplied by the probability that 𝑋 equals 𝑥 sub one
plus 𝑥 sub two multiplied by the probability that 𝑋 equals 𝑥 sub two and so on
all the way up to 𝑥 sub 𝑛 multiplied by the probability that 𝑋 equals 𝑥 sub
𝑛. Our second step is to compute 𝐸 of
𝑋 squared. This can be calculated in a similar
way as 𝐸 of 𝑋, except we use 𝑥 sub one squared, 𝑥 sub two squared, and so
on. We square the individual 𝑥-values
before multiplying them by the corresponding probabilities.

Our third step is to compute the
variance of 𝑋. We do this using the formula
above. The variance of 𝑋 is equal to 𝐸
of 𝑋 squared minus 𝐸 of 𝑋 all squared. Finally, we’re in a position to
compute the standard deviation 𝜎 by square rooting the variance of 𝑋. We will now look at some examples
where we need to follow these four steps in different contexts.

The function in the table is a
probability function of a discrete random variable 𝑋 find the standard deviation of
𝑋. Give your answer to two decimal
places.

In order to answer this question,
we recall the four-step process used to obtain the standard deviation 𝜎. Firstly, we compute the mean or
expected value 𝐸 of 𝑋. Secondly, we compute 𝐸 of 𝑋
squared. Thirdly, we compute the variance or
var of 𝑋. This is equal to the 𝐸 of 𝑋
squared minus the 𝐸 of 𝑋 all squared. Our fourth and final step is to
compute the standard deviation 𝜎, which we recall is equal to the square root of
the var of 𝑋.

Let’s begin by recalling how we
calculate the 𝐸 of 𝑋. We do this by multiplying each
value of 𝑋 by its corresponding probability and then finding the sum of these
values. We multiply negative five by
one-third. We then add the product of negative
four and one-eighth, negative three and one-quarter, and finally negative one and
seven twenty-fourths. Whilst we could work out each of
the four products individually, typing the whole calculation into our calculator
gives us negative 77 over 24.

Our second step is to calculate 𝐸
of 𝑋 squared. In order to do this, it is worth
adding an extra row to our table to calculate the 𝑋 squared values. Negative five squared is 25, as
multiplying a negative number by a negative number gives a positive answer. In the same way, squaring negative
four, negative three, and negative one gives us values of 16, nine, and one. We can now repeat the process we
used to calculate 𝐸 of 𝑋. This time, we multiply the 𝑋
squared values by their corresponding probabilities. This gives us 25 multiplied by
one-third plus 16 multiplied by one-eighth plus nine multiplied by one-quarter plus
one multiplied by seven twenty-fourths. Once again, we can type this
directly into our calculator, giving us 103 over eight.

The third step of our process is to
calculate the variance or var of 𝑋. This is equal to 𝐸 of 𝑋 squared
minus 𝐸 of 𝑋 all squared. Substituting in the values we have
calculated, we have 103 over eight minus negative 77 over 24 squared. Typing this into our calculator
gives us 1487 over 576. As the standard deviation is the
square root of the variance, we can calculate this by square rooting 1487 over
576. Noting that we need to give our
answer to two decimal places, this is approximately equal to 1.61. The standard deviation of our
function to two decimal places is 1.61.

In our next question, one of the
values of 𝑋 in our table will be unknown.

The function in the given table is
a probability function of a discrete random variable 𝑋. Given that the expected value of 𝑋
is 6.5, find the standard deviation of 𝑋. Give your answer to two decimal
places.

In this question, we are given the
expected value 𝐸 of 𝑋, which is equal to 6.5. We can use this to help us identify
the unknown parameter 𝐴. We recall that we can calculate the
expected value by multiplying each of our 𝑋-values by the corresponding
probabilities. We then find the sum of all these
values. This means that 𝐸 of 𝑋 is equal
to three multiplied by 0.2 plus 𝐴 multiplied by 0.1 plus six multiplied by 0.1 plus
eight multiplied by 0.6. Simplifying this right-hand side,
we have 0.6 plus 0.1𝐴 plus 0.6 plus 4.8. And we know this is equal to
6.5. Subtracting 0.6, 0.6, and 4.8 from
both sides of our equation gives us 0.5 is equal to 0.1𝐴. We can then divide both sides of
this equation by 0.1 giving us 𝐴 is equal to five. The missing value in our table is
five such that the probability that 𝑋 equals five is 0.1.

Next, we recall that to compute the
standard deviation, we need to follow four steps. Firstly, we compute 𝐸 of 𝑋. Secondly, we compute 𝐸 of 𝑋
squared. Our third step is to compute the
variance or var of 𝑋 which is equal to 𝐸 of 𝑋 squared minus the 𝐸 of 𝑋 all
squared. Finally, we can compute the
standard deviation 𝜎 by square rooting the variance of 𝑋.

Clearing some space, we already
know that the expected value or mean 𝐸 of 𝑋 is equal to 6.5. We calculate 𝐸 of 𝑋 squared in a
similar way to 𝐸 of 𝑋. This is equal to three squared
multiplied by 0.2 plus five squared multiplied by 0.1 plus six squared multiplied by
0.1 plus eight squared multiplied by 0.6. Typing this into our calculator
gives us 46.3. We now have values of both 𝐸 of 𝑋
and 𝐸 of 𝑋 squared. The var of 𝑋 is equal to the 𝐸 of
𝑋 squared minus the 𝐸 of 𝑋 all squared. So, in this case, we have 46.3
minus 6.5 squared. This is equal to 4.05. Finally, we can calculate the
standard deviation by square rooting this variance. To two decimal places, this is
equal to 2.01. The standard deviation of the
function in the given table to two decimal places is 2.01.

Before looking at one final
example, we will consider the coefficient of variation. The coefficient of variation,
written 𝐶 sub 𝑉, gives the standard deviation as a percentage of the expected
value. If we let 𝑋 be a discrete random
variable with mean 𝐸 of 𝑋 and standard deviation 𝜎 sub 𝑋, if we assume further
that 𝜇 is not equal to zero, then the coefficient of variation 𝐶 sub 𝑉 is given
by 𝐶 sub 𝑉 of 𝑋 is equal to 𝜎 sub 𝑋 divided by 𝐸 of 𝑋 multiplied by 100. We assume that 𝜇 is not equal to
zero as 𝐶 sub 𝑉 is not defined when the mean equals zero. As the standard deviation is always
positive, the coefficient of variation will be negative when 𝐸 of 𝑋 is negative
and positive when 𝐸 of 𝑋 is positive.

It is important to note that while
the standard deviation is an absolute measure of spread, the coefficient of
variation is a relative measure of spread. This is useful as when we deal with
variables with larger expected values, they’re more likely to be more spread
out. It therefore makes sense to use a
relative measure when comparing spreads. The coefficient of variation is
also useful when comparing data sets with different means and standard
deviations. The coefficient of variation
therefore represents how far on average data points are from the mean relative to
the size of the mean. We will now look at an example
where we need to calculate this coefficient of variation.

Work out the coefficient of
variation of the random variable 𝑋 whose probability distribution is shown. Give your answer to the nearest
percent.

We know that our figure is a
probability distribution graph. And we recall that the coefficient
of variation, written 𝐶 sub 𝑉, is equal to the standard deviation 𝜎 divided by
the expected value or mean 𝐸 of 𝑋 multiplied by 100 percent. This coefficient of variation
represents how far on average data points are from the mean relative to the size of
the mean. We will begin by calculating the
mean or expected value 𝐸 of 𝑋. We do this by multiplying each of
our 𝑋-values by the corresponding 𝑓 of 𝑥 value or probability. We then find the sum of all these
products.

From the graph, we begin by
multiplying one by one-tenth. Next, we multiply three by
two-tenths. We also need to multiply five by
three-tenths and seven by four-tenths. Calculating each of these products
gives us 0.1, 0.6, 1.5, and 2.8. 𝐸 of 𝑋 is therefore equal to
five. As we also need to calculate the
standard deviation, our next step is to calculate 𝐸 of 𝑋 squared. This is equal to one squared
multiplied by one-tenth plus three squared multiplied by two-tenths plus five
squared multiplied by three-tenths plus seven squared multiplied by four-tenths. This is equal to 0.1 plus 1.8 plus
7.5 plus 19.6. 𝐸 of 𝑋 squared is therefore equal
to 29.

Next, we recall that the variance
or var of 𝑋 is equal to 𝐸 of 𝑋 squared minus 𝐸 of 𝑋 all squared. In this question, we have 29 minus
five squared. This is equal to four. Clearing some space, we have the
following three values. We know that the standard deviation
𝜎 is equal to the positive square root of the variance of 𝑋. This means that in this question,
the standard deviation is the positive square root of four, which equals two. We can now substitute our values
into the formula for the coefficient of variation. We need to multiply two-fifths or
0.4 by 100. This is equal to 40 percent. The coefficient of variation of the
random variable 𝑋 shown in the graph is 40 percent.

We will now finish this video by
summarizing the key points. Given the probability distribution
of a random variable 𝑋, we can compute the standard deviation 𝜎 using the
following steps. (i) Compute the mean or expected
value 𝐸 of 𝑋, (ii) compute 𝐸 of 𝑋 squared, (iii) compute the variance, the var,
of 𝑋, which is equal to 𝐸 of 𝑋 squared minus 𝐸 of 𝑋 all squared, and (iv)
compute 𝜎 the standard deviation by finding the positive square root of the var of
𝑋.

We also saw that the coefficient of
variation, 𝐶 sub 𝑉, represents the standard deviation 𝜎 as a percentage of 𝐸 of
𝑋, the expected value, such that 𝐶 sub 𝑉 is equal to 𝜎 divided by 𝐸 of 𝑋
multiplied by 100 percent. We note that standard deviation is
an absolute measure of spread, and the coefficient of variation is a relative
measure of spread.