Video Transcript
Calculate the standard
deviation. Round your answer to two decimal
places.
The table tells us that three items
had a price of 10 dollars, two items had a price of 20 dollars, and four items had a
price of 30 dollars. We have been asked to find the
standard deviation of this data set. Let’s recall how we can find the
standard deviation of a data set.
We have that the standard deviation
is equal to the square root of the mean of the squares minus the square of the
mean. In equation form, this looks like
the square root of the sum of the frequencies multiplied by the squares of the
prices over the sums of the frequencies minus the square of the sum of the
frequencies multiplied by the prices over the sum of the frequencies.
In order to use this formula, we
can see there are a few things we need to calculate. Let’s start by labeling our price
as 𝑥 and our frequency as 𝑓. We can start by finding the sum of
the frequencies. We do this by simply adding three,
two, and four, which is equal to nine.
Next, we can extend our table to
help us find the other quantities. We need to find the sum of the
frequencies multiplied by the corresponding prices and the sum of the frequencies
multiplied by the squares of the corresponding prices. We can start by finding the squares
of 𝑥, or the squares of the prices. The first price is 10, so its
square is 100. The second price is 20, so its
square is 400. And the final price is 30, so its
square is 900.
Next, we can find 𝑓𝑥. We do this by multiplying the
frequency with the corresponding price. The calculations we have to perform
are three multiplied by 10, two multiplied by 20, and four multiplied by 30, giving
us 30, 40, and 120. We can then find the sum of 𝑓𝑥 by
adding these values together, giving us 190. Next, we calculate the frequencies
multiplied by the squares of the prices. We get 300, 800, and 3600.
We now have all the parts we need
to find the sum of 𝑓𝑥 squared. We find that it is equal to
4700. We’ve now found all the parts we
need to find the standard deviation, and we are ready to calculate it. We substitute the values we have
just found into the formula. This will give us that the standard
deviation is equal to 8.7488 and so on.
Let’s not forget that the question
has asked us to round our answer to two decimal places. Hence, our solution is that the
standard deviation is 8.75.
