### Video Transcript

Calculate the standard deviation of
the following data. Round your answer to two decimal
places.

For the data, we have scores of
one, two, three, four, and five and frequencies of three, nine, 12, five, and four,
respectively. To find the standard deviation from
a frequency table, there are a couple of methods available to us. One formulation we might be
familiar with is the square root of the sum of each score π₯ π minus the mean π
all squared times each frequency π π and then divided by the sum of the
frequencies. Equivalently, this is equal to the
square root of the sum of each frequency times its score squared divided by the sum
of the frequencies minus π squared. Although we wonβt prove it here,
these two formulations are exactly the same. But the second version is often
slightly easier to use, so letβs use it.

Incidentally, itβs also simpler to
just write this expression without the πβs as long as we remember that we just want
to multiply entries in each column together. So the first values we will need to
find are the squares of the scores, that is, π₯ squared, which we can get by just
squaring the scores in the corresponding columns above. So the first entry is one squared,
which is just one. The second entry is two squared,
which is four, and we can continue doing this to get nine, 16, and 25.

Now, weβre also going to need to
find this π squared term. And we can recall that π is the
mean, which for a frequency table is the sum of the frequencies times the scores
divided by the sum of the frequencies. So, in particular, we will need to
find the product of each frequency with its corresponding score.

For the first value, we multiply
one by three, giving us three. And we can do this for all the
other entries, giving us two times nine is 18, three times 12 is 36, and four times
five or five times four are both 20. So now we can actually work out the
mean if we sum up these ππ₯ values and divide by the sum of the π values. The sum of the ππ₯ values is 97,
and the sum of the frequencies is 33, which means the mean is 97 over 33. We donβt need to simplify this, but
this is approximately 2.93.

Now, we have the mean and the π₯
squared values, but we will also need π times π₯ squared. We can find these values by
multiplying the π₯ squared values we previously found by the π-values. So the first entry is three times
one, which is three. Then, we have nine times four,
which is 36, and we continue this process, giving us 108, 80, and 100.

Now, we can finally calculate the
standard deviation. We take the sum of the ππ₯ squared
values, which is equal to 327. And we already found that the sum
of the πβs is 33. Therefore, the standard deviation
will be the square root of 327 over 33 minus 97 over 33 squared. And putting this all into a
calculator gives us 1.1265 et cetera, which we can round to two decimal places as
specified.

Thus, our final answer is that the
standard deviation of our data set to two decimal places is 1.13.