Video Transcript
Calculate the standard deviation of
the values 45, 35, 42, 49, 39, and 34. Give your answer to three decimal
places.
Let’s begin by recalling what we
mean by the standard deviation. The standard deviation is a measure
of how spread out a set of numbers are. And to find it mathematically, we
have to take the square root of the variance. We might see the standard deviation
written as the Greek letter 𝜎 and the variance as 𝜎 squared.
Next, we recall what we mean by the
variance. This is the average of the squared
differences from the mean. We will therefore begin by
calculating the mean. To calculate the mean of a set of
numbers, we add them all together and then divide by how many there are. In this question, we need to find
the sum of 45, 35, 42, 49, 39, and 34 and then divide this total by six. The six numbers sum to 244, and
dividing this by six gives us 40.6 recurring. However, for accuracy, we will
leave our answer as the fraction 122 over three. The mean of the six values is 122
over three.
We are now in a position to find
the variance. And as already mentioned, this is
the average of the squared differences from the mean. We begin by subtracting each of our
six values from the mean. We then square each of these
values, add them together, and divide by how many values we have. The variance in this question is
therefore equal to the following equation. Subtracting 45 from 122 over three
and then squaring the answer gives us 169 over nine. Squaring 122 over three minus 35
gives us 289 over nine. And repeating this for the four
other terms on the numerator, we have 16 over nine, 625 over nine, 25 over nine, and
400 over nine.
The variance is the sum of these
six values all divided by six. The numerator is equal to 1524 over
nine. And dividing this by six, we have
254 over nine. Once again, we will leave this as a
fraction. The variance of our six numbers is
254 over nine. We are now in a position to
calculate the standard deviation. It is equal to the square root of
254 over nine. Typing this into our calculator
gives us 5.31245 and so on.
We are asked to give our answer to
three decimal places. Since the fourth digit after the
decimal point is a four, we round down. And we can therefore conclude that
the standard deviation of the six values, correct to three decimal places, is
5.312. Whilst it is not covered in this
video, it is worth noting that some scientific calculators will calculate the mean,
variance, and standard deviation for us by simply typing in the values.
