If the sum of all the 𝑥-values
minus the mean squared for a set of six values equals 25, find the standard
deviation of the set and round the result to the nearest thousandth.
Before we go any further, I want us
to think about and examine the formula for standard deviation. Standard deviation is given by the
formula 𝜎 equals the square root of the summation of all 𝑥-values minus the mean
of the 𝑥-values squared divided by 𝑛, the number of 𝑥-values.
How can we use the information
we’re given to solve for the standard deviation? If we look closely, we’ll see that
we’re already given the summation we need. The sum of all the 𝑥-values minus
the mean of the 𝑥-values squared equals 25.
We now know that the numerator
inside the square root is 25. We also know that the 𝑛 value is
the number of 𝑥-values. In this case, we have six
𝑥-values. So we plug in six for 𝑛. We can simplify this by taking the
square root of the numerator and the square root of the denominator. The square root of 25 equals five
and we can’t simplify the square root of six any further.
The standard deviation for these
six values will be equal to five over the square root of six. However, we want the value rounded
to the nearest thousandth. To do that would mean we would need
to divide five by the square root of six. We can use a calculator for this
process by plugging in five divided by six. And the calculator returns an
irrational number to us: 2.041241452.
We want to round to the thousandth
place. The thousandth place is three
spaces to the right of the decimal. The digit to the right of the
thousandth place is a two. Therefore, we round down. The one in the thousandth place
stays the same. And we bring down everything to the
left of the thousandth place: 2.041.
The standard deviation of the set
of six values is equal to 2.041.