We want to calculate the sample standard deviation of the data shown below,
to two decimal places. The standard deviation is a measure of variation
based on measuring how far each data value deviates or is different from the
mean.
And here are a few important characteristics. The standard deviation is always positive.
The standard deviation will be zero if all the data values are equal,
and will get larger as the data spreads out. Standard deviation has the same units
as the original data.
And standard deviation, like the mean, can be highly influenced by out outliers.
In order calculate standard deviation, we'll follow these steps here.
Step one: we find the deviation of each data value from the mean,
in other words we subtract the mean from the data value.
Step two: we square the deviation.
Step three: we add the square deviations.
Step four: we'll divide by n,
the number data values, if the data represents the whole population;
we divide by N minus one, if the data is a sample.
Because we're asked to find the sample standard deviation, we'll be dividing by N minus one
shown in the formula here below. And then finally, we compute the square root
of the result, which gives us the standard deviation.
So going back to our example, I've already done some work to save some time.
So this first column are the data values.
So we need to find the data values.
So the first up is the find the deviations, which would be the data
value
minus the mean. So notice below I determined the
mean, or X bar, by summing the data values,
and dividing by the number of data values which was six,
which gave a mean of twenty point five. If the mean doesn't terminate,
we should include at least three or four decimal places for accuracy.
So notice the second column, X minus X bar,
we started with the data values, and then subtracted
the mean.
The next step is to square the deviation, so we took the deviations here,
and then squared them in the next column. From here
let's set up the sample standard deviation formula. We would have s
equals, and then we'll have the square root. We'll have a fraction under the square
root,
where the numerator is going to be
the sum of the squares of the deviations.
So we'd have zero point two, five.
Plus seventy-two point two, five.
Plus twelve point two, five.
Plus
six point two, five.
Plus
twenty point two, five.
And then finally, plus
thirty point two, five. Now we're determining the sample standard deviation.
So we'll divide this sum by n minus one,
again there are six data values, six minus one would be five.
And then we take the square root of this fraction,
which will give us our sample standard deviation.
And now we'll go to the calculator. So we'll have
square root. And now we'll put the numerator of the fraction in parenthesis. So open
parenthesis,
point two, five. Plus seventy-two
point two five. Plus twelve point two, five.
Plus six point two, five.
Plus twenty point two, five. Plus
thirty point two, five. There's the numerator so we'll have closed parenthesis.
And notice how the denominator of five is still under the square root,
so we'll press divided by five, right arrow.
Again notice how the entire fraction is under the square root, so we'll press enter.
And to two decimal places we have approximately five point three, two
for our sample standard deviation.
So going back to our first slide, we enter
five point three, two.
I hope you found this helpful.