Summary Statistics Examples

What is the standard deviation of {643, 551, 399, 97, 174, 11}?

To find the standard deviation, we first need to find the mean.

= 312.5

Now we can work towards the standard deviation. Its formula is:

We like to make a table at this point that includes the deviations from the mean and the squared deviations. It's easier to keep track of everything this way.

Ugh, those are some big, nasty numbers. Now we have to add them together, which equals…

carry the one…

330119.5

Yuck. The next step is to divide by n – 1, which is 5 in this case. Then we take the square root of our result, which finally gets us the standard deviation.

That is our final answer. When finding the standard deviation by hand, we have to be very careful with our calculations. Possible dangers include displaced decimals, gnostic numbers, and the plague.

Find the mean, median, and standard deviation of this dataset: {23, 19, 29, 32, 34, 30, 29, 27, 24, 25}.

We'll find the median first, then we'll tackle the mean and standard deviation. First, though, we have to tidy up a bit.

{19, 23, 24, 25, 27, 29, 29, 30, 32, 34}

Much better. We have 10 terms here, so we have to take the average of the middle two. That's . That's our median.

We'll take the average of everything to find the mean.

= 27.2

Now to throw down with the big one, the standard deviation. Setting up our table:

We add up the last column to finish off the numerator of the standard deviation formula. That equals 183.6. Dividing by n – 1, or 9, gives us 20.4. The final step is taking the square root, which gives a final answer of:

s = 4.52

Which summary statistic—the mean, median, or standard deviation—is most strongly affected by outliers?

The median isn't affected by outliers at all. It's the middle term of the dataset, so it doesn't care what the actual values of all the other numbers are.

The mean is pulled by outlier values. They sit up in the denominator of the formula for the average saying, "Hey, we have cake. Come join us." And the mean listens.

The standard deviation, though, is affected the most by outliers. Every data point subtracts the mean, which already has an effect, and then the results are squared, which makes it even more ridiculous. The standard deviation follows the outliers around on Facebook, Twitter, Tumblr, and in the bushes outside their house.