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Standard Deviation

Standard Deviation

Description:

This lesson will explain standard deviation.

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Tutorial

What's Covered

This tutorial is going to explain to you the concept of standard deviation. It will cover:

  1. Standard Deviation
  2. Interpreting Standard Deviation
  3. Using Technology

1. Standard Deviation

Standard deviation is a measure of variation that we use quite often in statistics.

Term to Know

Standard Deviation

A typical amount by which we would expect a data point to differ from the mean. Typically, about half to two-thirds of the data points fall within one standard deviation of the mean.

Standard deviation measures spread. You will interpret the standard deviation as the typical amount that you would expect data to be within the mean. The full name for standard deviation is "standard deviation from the mean." And if you break that down, "standard" just means it's typical. "Deviation" means that you expect it to be off from the mean, just by chance. And "standard deviation from the mean" states that the data will be away from the mean.

Here is what the formula looks like:

For as ugly as the formula looks, it's actually the one preferred most, provided that the distribution that you’re looking for is roughly symmetric and doesn’t have outliers. If the data isn't roughly symmetric or has outliers, you will use a different measure of spread, the interquartile range.


2. Interpreting Standard Deviation

These are the heights of the Chicago Bulls basketball team.

Each of these items are the Xᵢs (X sub 1) in the original formula. To find the standard deviation of this data set, follow these steps:

Step by Step

1. First, subtract the mean, which means you have to calculate the mean first. A thorough explanation of how to calculate the mean is in another tutorial. So for today, know that the mean is around 78.33 inches. So subtract 78.33 from each of these values.

2. Square those values, resulting in x minus mean-squared.

3. Use this sigma notation, which is the same as summation notation, to add these values up. They sum up to 205⅓.

4. Divide that sum by n minus 1. n in this case is 15 because there were 15 players. This is almost like averaging by dividing. But you’re not dividing exactly by n., but by n minus 1. So dividing that by 14 gets you 14⅔.

If you stopped here, this 14.6666 number would measure a kind of variation called "variance."

Variance isn’t used very often mainly because it is still a squared value. The units on variance are not the same as the units for the ones that we measured in. The measurement is 14.66667 inches squared, not inches.

5. So the final step is to take the square root of that number, which gets you 3.83.

Here is an image of that calculation:

For this data set, you would expect a good portion of the heights to be within 3.83 inches of 78.33.

Term to Know

  • Variance
  • The square of standard deviation. While it has some uses in statistics, it is not a practical unit of measurement. It is calculated the same way as standard deviation, but without the square root.

Try It

Considering the original list of heights, which fall into one standard deviation of the mean?

You'll notice about ⅔ of the players had heights between 78.33 minus 3.83 or 78.33 plus 3.83. So that's how you interpret the standard deviation.

Big Idea

Standard deviation is a typical amount by which we would expect values to vary around the mean.


3. Using Technology

The standard deviation is almost always found on a calculator or a spreadsheet or some kind of applet on the internet that you find. Typically, it is not solved by hand. So if you're frustrated with the calculation you just practiced, you can use your calculator or a spreadsheet in the future.

Here is how to use spreadsheets to calculate standard deviation:

Step by Step


1. Enter your list of data into the spreadsheet.

2. For the formula, select “=stdev.”

3. Select the list that you want to find the standard deviation for and hit Enter.

This spreadsheet formula finds the same number as the calculation you did by hand: 3.83.

Big Idea

If you have Excel or some spreadsheet program, there should be a standard deviation formula that you can use.


Summary

Interpreting standard deviation is important, but the standard deviation can be difficult to calculate. For this reason, calculating this number is typically done using technology. It's a measure of how far we would expect a typical data point to be from the mean.

Standard deviation is the square root of a value called variance, which is a little bit easier to calculate but not as useful as a standard deviation number.

Since the standard deviation is based on the mean, the standard deviation should only be reported as the measure of spread when you're reporting the measure of center to be the mean. You shouldn't mix and match the standard deviation with the median, and you shouldn't mix and match the mean with the interquartile range, either.

Thank you and good luck!

Source: THIS WORK IS ADAPTED FROM SOPHIA AUTHOR JONATHAN OSTERS

TERMS TO KNOW
  • Variance

    The square of standard deviation. While it has some uses in statistics, it is not a practical unit of measurement. It is calculated the same way as standard deviation, but without the square root.

  • Standard Deviation

    A typical amount by which we would expect a data point to differ from the mean. Typically, about half to two-thirds of the data points fall within one standard deviation of the mean.