## Standard Deviation

 Getting Started General Instructions | Introduction to Your Study Descriptive Statistics Histograms | Scatter Plots | Central Tendency | Standard Deviation | Confidence Intervals Relating Variables Correlation Important Concepts The Normal Distribution | Z Scores | Probability Distributions Levels You are currently on Standard Deviation at level 1. Level 1 | Level 2 | Level 3 Next Topic Central Tendency | Confidence Intervals

### Explanation

The average of your data summarises it all in a single value. That certainly throws away a lot of information. If you were to know one more thing about the data, after the average, what would be the most useful thing? The range (largest and smallest) might be useful, but there is a different measure that is even better - the standard deviation.

The standard deviation measures how much the data varies:

• A large number means the data varies a lot
• A small number means the data varies a little
• A standard deviation of zero indicates that all the values in the data are identical

For example, the standard deviation of temperature in Florida is less than it is in New York.

The standard deviation tells you something more about the average too, as it measures variation in terms of how far from the mean all the values in the sample fall. Values are further from the mean on average when standard deviation is large than they are when it is small.

The symbol for the standard deviation is of a sample of data is s.

### Exploration

Use the game on the left to explore how standard deviation works. Imagine you are an archer - your target is shown on the screen. You are quite good and you hit the centre more often than not, so that is your average. To know whether you are better than somebody else whose average is also the centre, you need to know the standard deviation of your shots.

The target shows where your shots have landed and the standard deviation of your shots is shown above the target. Click anywhere on the left of the screen to reduce your standard deviation and click anywhere on the right to increase it (you will need repeated clicks to see much of a change).

• What is the relationship between the standard deviation (s) and how spread out your shots are?

• How could you move the shots in towards the average (the centre)?

• Are the shots more spread out when standard deviation is large or small?

• Would the better archer have a larger or smaller standard deviation?

( You need to enable Java to see this applet. )

You can see now why standard deviation is more useful than range. If you had one bad shot and hit the very edge of the target, the range of your shots would be from that point, no matter how close the rest of your shots are. Standard deviation takes into account how far apart all the values are from each other, not just the distance between the extremes.

### Application

• The standard deviation for no. apps is 30.64
• The standard deviation for cost is 298.47
Which variable has the most variation between its values?
 Central Tendency | Confidence Intervals