## 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 3. Level 1 | Level 2 | Level 3 Next Topic Central Tendency | Confidence Intervals

### Explanation

At this level, we will look at three key aspects of the standard deviation:
• The standard deviation is useful as a distance measure, for example you can say that a value is 1 standard deviation above the mean;
• The standard deviation appears in many statistical measures;
• The standard deviation of a sample can be used to calculate the standard error of the sampling distribution of sampling means.
Standard Deviation as a Measure of Distance
As the standard deviation is a measure of distance from the mean, you can talk about any value in your data in terms of how many standard deviations from the mean it lies. A value that is one standard deviation from the mean is obviously closer than a value that is two, or three standard deviations from the mean. This fact is the basis of z scores, which you will learn about later. For example, if IQ has an average of 50 and a standard deviation of 10, and you have an IQ of 70, then you are two standard deviations above the average!

Uses of the Standard Deviation
The standard deviation appears in many statistical measures, for example:

• Z scores are measured in units of the number of standard deviations away from the mean
• You will have noticed that the final step in calculating the standard deviation is to take a square root. If you did not perform that final step, you would have a measure known as the variance of the data. That is to say:
• The variance of a sample of data is the standard deviation squared
• The standard deviation of a sample of data is the square root of the variance.
The two points above are, of course, two ways of saying the same thing.
• t-tests use variance to test whether or not two samples are significantly different from each other
The Standard Deviation of the Sampling Distribution of Sample Means - Standard Error
In level three of the topic on central tendency, we saw that the mean of a sample is just one of the many possible sample means that might have been found. All these possible sample means form what is know as the sampling distribution of sample means. We also saw that the mean of this theoretical distribution is the same as the population mean. Now we will learn about the standard deviation of the sampling distribution of sample means.

The standard deviation of the sampling distribution of sample means tells you about how representative of the population your sample is likely to be. A very tight distribution indicates that the mean of any sample will be close to the population mean. A very wide distribution, on the other hand, indicates that any single sample mean has a higher chance of being far from the population mean. For these reasons, the standard deviation of the sampling distribution of sample means is given the far easier name: Standard Error

Standard error tells you the average distance between each possible sample mean and the population mean. Standard error is affected by your sample in a number of ways:

• Sample size - Larger samples have a smaller standard error;
• Sample variation - Samples with larger standard deviation have a larger standard error.

### Exploration

In this section we will look at the factors affecting the spread of the sampling distribution of sample means, in other words, the standard error. The game below generates 1000 random samples each time you click the [1000 Samples] button. It Plots the histogram of the means of those samples and reports the standard error.

You can choose whether the standard deviation of the sample is high or low by clicking the [Sample Standard Deviation] button and you can change the size of the samples by entering a number in the box provided.
• How does the standard deviation (S) of the sample effect the standard error (SE)?

• How does sample size effect Standard Error (SE)?

• How does the spread of the sampling distribution of sample means relate to Standard Error (SE)?

( You need to enable Java to see this applet. )
Remember that the histogram shows the number of times the mean of each sample was found to be a certain value (in this case, from 0 to 8).
What is the population mean most likely to be?
When standard error is low, are the sample means in the histogram on average closer to or further away from the population mean?

### Application

We have already calculated the standard deviation for your variable written. It is 24.74.

Now we will practice using the standard deviation as a distance measure.

If the mean of written = 62.68, what value is one standard deviation above the mean?
If the mean of written = 62.68, what value is one standard deviation below the mean?
If the mean of written = 62.68, how many standard deviations above the mean is 112.16?
(it is a whole number)

The standard error of written is 4.68. What does that value reflect?
Calculator
( You need to enable Java to see this applet. )
Help
 Central Tendency | Confidence Intervals