ExplanationAt this level, we will look at three key aspects of the standard deviation:
Standard Deviation as a Measure of Distance
- 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.
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:
The Standard Deviation of the Sampling Distribution of Sample Means - Standard Error
- 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 two points above are, of course, two ways of saying the same thing.
- 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.
- t-tests use variance to test whether or not two samples are significantly different from each other
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.