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Devin has taught psychology and has a master's degree in clinical forensic psychology. He is working on his PhD.

Looking specifically at range, variance, and standard deviation, this lesson explores the relationship between these measures and samples, populations, and what it says about your data.

Measures of Variability

I was going to write this about intelligence and intelligence quotients, but that got really complicated really fast. So, let's talk about obesity instead, because you're more likely to hear about the rising rates of obesity rather than the rising IQs. (Unless you go on to do psychological assessments. Then I would recommend you look up the Flynn Effect and the Idiocracy Theory.) But this lesson is about weight and understanding the descriptions of it.

Weight, like so many other things, is not static or unchanging. Not everyone who is 6 feet tall is 200 pounds - there is some variability. When reporting these numbers or reviewing them for a project, a researcher needs to understand how much difference there is in the scores. This is where we will look at measures of variability, which are statistical procedures to describe how spread out the data is. They are:

Range: defined as a single number representing the spread of the data

Standard deviation: defined as a number representing how far from the average each score is

Variance: defined as a number indicating how spread out the data is

When trying to understand how spread out the data is, we, as researchers, need to differentiate and know the difference between population and sample. A population is defined as the complete collection to be studied, like all the police officers in your city. A sample is defined as a section of the population and would be a selection of police officers you are studying. This can be anywhere from 1% to 99% of them.

When researchers do psychological experiments, they often must work with samples, because to find everyone in the population is nearly impossible. If you want a population data set, such as the world's weight, for example, that would be about seven billion data points. If you wanted the population data set of everyone in California, then that means you need about 33 million data points. In my own town, this is about 100,000 people. The trick is trying to make your sample data look like the population, which means you need to find measures on how variable your data is compared to the estimated population.

Range

Let's go back to our study on obesity. What's the range of weights we'll be looking at? Range is simply taking the highest score and subtracting the lowest score from it. It's pretty simple to find. If the heaviest person is 800 pounds and the smallest person is 100, then our range is 700 pounds. Remember: to do range, you will need to have scores that have some variability. For example, weight has a large variability in the scores and has a meaningful range. A five-question quiz would not have a very meaningful range because the largest possible range is five.

Range has a simple and easy to understand purpose as well: to quickly and easily inform us on how wide (no pun intended) the scores are. If we're doing a study and using a sample, we need to know how representative of the population our sample is. For example, if we are looking at weight and depression and our range is 50 pounds, then we don't have a very wide range, and it's not representative of the population. This may limit the findings on how depression affects weight because we're only looking at either the super skinny or the overweight instead of comparing the two. If our range is 500 pounds, now we're looking at a broader sample and a likely more representative sample of weight and how it affects depression.

Standard Deviation

While range is about how much your data covers, standard deviation has to do more with how much difference there is between the scores. If all of the scores are grouped around the average, then your standard deviation will be lower. If your scores are all over the map and not grouped together at all, then your standard deviation will be huge. The steps for calculating it are:

Calculate the average

Calculate the deviations, which are the scores minus the average

Square the deviations

Sum up the squared deviations

Divide this by the number of scores in your data set (or multiply by 1/N, same thing)

Take the square root

Formula for finding standard deviation

The formula takes advantage of statistical language and is not as complicated as it seems. The part in the parentheses above is the first two steps, subtracting the average (the x with the line over it) and the score (represented by xi). Then you square each result. The big, funny E (called sigma) means that you add up all the squared deviations. Then you multiply the sum by one divided by the number of scores in your sample. The last step is square rooting to get your standard deviation, which is represented on the left side of the equation by the Sn.

If you have a group of scores and they're all clustered around the mean, then our second step of calculating the squared deviations would result in a smaller number. This would make all the math later much smaller, and thus our standard deviation smaller.

When all our scores are clustered around the middle, it would look like the graph below, with all the scores making a huge bump in the middle.

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When the scores are clustered around the middle, the graph shows a huge bump around the middle.

If the scores are all spread out or clumped in weird places, then the standard deviation will be really high.

Standard deviation is important to understanding samples and populations because it lets you know how varied the scores are. First off, if you're looking at a study involving weight with the average being 200 and the standard deviation being 50 pounds, then that means about 68% of the data is between 150 and 250 pounds. That's not bad, depending on how big of a weight difference you want. Many statistical tests could be compromised because the data set is too widely spread.

To make things a little more complicated, the standard deviation formula can vary depending on if you have collected all the people in the group (a population) or a few people in the group (a sample). The reason behind this is there is an assumed bias, or skew, in the sample. If you have a population, you have everyone. If you have a sample, you have missed a group that might change your results.

Variance

Variance is extremely similar to standard deviation mathematically. In fact, it's the same math except for one step. Can you guess which one?

First, you calculate the average

Then you calculate the deviations, which is the score minus the average

You square the deviations

You sum up the squared deviations

Then you divide your squared deviations sum by the number of scores in your data set

The last step, square rooting, is missing. See the formula?

Formula for finding variance

All that is different is you don't take the square root of it. This translates into a larger score than standard deviation and not one that is readily usable.

Variance is used to attempt to elucidate, or make an estimated guess, at what the parameter is. A parameter is defined as a numerical value representing the total variability of the population. If you remember, most studies are done looking at samples with the hopes of saying something about the larger population. With variance as an estimate, we can begin to make educated guesses at understanding and predicting what the wider population looks like without having to make uneducated or wild guesses. Because of this, variance is not often used much.

Lesson Summary

Measures of variability are statistical procedures to describe how spread out the data is. There are three main ways to measure variability in a data set. They are:

Range: defined as a single number representing the spread of the data

Standard deviation: defined as a number representing how far from the average each score is

Variance: defined as a number indicating how spread out the data is

A researcher often uses a sample, which is defined as a section of the population in an experiment. The hope is that in understanding a small sample, we can predict something about the population, which is defined as the complete collection to be studied. With a sample, we are attempting to predict what the population really is. To this end, a variance is often used to help estimate a parameter, which is defined as a numerical value to represent the variability of the population.

Learning Outcomes

Completing the video lesson could enable you to:

Detail the three measures of variability: range, standard deviation, and variance

Illustrate the formulas for standard deviation and variance

Recall the definitions of sample, population, and parameter, and explain the importance of these terms to research

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