Measures of Variability: Range, Variance & Standard Deviation

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  • 0:07 Measures of Variability
  • 2:26 Range
  • 3:48 Standard Deviation
  • 6:33 Variance
  • 7:56 Lesson Summary
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Lesson Transcript
Instructor: Devin Kowalczyk

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
formula for 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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