# Range in Statistics, Variance in Statistics, Standard Deviation in Statistics, and Standard Error of the Mean in Statistics

Posted January 2, 2009

I spoke with someone today about a few things that I just take for granted as a dissertation consultant and statistician. This particular client, ever so politely asked that I explain the terminology associated with her descriptive statistics. Specifically, the terms in question were range, variance, standard deviation, and standard deviation of the mean. There are so many terms out there like these that are thrown around in research papers, journal entries and the such, without many of the readers – or even the authors – really knowing what they mean.

- What exactly is range and how is it used in a dissertation or thesis?
- What about variance and standard deviation? Are there numbers I should be looking for when I see the results of these tests?
- What in the world is standard error and is there a difference between just standard error and standard error of the mean?

For me, and any other statistician, these questions are easy, but what about the nursing student trying to finish up nursing school or the school teacher trying to become an assistant principle or principle so he or she can shape the lives or our young people? These people don't really have to be experts on this stuff and will probably never use it again.

So it's my job to make sure I can help these wonderful people move on to the truly important things, like saving lives and teaching children. Since you don't have to be an expert and you really just want to know how and why you are presenting this information, here is a quick and dirty explanation of those terms I mentioned at the top.

What is range in statistics?

Also commonly called range score, statistics range, or range value (these aren't formal terms, just what they're commonly called by my clients), this is in the most simple terms, the difference between the largest and smallest values in variable in a dataset. To calculate this by hand…you guessed it…subtract the smallest value in the variable from the largest value in the variable. You shouldn't have to calculate this by hand, since Excel will do this for you and other statistics software packages like SPSS and SAS will also calculate range for, by simply checking a box or choosing a command. I would like to clarify before I continue that I specify variable, because the range measurement will only be for one variable in your dataset. Receive professional assistance with using range in your Master's thesis, Master's dissertation, Ph.D. thesis, or Ph.D. dissertation.

What is variance in statistics?

Variance is used in so many ways that I could dedicate a blog just to variance as it pertains to statistics. What do you need to know to stand before your committee and defend this stuff? It pretty much tells you how the data is dispersed around the mean of your variable. If the variance is small, then the data is very closely dispersed around the mean, i.e. the data points are close in value to the mean. If the variance is large, then the data is more widely dispersed around the mean, i.e. the data points are different from the mean.

You would be crazy to calculate by hand since there are so many programs out there that will do it for you, but should you choose to, you are going to average the squared differences between each value in the variable and the mean. To illustrate more plainly, let's start with just one value. I am going to refer to the dataset 3, 5, 8, 4, 2, and 9.

- To start, we are going to calculate the mean of the dataset. We calculate the mean or average of the variable by adding the values and dividing by the number of values – just like grade school. When we finish we have an average or mean of 5.1667.

- We are then going to subtract that mean from the first value in the dataset, which is 3. This gives us -2.1667.

- We then square this value to obtain 4.6946 rounded.

- This value of 4.6946 is then divided by the number of observations or numbers in our variable minus 1, which is 0.9389.

- Do the same thing for each of the other variables in the variable, 5, 8, 4, 2, and 9.

- You should end up with 0.9389 + 0.0056 + 1.6055 + 0.2722 + 2.0056 + 2.9388 = 7.7666. This is our variance for the variable in our dataset consisting of 3, 5, 8, 4, 2, and 9.

What is standard deviation in statistics?

Standard deviation in statistics is also presented in the descriptive statistics results of any graduate thesis or dissertation. For the purposes of what we are doing, the standard deviation tells us how all the observations in the variable are distributed or clustered about the mean of the variable. After calculating the variance, the rest is all downhill. You are going to love this! Ready? The standard deviation is simply the square root of the variance, which is 2.7869. That's it!

The same rules apply to standard deviation as apply to variance: when the data is very closely dispersed around the mean, i.e. the data points are close in value to the mean, the standard deviation will be small. When the data is more widely dispersed around the mean, i.e. the data points are different from the mean, the standard deviation will be large. To receive help calculating standard deviation for the variables used in your Master's thesis, Master's dissertation, Ph.D. thesis, or Ph.D. dissertation, click here.

What is standard error of the mean in statistics?

Standard error of the mean in statistics is often defined as the standard deviation of all possible sample means. Basically, if we had a population from which to draw, we could draw many 6 observation samples from our population and each of the samples could have a different mean. The standard error tells us how much these means vary. With one sample, we often still want to know how much the mean could vary, therefore we estimate the standard error by dividing the standard deviation we just calculated (2.7869) by the square root of the number of observations (6) in our variable. We end up with 1.1377. Another easy one.

Interpreting the estimated standard error of the mean is a little different. In this case, the number is going to fluctute with the number of observations we have in the variable. The more observations we have, the smaller the standard error. The fewer the observations, the larger the standard error. Click here to receive help with standard error as it pertains to the statistical analysis in your Master's thesis, Master's dissertation, Ph.D. thesis, or Ph.D. dissertation.

It is amazingly rewarding to help graduate students understand this stuff. It is the moment the light switch comes on that makes it all worthwhile. If you are having some difficulty or just want another set of eyes to see your Master's or Ph.D., thesis or dissertation, I can help you. Please click here to schedule an appointment for a free consultation.

Hi James,You’ve hit on an area that I’ve always sort of glossed over in my understanding, but now that I’m mid-way through my PhD in Ecology and have been teaching students stats, I find myself unable to ignore the issue any longer. I was hoping you could elaborate on your previous descriptions. Specifically, what exactly is the difference between variance and standard deviation, interpretation-wise? I understand how to calculate both, and that fundamentally they both describe the spread of data around a mean. But what is the difference between them and when would using one over the other be appropriate.Similarly, I am starting to really understand standard error. But how does one know when to report variance, std.dev, std.error, or confidence interval with their mean? What are the ‘rules’ concerning this? Furthermore, if I’m to read a study where one of these error or dispersion values is reported, how should I interpret the fact that they’ve reported confidence intervals instead of std. error, for example?Thanks sooo much!