# Large vs. Small Standard Deviation

I understand how to calculate the mean, variance and standard deviation of a given set of numbers, and I also understand the standard deviation is a measure of spread from the mean.

In most texts (and blogs, and articles), we learn that a "small standard deviation" means most of the data values fall on or near the expected value and a "large standard deviation" means that there is more spread. Got it. What are the definitions of "small" and "large" in this context?

Do you take the value of the standard deviation and compare it to the mean? The median? Something else?

Here's a real-life example: I have 28 college students and I just calculated their final grades using Excel. Here are the summary statistics:

So, based on the data presented, is the standard deviation "large" or "small"? What are you comparing it to to make this determination?

Thank you, John

• "Small" and "large" is completely arbitrary and dependent on the data. There is no formal definition. Commented Dec 18, 2020 at 13:09
• @user332577 Why do you say that MAD and IQR are scale-invariant? The are in the original units, just like standard deviation.
– Dave
Commented Dec 18, 2020 at 17:00
• @user332577 What is robustness to differences in scale?
– Dave
Commented Dec 18, 2020 at 17:13
• @user332577 It's a term you're using, and I am curious what you mean.
– Dave
Commented Dec 18, 2020 at 17:20
• @user332577 "Scale-invariant dispersion" strikes me as an oxymoron. It's like a speed-invariant measure of how fast an object travels. Is something fast because it is traveling 50 km/h? If it's a sprinter, then yes. If it's a comet, then no.
– Dave
Commented Dec 18, 2020 at 17:51