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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:

Summary Stats

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

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    $\begingroup$ "Small" and "large" is completely arbitrary and dependent on the data. There is no formal definition. $\endgroup$
    – user2974951
    Commented Dec 18, 2020 at 13:09
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    $\begingroup$ @user332577 Why do you say that MAD and IQR are scale-invariant? The are in the original units, just like standard deviation. $\endgroup$
    – Dave
    Commented Dec 18, 2020 at 17:00
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    $\begingroup$ @user332577 What is robustness to differences in scale? $\endgroup$
    – Dave
    Commented Dec 18, 2020 at 17:13
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    $\begingroup$ @user332577 It's a term you're using, and I am curious what you mean. $\endgroup$
    – Dave
    Commented Dec 18, 2020 at 17:20
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    $\begingroup$ @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. $\endgroup$
    – Dave
    Commented Dec 18, 2020 at 17:51

2 Answers 2

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As other users have mentioned in the comments, "small" and "large" are arbitrary and depend on the context. However, one very simple way to think about whether a standard deviation is small or large is as follows. If you assume that your data is normally distributed, then approximately 68% of your data points fall between one standard deviation below the mean, and one standard deviation above the mean. In the case of your data, this would mean 68% of students scored between roughly 63 and 95, and conversely 32% scored either above 95 or below 63. This gives a practical way to understand what your standard deviation is telling you (again, under the assumption that your data is normal). If you would have expected a greater percentage to fall between 63 and 95, then your standard deviation may be considered large, and if you would have expected a smaller percentage, then your standard deviation may be considered small.

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I believe that standard deviation is a tool to compare two data sets or more. Thus, the higher standard deviation of dataset will be the one considered large where data are more spread-out in relationships to the mean. On the other hand, a lower standard deviation will be considered small.

Also, it is a tool to evaluate how the numbers are spread-out from one data set.

the standard deviation could be considered big or small based on whatever reason the data set is serving. Example salaries of entry-level jobs, run-time of one mile for a particular sport team. for the sport team, you may have one athlete that is way faster than the others. Thus, we can use standard deviation to see how far he is above the mean. bottom line. it depends on how you want to use your data. If you think it is small, it is small. if you think it is big, it i

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