# Standard Deviation vs Standard Error

I am confused about the difference between standard deviation and error. When do I use which? From my lecture:

• Standard error measures spread of all means

I actually learn this from my physics data analysis lesson. So suppose I have a set of measurements of distance ($D$) verses time ($t$) and I want to compute avg velocity:

$t_{avg}$ is calculated as $(t_1 + t_2 + t_3)/3$ (of course I used excel's AVERAGE here). Then avg velocity was $t_{avg}/D$.

Then I was asked which measurement #1-10 I think is the closest approximate the the actual velocity. So I was thinking I find the standard error and the one with the lowest is the closest approximate? Is this the reasonable thing to do? I found standard error by STDEV(t_1, t_2, t_3)/SQRT(3). Is this correct?

Heres my confusion: STDERR is defined as the spread of all means. I am calculating the spread of all measured data? Not the mean?

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In the context of your example: Does it really make a difference which you use? If standard deviation is 's'. The standard error is just s/SQRT(n). This is a linear transformation; therefore for the purpose of comparison it makes no difference which you use.

Now, for the purpose of making a statistical test. If you calculate a group of 'sample means' all independent and identically distributed. Then that sample of 'sample means' would have standard deviation given by s/SQRT(n).

Its intuitive that using a sample mean would give more information of the data, therefore s/SQRT(n) < s; that is to say the variability in the sample of 'sample means' is less than the variability in the individual sample.

TL;DR

-use 's' for the sake of comparison.

-For statistical tests use 's/SQRT(n)' when say finding the distribution or confidence interval for the sample means.

-For statistical tests use 's' when say finding the distribution or confidence interval for individual samples.

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In other words, the above excel image seems right to you? – Jiew Meng Sep 9 '12 at 11:41
Also to find the closest approximate to actual velocity, is it right to conclude that #10 (D is smallest) is the closest approximate as STDERR is smallest? – Jiew Meng Sep 9 '12 at 11:44
I should add that I'm a student myself- and therefore not a reliable expert! my name gives it away :). Whether you use STDEV or STDERR, #10 will always be the smallest. In fact the order of highest to lowest STDEV is the same as the order for highest to lowest STDERR (explained in paragraph 1). So u can use either 's' or 's/SQRT(n)' for comparing as you are. – student101 Sep 9 '12 at 13:13
Your assumption '# 10 is the closest estimate...since STDERR is smallest,'... is difficult for me to conclude on as I'm unsure of what the data represents. But prima facie yes, your statement is correct since low STDEV or STDERR generally indicates more reliable data- so #10 is reliable. – student101 Sep 9 '12 at 13:21
Just checked some of the excell calculations for STDERR of time, they seem right. – student101 Sep 9 '12 at 13:34