standard deviation help

[rusty JEDI]

is it fair to say that standard deviation is the average from the average? or would this statement be wrong.

if it is wrong could you please give a simple to understand definition of what standard deviation represents

[BruceZ]

The variance is the average squared distance from the average. Standard deviation is the square root of the variance. So in a sense you can think of it as kind of an average distance from the average, though it is not a simple arithmetic average because the square root of the average of squared distances is not the same as the average of the distances. That is, the average of squares is not the same as the square of the average. The square is taken so positive and negative distances are treated equally. The values will lie within +/- 1 standard deviation of the average 68% of the time.

So if x[n] are values of N measurments, and the average value is u, then:

variance = 1/N*sum[i=1 to N](x[n]-u)^2
standard deviation = sqrt(variance)

average distance from average would be:
1/N*sum[i=1 to N](|x[n]-u|)

If you're an electrical engineer, variance can be thought of as power, while standard deviation is rms amplitude. rms means "root mean square" or the square root of the mean of the square".

[BruceZ]

Here's an everyday example of rms average. In the US we say the AC coming from your wall is 120 volts. This is a sine wave, but if we measured the amplitude we would see it actually goes up to a peak of 170 volts. The 120 volts is an rms average. It is the result of integrating [170*sin(x)]^2 over a period of 2*pi, dividing by 2*pi, and taking the square root. This is the square root of the average of the squares of all the amplitudes which turns out to be 170/sqrt(2) = 120 rms. It will produce the same power as a constant DC voltage of 120 volts. It is not the same as the average excursion from 0 ignoring sign, which turns out to be 170*2/pi = 108 volts.

[Carl_William]

rusty JEDI

You wrote:

"Is it fair to say that standard deviation is the average from the average?"

A simple definition might be:

Standard deviation describes the dispersion about the mean (or average).

Regards,

Carl William

[BruceZ]

<pre><font class="small">code:</font><hr>



170----*-*------------------
v * * &lt;--- rms level = 120 volts
o * *
l 0---*-------*-------*-------&gt; time
t * *
s * *
-170-----------*-*----------

</pre><hr>
120 volts is the standard deviation of the voltage (how many people realized that?) however, in this case it is not true that 68% of the voltages lie within 1 standard deviation of average because that is generally true only for normally distrubuted data (like your poker results). If your poker results look like the above, you've got a problem.

I had a slight indexing problem with the above formulas which I might as well correct here:

If x[n] are values of N measurments, and the average value is u, then:

variance = 1/N*sum[n=1 to N](x[n]-u)^2
standard deviation = sqrt(variance)

average distance from average would be:
1/N*sum[n=1 to N](|x[n]-u|)