# Adding two set of samples with standard deviation confusions

#### mattkunq

Hello People,
Lets say I have a sample set A with sample size= 101 mean =10 and sample stdv of 1

then sample set B with sample size= 100 Mean=15 with sample stdv of 2.

If i add these two samples sets together I should get a new stdv that is no smaller than the smallest stdv of the two. Right?

If you don't agree please tell so I know Im wrong but if you do think that it is true, then here is my confusion.

Fourier Series and make a straight line say(y=1) a sum of different sine/cosine waves.
Now if I take the x values as the sample names and the y axis values as the test values of that sample. I can calculate a sample stdv for each wave. If i add the whole Fourier series together and get a straight line I would get a Stdv of zero. Which is less thant whatever the individual cosine/sine waves stdv were.

Thank you =)

I took one engineering stats course.

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#### Stephen Tashi

Let's skip the fourier series and look at a simpler example.

Suppose the measurements in set A are {A1=-2, A2=0, A3=2}
and the measurements in set B are {B1 = 2, B2 = 0, B3= -2}
Then the measurements like the ones you want to consider are those in the set C given by:
{C1 = A1+B1 = 0, C2 = A2+B2 = 0, C3 =A3 + B3 = 0 }

"Standard deviation" is an ambiguous term. Among it's possible meanings are:

1. The standard deviation of a probability distribution, also called the "population standard deviation".

2. The sample standard deviation, which is a formula that specifies a function of the values of a random sample from a probability distribution.

3. A particular value of the sample standard deviation that resulted from one particular sample (e.g. a specific number instead of a function).

4. An estimator of the poplation standard deviation, wihich is some function of the values in a random sample.

5. A particular value of an estimator of the population standard deviation

6. The mean square deviation of a function f(x) computed over an interval (such as the interval of one period of a periodic function).

Is your question using the term "standard deviation" in two different ways?

#### mathman

Hello People,
Lets say I have a sample set A with sample size= 101 mean =10 and sample stdv of 1

then sample set B with sample size= 100 Mean=15 with sample stdv of 2.

If i add these two samples sets together I should get a new stdv that is no smaller than the smallest stdv of the two. Right?

If you don't agree please tell so I know Im wrong but if you do think that it is true, then here is my confusion.

Fourier Series and make a straight line say(y=1) a sum of different sine/cosine waves.
Now if I take the x values as the sample names and the y axis values as the test values of that sample. I can calculate a sample stdv for each wave. If i add the whole Fourier series together and get a straight line I would get a Stdv of zero. Which is less thant whatever the individual cosine/sine waves stdv were.

Thank you =)

I took one engineering stats course.
The best way to get the combined mean and standard deviation is to reconstruct the original sum and sum of squares for each set of samples and then combine them to compute a mean and standard deviation for the totality of samples. My calculation led to a mean of 12.49 and a standard deviation of 2.957. Qualitatively the standard deviation is larger because of the significant difference in the means of the two sets of samples.

#### mattkunq

Hello People,
I have figured it out. When I say standard deviation I mean the follow equation is applied for the sample:
[1/(n-1)]*[(sum of x^2)-n*samplemean^2)]

So my confusion was the difference between combining two sample sets together and adding the two sample outputs respectively. Essentially stacking them. Like adding two inverted colored checker boards to one homogenous sheet of color.

As appose to puting the two different checkers side by side and evaluate the grey level sample standard deviation of that.

Thanks though! =)

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