How to 'sum' a standard deviation?

Question

I have a monthly average for a value and a standard deviation corresponding to that average. I am now computing the annual average as the sum of monthly averages, how can I represent the standard deviation for the summed average ?

For example considering output from a wind farm:

Month        MWh     StdDev
January      927     333 
February     1234    250
March        1032    301
April        876     204
May          865     165
June         750     263
July         780     280
August       690     98
September    730     76
October      821     240
November     803     178
December     850     250

We can say that in the average year the wind farm produces 10,358 MWh, but what is the standard deviation corresponding to this figure ?

Answer 1

Short answer: You average the variances; then you can take square root to get the average standard deviation.

---

Example

Month          MWh  StdDev  Variance
==========   =====  ======  ========
January        927    333     110889
February      1234    250      62500
March         1032    301      90601
April          876    204      41616
May            865    165      27225
June           750    263      69169
July           780    280      78400
August         690     98       9604
September      730     76       5776
October        821    240      57600
November       803    178      31684
December       850    250      62500
===========  =====  =======  =======
Total        10358            647564
÷12            863    232      53964

And then the average standard deviation is sqrt(53,964) = 232

---

From Sum of normally distributed random variables:

If $X$ and $Y$ are independent random variables that are normally distributed (and therefore also jointly so), then their sum is also normally distributed

...the sum of two independent normally distributed random variables is normal, with its mean being the sum of the two means, and its variance being the sum of the two variances

And from Wolfram Alpha's Normal Sum Distribution:

Amazingly, the distribution of a sum of two normally distributed independent variates $X$ and $Y$ with means and variances $(\mu_X,\sigma_X^2)$ and $(\mu_Y,\sigma_Y^2)$, respectively is another normal distribution

$$ P_{X+Y}(u) = \frac{1}{\sqrt{2\pi (\sigma_X^2 + \sigma_Y^2)}} e^{-[u-(\mu_X+\mu_Y)]^2/[2(\sigma_X^2 + \sigma_Y^2)]} $$

which has mean

$$\mu_{X+Y} = \mu_X+\mu_Y$$

and variance

$$ \sigma_{X+Y}^2 = \sigma_X^2 + \sigma_Y^2$$

For your data:

• sum: 10,358 MWh
• variance: 647,564
• standard deviation: 804.71 ( sqrt(647564) )

So to answer your question:

• How to 'sum' a standard deviation?

• You sum them quadratically:

  s = sqrt(s1^2 + s2^2 + ... + s12^2)
  

Conceptually you sum the variances, then take the square root to get the standard deviation.

---

Because i was curious, i wanted to know the average monthly mean power, and its standard deviation. Through induction, we need 12 normal distributions which:

• sum to a mean of 10,358
• sum to a variance of 647,564

That would be 12 average monthly distributions of:

• mean of 10,358/12 = 863.16
• variance of 647,564/12 = 53,963.6
• standard deviation of sqrt(53963.6) = 232.3

We can check our monthly average distributions by adding them up 12 times, to see that they equal the yearly distribution:

• Mean: 863.16*12 = 10358 = 10,358 (correct)
• Variance: 53963.6*12 = 647564 = 647,564 (correct)

Note: i'll leave it to someone with a knowledge of the esoteric Latex math to convert my formula images, and formula code into stackexchange formatted formulas.

Edit: I moved the short, to the point, answer up top. Because i needed to do this again today, but wanted to double-check that i average the variances.

Answer 2

This is an old question but the answer accepted is not actually correct or complete. The user wants to calculate the standard deviation over 12 month data where the mean and standard deviation is already calculated over each month. Assuming that the number of samples in each month is the same, then it is possible to calculate the sample mean and variance over the year from each month data. For simplicity assume that we have two sets of data:

$X=\{x_1,....x_N\}$

$Y=\{y_1,....,y_N\}$

with known values of sample mean and sample variance, $\mu_x$, $\mu_y$,$\sigma^2_x$,$\sigma^2_y$.

Now we want to calculate the same estimates for

$Z=\{x_1,....,x_N, y_1,...,y_N\}$.

Consider that $\mu_x$,$\sigma^2_x$ are calculated as:

$\mu_x = \frac{\sum^N_{i=1} x_i}{N}$

$\sigma^2_x = \frac{\sum^N_{i=1} x^2_i}{N}-\mu^2_x$

To estimate mean and variance over the total set we need to calculate:

$\mu_z = \frac{\sum^N_{i=1} x_i +\sum^N_{i=1} y_i }{2N}= (\mu_x+\mu_y)/2$ which is given in the accepted answer. For variance however the story is different:

$\sigma^2_z = \frac{\sum^N_{i=1} x^2_i +\sum^N_{i=1} y^2_i }{2N}-\mu^2_z$

$\sigma^2_z = \frac{1 }{2}(\frac{\sum^N_{i=1} x^2_i}{N}-\mu^2_x + \frac{\sum^N_{i=1} y^2_i}{N}-\mu^2_y )+\frac{1 }{2}(\mu^2_x+\mu^2_y) -(\frac{\mu_x+\mu_y}{2})^2$

$\sigma^2_z = \frac{1 }{2}(\sigma^2_x+\sigma^2_y )+(\frac{\mu_x-\mu_y}{2})^2$

So if you have the variance over each subset and you want the variance over the whole set then you can average the variances of each subset if they all have the same mean. Otherwise you need to add the variance of mean of each subset.

Lets say that over the first half of the year we produce exactly 1000 MWh per day and in the seconds half we produce 2000 MWh per day. Then the mean and variance of energy production in first and seconds half are 1000 and 2000 for mean and variance is 0 for both halves. Now there are two different things that we may be interested in:

1-We want to calculate the variance of energy production over the whole year: then by averaging the two variance we arrive at zero, which is not correct since the energy per day over the the whole year is not constant. In this case we need to add the variance of all the means from each subset. Mathematically in this case the random variable of interest is energy production per day. We have sample statistics over subsets and we want to calculate the sample statistics over a longer time.

2-We want to calculate the variance of energy production per year: In other words we are interested in how much energy production changes from one year to another year. In this case averaging the variance leads to the correct answer which is 0, since in each year we are producing exactly 1500 MHW on average. Mathematically in this case the random variable of interest is average of energy production per day where the averaging is done over the whole year.

Answer 3

I believe what you may be really interested in though is the standard error rather than the standard deviation.

The standard error of the mean (SEM) is the standard deviation of the sample-mean's estimate of a population mean, and that will give you a measure how how good your yearly MWh estimate is.

It's very easy to compute: if you used $n$ samples to obtain your monthly MWh averages and standard deviations, you would just compute the standard deviation as @IanBoyd suggested and normalize it by the total size of your sample. That is,
$$ s = \frac{\sqrt{s_1^2 + s_2^2 + \ldots + s_{12}^2}}{\sqrt{12 \times n}} $$

Answer 4

I'd like to stress again the incorrectness in part of the accepted answer. The wording of the question lead to confusion.

The question have Average and StdDev of each month, but it's unclear what kind of subset is used. Is it the average of 1 wind turbine of the whole farm or the daily average of the whole farm? If it's the daily average for each month, you can't add up the monthly average to get the annual average because they do not have the same denominator. If it's the unit average, the question should state

We can say that in the average year each turbine in the wind farm produces 10,358 MWh,...

Instead of

We can say that in the average year the wind farm produces 10,358 MWh,...

Further more, The Standard deviation or variance is the comparison against the set's own average. It does NOT contain any information regarding the average of the whole set.

The image's not necessary very correct but it conveys the general idea. Let's imagine the output of 1 wind farm as in the image. As you can see, the "local" variance has nothing to do with the "global" variance, no matter how you add or multiply those. You cannot predict the variance of the year using variance of 2 half year. So, in the accepted answer, while the sum calculation is correct, the division by 12 to get the monthly number means nothing.. Of the three section, the first and the last section is wrong, the second is right.

Again, it's very wrong application, please do not follow it or it would get you into trouble. Just calculated for the whole thing, using total yearly/monthly output of each unit as data points depending whether you want yearly or monthly number,that should be the correct answer. You probably want something like this. This is my randomly generated numbers. If you have the data, the result in cell O2 should be your answer.