Standard deviation of mean of a set of numbers, which are imprecise

Question

I have a problem which seems very simple, but for some reason I can not find out what I have to do exactly.

Let's say I have a set of derived values, where each of them has an individual error: $$X_{all}=(x_1 \pm \sigma_{x_1}, x_2 \pm \sigma_{x_2}, ..., x_n \pm \sigma_{x_n})$$ (where $\sigma_i$ stands for the standard deviation).

Now I want to have the average value of $X_{all}$ and some confidence of that value. The average is of course: $X_{avg}=\frac{1}{N}\sum x_i$

But for the standard deviation of $X_{avg}$, I dont know what I should use. There are two possibilities that seem neccesary:

1) From error-propagation: $$\sigma_{X_{avg}} = \sqrt{\frac{1}{N} \sum_i \sigma_{x_i}^2}$$

2) Normal standard-deviation when creating a mean value: $$\sigma_{X_{avg}} = \sqrt{\frac{1}{N} \sum_i (x_i-X_{avg})^2}$$

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Concrete Example:

I perform a measurement to get the value $X$. In order to get a statistical significant knowlegde about $X$ and its standard deviation, I perform the measurement n times, leading to the results $x_i$.

However, I can not measure $x_i$ directly, but only $y_i=x_i+BG$, where BG is a background value. For each measurement of $y_i$, I automatically get the information about $BG$ 1000 times, which gives me $BG_{avg}$ and $\sigma_{BG}$ for each $y_i$ ($BG$ is gauss distributed). Now I have $x_i = y_i - BG_{avg}$, thus I get a $\sigma_{x_i}=\sigma_{BG}$.

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Other Concrete Example:

I want to know the average of the lap-time of a racing car. I measure 100 laps. However, I know my clock has an uncertainty $\sigma_{clock}$. Moreover, for some reason I use for each lap a different clock with a different uncertainty $\sigma_{clock_i}$.

So I get 100 times $t_i$ for the time in lap $i$, with an uncertainty of $\sigma_{clock_i}$, corresponding to the uncertainty I introduce due to the clock itself.

What is my uncertainty of the average lap-time of the racing car?

Answer 1

You have defined two distinct standard deviations, describing different things.

The first one describes the standard deviation of the average itself - that is, a measure of how accurately we know the average, ignoring the spread of the set of measurements.

The second one describes the standard deviation of the set of measurements, ignoring the confidence of each measurement.

I assume that what you seek is the standard deviation of the set of all possible sets of measurements, where each measurement has a distribution described by the individual mean and standard deviation. That's a slightly more complicated problem.

Recall that the variance is defined as $$ \text{Var}(X) = E(X^2)-E(X)^2 $$ Now, if $$ X=\frac1N\sum_{i=1}^N X_i $$ where each $X_i\sim \mathcal{N}(x_i,\sigma_i)$, then $E(X)$ is just the average of the $x_i$ values. However, $E(X^2)$ is the average of the expected values of $X_i^2$. And so, we have $$ E(X_i^2) = \text{Var}(X_i)+E(X_i)^2 = \sigma_i^2+x_i^2 $$ and the average value is the sum of the average variance and the average of the $x_i^2$ values.

From here, it is easy to see that the final variance is quite simply the sum of the average of the measurement variances and the variance in the measurement values. That is, taking the square root to get the final standard deviation, $$ \sigma = \sqrt{\frac1N\left(\sum_i \left[\sigma_i^2+(x_i-\mu)^2\right]\right)} $$ where $\mu=\frac1N \sum_i x_i$.

Answer 2

I think you want the first solution, but you don't have it quite right. With the definition $$\bar x=X_{avg} ={{1} \over {N}} \sum_{i=1}^n x_i$$ we will have variance given by $$\sigma^2_{\bar x}={{1} \over {N^2}} \sum \sigma^2_{x_i}$$ So the standard deviation is $$\sigma_{\bar x} = {{1} \over {N}} \sqrt {\sum_i \sigma^2_{x_i}}$$ The reasoning is just the definition of variance and the independence of the random variables being summed.

Answer 3

The exact answer may depend. If your $x_i$ are supposed to be identically distributed (and the $\sigma_i$ are estimates of the standard derivation of the underlying distribution), then the second equation may be useful (though more precisely with $\frac 1{N-1}$ in place of $\frac 1N$); it determines a better estimation of the standard deviationfrom the observations. Example: You make repeated measurements of lap times of the same racing car and want to determine the avearage lap time of that car.

If on the other hand it makes little sense to view the $x_i$ do not follow the same distributon. Example: In a family one parent makes $5000\pm 10\$$ a month, another parent make $2000\pm 50\$$ and the three kids make $0\$$ a month each. Then the income of a randomly picked familiy member is $1400\$$ with a very large variance (per second formula); but the average income per family membre is quite precisely $1400\$$ (per first formula, error propagation).

You will notice a difference between the two approaches only in those cases where the $\sigma_i$ are much smaller and known from other sources (such as measurement or quantification errors) than the random nature of the underlying process ...