The Error in the Estimated Mean

So we see that if we have a finite data ``sample'', we can get an estimate of the true values of $\bar x$ and $\sigma$. But how far is our estimate $\bar x^{*}$ from the true value $\bar x$? This is the central question of every measurement, because it tells us how much confidence we may put in our result. Measurements without error ranges are meaningless! For example, there is really no meaning to the statement that the length of the table top is 3 meters, because the associated error might be a kilometer. There is meaning only if we can associate an error with this figure and say, for example, that the length is 3.00 meters with an error of plus or minus 0.01 meter.

Now suppose we make $N$ measurements to make up one data ``sample'' on one day and make another $N$ measurements to make a second sample on the next, and so collect a large number of samples. We determine the estimated mean value $\bar x^*$ for each sample. What is the probability distribution for this estimated mean value? Note that it is not the same as the probability distribution of the population. One way to see this is to realize that if we take larger and larger samples almost all of our values would be expected to be closer and closer to the true mean $\bar x$. In fact, a famous theorem of statistics, called the ``central limit'' theorem states that the probability distribution of the mean value approaches a Gaussian normal distribution as the sample size increases, regardless of whether the underlying population distribution $P(x)$ is itself Gaussian. The standard deviation of the mean value is estimated by

$$\sigma^*_{mean} = \sigma^*/\sqrt{N},$$ (14)

where $\sigma^*$ is given by Eq (13). As the sample size grows, $\sigma^*$ stabilizes, and the standard deviation of the mean shrinks as $1/\sqrt{N}$, so that the distribution of sample means $\bar x^*$ gets sharper around the true value $\bar x$.

So as a result of measuring one sample, we estimate the true mean value to be

$$\ \bar x^* \pm \sigma^*_{mean} = \bar x^{*} \pm \sigma^{*}/\sqrt{N}.$$

This is a practical formula. With it we need only make $N$ measurements, then estimate the population mean from Eq. (11) and the population standard deviation from Eq. (12). Then we compute the error in the mean from Eq. (14). Please bear in mind the difference between $\sigma^{*}$, which is the estimate of the error in a single measurement, i.e. the ``population'' standard deviation, and $\sigma^{*}_{mean}$, which is the estimate of the error in our estimated mean value $\bar x^{*}$.