Analysis 1

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Mean and Variance

The measurements you make in the lab are subject to error. We can formalize this idea by thinking of a measurement of<math>x</math>. This may be the number of counts per minute from the ML or the RS labs, or it might be a reading of the Hall voltage, etc etc. Nature determines the expectation value of the measurement. We'll use the mean as the expectation value (instead of, say, the median):

<math>\mu = \langle x \rangle. </math>

If you take a lot of data, such that you have <math>N</math> measurements of <math>x</math> -- <math>x_i</math>, then the sample mean is given by

<math>m = \frac{1}{N} \sum^N_i x_i </math>

which should be familiar. In order to quantify the error associated with our data, we introduce the variance $\sigma^2$

<math>\sigma^2 = \langle (x-\mu)^2 \rangle. </math>

This is the "second moment" of the distribution of data. Note that the square is essential because, from the definition of the mean, <math> \langle (x-\mu) \rangle = 0 </math>. The square root of the variance is the standard deviation <math>\sigma</math>. Analogously to the mean, we can compute the sample variance from our dataset <math>x_i</math>:

<math>s^2 = \frac{1}{N-1} \sum^{N}_i (x_i - m)^2 </math>.

Why do we use <math>N-1</math> instead of <math>N</math> in the denominator for the sample variance? We'll come back to that later.

Simple Error Propagation

Given a function <math>f(x)</math>, what is the error on <math>f</math> if you know the mean <math>\mu</math> and the variance <math>\sigma^2</math> of <math>x</math>? Let's make things interesting and introduce another variable <math>y</math>. Let's start with our expression for variance

<math>\sigma_f^2 = \langle (f(x) - \mu_f)^2 \rangle </math>,

where <math>\sigma_f^2</math> and <math>\mu_f^2</math> are the variance and the mean of $f(x)$. As you might

Error on the Sample Mean

Armed with the above derivations, we can now address the variance of the sample mean.