Analysis 1: Difference between revisions
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==Mean and Variance== | ==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): | 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 | <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 | 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 | ||
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<math>m = \frac{1}{N} \sum^N_i x_i </math> | <math>m = \frac{1}{N} \sum^N_i x_i </math> | ||
which should be familiar. In order to | which should be familiar. In order to quantify the error associated with our data, we introduce the ''variance'' $\sigma^2$ | ||
<math>\sigma = \langle (x-\mu)^2 \rangle | <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>. 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> | |||
Revision as of 14:17, 8 February 2012
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>. 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>