Statistics and uncertainty

Mean value - formula in LaTeX

The best estimate of a quantity measured repeatedly under the same conditions. The bar comes from \bar{x}, or \overline over a longer expression. The limits go in _{i=1} and ^{n}; in a displayed equation they sit above and below the sign.

$$
\bar{x} = \dfrac{1}{n}\sum_{i=1}^{n} x_i
$$

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What the symbols mean

$\bar{x}$

arithmetic mean

$n$

number of measurements

$x_i$

result of the i-th measurement

A statistics and uncertainty formula - a required part of the data analysis in every lab report.

Worked example

The mean of four stopwatch readings. The unit can be written once, after the whole fraction, instead of after every reading.

Other ways to write it

• Inside a sentence ($...$) the limits sit beside the sum sign; \limits forces them above and below.

  $$\bar{x} = \frac{1}{n}\sum\limits_{i=1}^{n} x_i$$

  \bar{x} = \frac{1}{n}\sum\limits_{i=1}^{n} x_i

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• The mean of a whole population, written with the Greek μ instead of x-bar.

  $$\mu = \dfrac{1}{N}\sum_{i=1}^{N} x_i$$

  \mu = \dfrac{1}{N}\sum_{i=1}^{N} x_i

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• The angle-bracket notation used in physics; \langle and \rangle are brackets, not less-than signs.

  $$\langle x \rangle = \dfrac{1}{n}\sum_{i=1}^{n} x_i$$

  \langle x \rangle = \dfrac{1}{n}\sum_{i=1}^{n} x_i

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• The weighted mean, when measurements carry different weights, for example different uncertainties.

  $$\bar{x}_w = \dfrac{\sum_{i} w_i x_i}{\sum_{i} w_i}$$

  \bar{x}_w = \dfrac{\sum_{i} w_i x_i}{\sum_{i} w_i}

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LaTeX commands in this formula

\bar

bar over a symbol (e.g. a mean)

\dfrac

fraction (display)

\sum

sum

x_{0}

subscript; wrap several characters in braces

x^{2}

superscript (power); wrap several characters in braces

How to use this formula in LaTeX?

1. Copy the code with the button above - $$…$$ is centered display math.
2. Paste it into the CamteX editor, Overleaf or any LaTeX document.
3. Inside a sentence, use single $…$ instead of double.
4. The formula needs the amsmath package - CamteX templates include it in the preamble.

Related formulas - Statistics and uncertainty

• Standard deviation
• Type A uncertainty
• Combined uncertainty
• Uncertainty propagation
• Expanded uncertainty