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Deviations from the mean:
Let \(x_{1}, x_{2}, x_{3}, … , x_{n}\) be the given data for \(n\) observations.
The mean of the given observations is given by \(\overline{x}\).
Then the deviations from the mean is given by, \(\left(x_{1} - \overline{x}\right)\), \(\left(x_{2} - \overline{x}\right)\), \(\left(x_{3} - \overline{x}\right)\), … , \(\left(x_{n} - \overline{x}\right)\).
Squares of deviation from the mean:
Let \(x_{1}, x_{2}, x_{3}, … , x_{n}\) be the given data for \(n\) observations.
The mean of the given observations is given by \(\overline{x}\).
Then the squares of deviation from the mean is given by, \(\left(x_{1} - \overline{x}\right)^2\), \(\left(x_{2} - \overline{x}\right)^2\), \(\left(x_{3} - \overline{x}\right)^2\), … , \(\left(x_{n} - \overline{x}\right)^2\) or \(\sum_{i = 1}^{n}\left(x_{i} - \overline{x}\right)^2\)
Variance:
The mean of squares of the deviation from the mean is called variance. It is denoted by the symbol \(\sigma^{2}\).
Let \(x_{1}, x_{2}, x_{3}, … , x_{n}\) be the given data for \(n\) observations.
The mean of the given observations is given by \(\overline{x}\).
Then, Variance \(\sigma^{2} =\) Mean of the squares of deviation.
\(\sigma^{2} = \frac{\left(x_{1} - \overline{x}\right)^2 + \left(x_{2} - \overline{x}\right)^2 + \left(x_{3} - \overline{x}\right)^2 + … + \left(x_{n} - \overline{x}\right)^2}{n}\)
\(=\) \(\frac{\sum_{i = 1}^{n}\left(x_{i} - \overline{x}\right)^2}{n}\)
Standard deviation:
The positive square root of variance is called standard deviation. It is denoted by the symbol \(\sigma\).
Standard Deviation, \(\sigma\) \(=\) \(\sqrt{\text{Variance}}\)
\(\sigma\) \(=\) \(\sqrt{\frac{\sum_{i = 1}^{n}\left(x_{i} - \overline{x}\right)^2}{n}}\)
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