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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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