## Statistics

# Variance and Standard Deviation

- Standard deviation (σ) = \tt \sqrt{variance}
- For the Individual series, the standard deviation is \tt \sigma = \sqrt{\frac{\sum (x_i - M)^{2}}{n}}

(general method) (Actual mean type) - For the individual series, the standard deviation is \tt \sigma = \sqrt{\frac{\sum x_i^{2}}{n} - \left(\frac{\sum x_i}{n}\right)^{2}}

(Short cut method - Actual mean type) - For the individual series, the standard deviation is \tt \sigma = \sqrt{\frac{\sum {d_i}^{2}}{n} - \left(\frac{\sum d_i}{n}\right)^{2}}

where d_{i}= x_{i}− A (Assumed mean type) - For the individual series, the standard deviation is \tt \sigma = h \sqrt{\frac{\sum {d_i}^{2}}{n} - \left(\frac{\sum d_i}{n}\right)^{2}}

where \tt d_i = \frac{x_i - A}{h} (Assumed mean type) - For grouped data, the standard deviation is \tt \sigma = \sqrt{\frac{\sum f_i \ {x_i}^{2}}{N} - \left(\frac{\sum f_i \ x_i}{N}\right)^{2}}

where N = Σ fi (Actual mean type) - For the grouped data, the standard deviation is \tt \sigma = \sqrt{\frac{\sum f_i \ {d_i}^{2}}{N} - \left(\frac{\sum f_i \ d_i}{N}\right)^{2}}

where d_{i}= x_{i}− A (Assumed mean type/short cut method) - For the grouped data, the standard deviation is \tt \sigma = h \sqrt{\frac{\sum f_i \ {d_i}^{2}}{N} - \left(\frac{\sum f_i \ d_i}{N}\right)^{2}}

where \tt d_i = \frac{x_i - A}{h} (Assumed mean type/short cut method) - For Symmetrical distribution

Mean deviation = \tt \frac{4}{5} (standard deviation) - Coefficient of standard deviation = \frac{\sigma}{M}

where σ = S.D

M = Mean

### View the Topic in this video From 11:37 To 54:20

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1. Variance and standard deviation for ungrouped data

\sigma^{2}=\frac{1}{n}\sum (x_{i}-\overline{x})^2, \ \ \ \sigma = \sqrt{\frac{1}{n} \sum (x_{i}-\overline{x})^2}

2. Variance and standard deviation of a discrete frequency distribution

\sigma^{2}=\frac{1}{N}\sum f_i(x_{i}-\overline{x})^2, \ \ \ \sigma = \sqrt{\frac{1}{N}\sum f_i(x_{i}-\overline{x})^2}

3. Variance and standard deviation of a continuous frequency distribution

\sigma^{2}=\frac{1}{N}\sum f_i(x_{i}-\overline{x})^2, \ \ \ \sigma = \frac{1}{N}\sqrt{N\sum f_i x_i^2-\left(\sum f_{i}x_{i}\right)^2}

4. Shortcut method to find variance and standard deviation

\sigma^{2}=\frac{h^{2}}{N^{2}}\left[N\sum f_i y_i^2-\left(\sum f_{i}y_{i}\right)^2\right], \ \ \ \sigma = \frac{h}{N}\sqrt{N\sum f_i y_i^2-\left(\sum f_{i}y_{i}\right)^2}

where y_{i} = \frac{x_{i} - A}{h}