## Statistics

# Analysis of Frequency Distributions

- Coefficient of Variance = \frac{\sigma}{M} \times 100

where σ = S.D

M = Mean - Variance of the combined series \sigma^{2} = \frac{n_{1}(\sigma_{1}^{2} + d_{1}^{2}) + n_{2}(\sigma_{2}^{2} + d_{2}^{2})}{n_{1} + n_{2}}

where n_{1}and n_{2}are the sizes

σ_{1}and σ_{2}are the standard deviations

d_{1} = \overline{x} - \overline{x_{1}}, d_{2} = \overline{x} - \overline{x_{2}}

\tt \overline{x} = combined \ mean - The variance of the combined series \tt \sigma^{2} = \frac{n_{1}\sigma_{1}^{2} + n_{2}\sigma_{2}^{2}}{n_{1} + n_{2}} + \frac{n_{1}n_{2}(\overline{x_{1}} - \overline{x_{2}})^{2}}{(n_{1} + n_{2})^{2}}

where n_{1}, n_{2 }are the sizes. σ_{1}, σ_{2}are the standard deviations - Variance of (X + a) = variance of X

where a is constant

Variance of (aX) = a^{2}. variance (X) - Var(aX + b) = a
^{2}Var (X) - For the series a, a + d, a + 2d, ...., a + (n − 1)d

Arithmetic Mean = a + \frac{(n - 1)d}{2} - For the series a, a + b, a + 2d, ...., a + (n − 1)d

Variance = \frac{(n^{2} - 1)}{12} \cdot d^{2} - For the series a, a + d, a + 2d, ....... a + (n - 1)d

Standard deviation = \sqrt{\frac{n^{2} - 1}{12}} \cdot d - Standard deviation of first n natural numbers is = \sqrt{\frac{n^{2} - 1}{12}}
- Variance of first n natural numbers is = \frac{n^{2} - 1}{12}
- Variance and standard deviation is independent change of origin
- For a, a + d, a + 2d, ....., a + 2nd, the Mean deviation from mean = \frac{n(n + 1)d}{2n + 1}
- For a, a + d, a + 2d, ......, a + 2nd, the standard deviation is = \sqrt{\frac{n(n + 1)}{3}} d
- In Statistical data, the sum of deviations of individual values from AM is always zero i.e. \tt \sum_{i = 1}^{n} f_i (x_i -\overline{x}) = 0
- In a statistical data, the sum of the squares of the deviations of individual values from AM is least i.e. \tt \sum_{i = 1}^{n} f_i (x_i -\overline{x})^{2} is least.
- C
_{0}+ C_{1}+ C_{2}+ ...... C_{n}= 2^{n}

where C_{0}, C_{1}, C_{2}, ...... C_{n}are binomial coefficients - If a variable takes the values 0, 1, 2, ........ n with frequencies proportional to binomial coefficients C
_{0}, C_{1}, C_{2}, ...... C_{n}, the mean of distribution is \tt = \frac{n}{2} - If a variable takes values 0, 1, 2, 3, ...... n with frequencies \tt q^{n}, \frac{n}{1} q^{n-1}p, \frac{n(n - 1)}{1.2} q^{n-2}p^{2}, ....... P^{n}
- where p + q = 1, then the mean is np

\tt S.D (x) = \sigma \Rightarrow S.D \left(\frac{ax + b}{c}\right) = |\frac{a}{c}| \sigma - If a variable x takes values 0, 1, 2, ...... n with frequencies proportional to
^{n}c_{0},^{n}c_{1}, ......^{n}c_{n}then the variance of x is \tt \frac{n}{4} - Sum of first n natural numbers 1 + 2 + 3 + ...... + n = \tt \frac{n(n + 1)}{2}
- Sum of the squares of first n natural numbers \tt 1^{2} + 2^{2} + 3^{2} + ..... + n^{2} = \frac{n(n + 1)(2n + 1)}{6}
- The A.M of the series
^{n}c_{0},^{n}c_{1},^{n}c_{2}, ......^{n}c_{n}= \tt = \frac{2^{n}}{n + 1} - In a series of 2n observations half of them equal to a and remaining half equal to −a . If the standard deviation of observations is λ. Then |a| = λ.

### View the Topic in this video From 00:30 To 54:20

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1. Coefficient of variation (C.V.) = \frac{\sigma}{\overline{x}}\times 100,\overline{x}\neq 0.

For series with equal means, the series with lesser standard deviation is more consistent or less scattered.