Name: Variance and Standard Deviation for Ungrouped Distribution
Uploaded: 2019-07-19
Duration: 6 min 21 s
Description: In this video, let us learn to find the variance and standard deviation of ungrouped data. We will also learn the formula to find the discrete frequency dist...

Question

The ages, $a$ years, of the five members of the boy-band AlphaArise are such that $Σ {(a - 21)}^{2} = 11.46$ and $Σ (a - 21) = - 6 .$
The ages, $b$ years, of the seven members of the boy-band BetaBeat are such that $Σ {(b - 18)}^{2} = 10.12$ and $Σ (b - 18) = 0 .$

Find the variance of the ages of the $12$ members of these two bands.

Accepted Answer

Given data:

The ages, $a$ years, of the five members of the boy-band AlphaArise are such that $Σ {(a - 21)}^{2} = 11.46$ and $Σ (a - 21) = - 6 .$
The ages, $b$ years, of the seven members of the boy-band BetaBeat are such that $Σ {(b - 18)}^{2} = 10.12$ and $Σ (b - 18) = 0 .$

Use the properties of mean, $Mean (x + k) = Mean (x) + k$ , i.e., $\frac{Σ (x + k)}{n} = \frac{Σ (x)}{n} + k$ and where $k$ is constant and $Var (x + k) = Var (x)$ and formula of variance is $Var (x) = \frac{\sum_{} x^{2}}{n} - {(\frac{\sum_{} x}{n})}^{2}$ .

Thus, $\bar{a} = (\frac{- 6}{5}) + 21 = 19.8, \bar{b} = (\frac{0}{7}) + 18 = 18$ .

Here, $Var (a) = Var (a - 21)$ , so $\frac{Σ a^{2}}{5} - 19 . 8^{2} = \frac{11.46}{5} - {(\frac{- 6}{5})}^{2} = 0.852$

$\Rightarrow Σ a^{2} = 5 \times (0.852 + 19 . 8^{2}) = 1964.46$

Here,

$Σ b = 7 \times 18 = 126$

$Var (b) = Var (b - 18)$

So, $\frac{Σ b^{2}}{7} - 18^{2} = \frac{10.12}{7} - {(\frac{0}{7})}^{2} = 1.4457$

$Σ b^{2} = 7 \times (1.4457 + 18^{2}) = 2278.12$

Formula of combined variance is $Var (x and y) = \frac{(\sum_{} x^{2} + \sum_{} y^{2})}{m + n} - {(\frac{\sum_{} x + \sum_{} y}{m + n})}^{2}$

Thus, $Var (a and b) = \frac{1964.46 + 2278.12}{12} - {(\frac{99 + 126}{12})}^{2} = 1.99 {years}^{2}$

Important Questions on Measures of Variation

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The ages, a years, of the five members of the boy-band AlphaArise are such that Σa-212=11.46 and Σa-21=-6. The ages, b years, of the seven members of the boy-band BetaBeat are such that Σb-182=10.12 and Σb-18=0. Find the variance of the ages of the 12 members of these two bands.

Important Questions on Measures of Variation

Learn and Prepare for any exam you want !