Data $5,7,9,11,13$ has mean $\bar x=9$. Deviations: $-4,-2,0,2,4$; squares $16,4,0,4,16$ summing to $40$. …

Mean $=\dfrac{65+70+75}{3}=70$. Deviations $-5,0,5$; squares $25,0,25$; sum $=50$. …

Combined mean $=\dfrac{5\cdot2+5\cdot4}{10}=3$. Deviations of group means: $d_1=2-3=-1$, $d_2=4-3=1$. Combined variance: …

[!TLDR]

Compute the mean, sum the squared deviations, divide by $n$, and take the square root to get $\sqrt{66}/2$.

Concept

For a data set the standard deviation is $\sigma = \sqrt{\dfrac{1}{n}\sum (x_i - \bar{x})^2}$, the root-mean-square deviation from the mean (standard NCERT/CBSE statistics).

Solution

The numbers are $-3, 0, 3, 8$ with $n = 4$.

$\bar{x} = \frac{-3+0+3+8}{4} = \frac{8}{4} = 2.$ …

A variance of zero means all observations are equal, and equal to the mean $\bar x=5$. Thus each $x_i=5$, so $x_i^2=25$. Since …

Concept and Intuition

Coefficient of variation CV = 100 * standard deviation / mean. With variances 25 and 36 the standard deviations are 5 and 6; invert the CV relation to get each mean.

Step-by-Step Solution

1. Dataset 1: sigma1 = sqrt(25) = 5, CV = 50 => mean1 = 100*5/50 = 10.
2. Dataset 2: sigma2 = sqrt(36) = 6, CV = 75 => mean2 = 100*6/75 = 8. …

Concept and Intuition

The coefficient of variation relates the spread to the mean: $\text{C.V.}=\frac{\sigma}{\bar{x}}\times 100$.

Step-by-Step Solution

1. Given $\text{C.V.}=75$ and $\bar{x}=44$.
2. Rearrange: $\sigma=\frac{\text{C.V.}\times \bar{x}}{100}$.
3. $\sigma=\frac{75\times 44}{100}=\frac{3300}{100}=33$. …