# How to calculate these step by step ?

May 29, 2018

mean is $19$
and the variance is $5.29 \cdot 9 = 47.61$

#### Explanation:

Since all the marks are multiplied by 3 and added by 7, the mean should be $4 \cdot 3 + 7 = 19$
The standard deviation is a measure of average squared difference from the mean and doesn't change when you add the same amount to each mark, it only changes when multiply all the marks by 3
Thus,
$\setminus \sigma = 2.3 \cdot 3 = 6.9$
Variance = $\setminus {\sigma}^{2} = {6.9}^{2} = 47.61$

Let n be the number of numbers where $\left\{n | n \setminus \in \setminus m a t h \boldsymbol{{Z}_{+}}\right\}$
in this case n= 5

Let $\setminus \mu$ be the mean $\setminus \textrm{v a r}$ be the variance and, let $\sigma$ be the standard deviation

Proof of mean: $\setminus {\mu}_{0} = \setminus \frac{\setminus {\sum}_{i}^{n} {x}_{i}}{n} = 4$
$\setminus {\sum}_{i}^{n} {x}_{i} = 4 n$
$\setminus \mu = \setminus \frac{\setminus {\sum}_{i}^{n} \left(3 {x}_{i} + 7\right)}{n}$

Applying the commutative property:

$= \setminus \frac{3 \setminus {\sum}_{i}^{n} {x}_{i} + \setminus {\sum}_{i}^{n} 7}{n} = \setminus \frac{3 \setminus {\sum}_{i}^{n} {x}_{i} + 7 n}{n}$

$= 3 \setminus \frac{\setminus {\sum}_{i}^{n} {x}_{i}}{n} + 7 = 3 \cdot 4 + 7 = 19$

Proof for standard deviation:
$\setminus {\textrm{v a r}}_{0} = \setminus {\sigma}^{2} = {2.3}^{2} = 5.29$

$\setminus {\textrm{v a r}}_{0} = \setminus \frac{\setminus {\sum}_{i}^{n} {\left({x}_{i} - \setminus {\mu}_{0}\right)}^{2}}{n} = \setminus \frac{\setminus {\sum}_{i}^{n} {\left({x}_{i} - 4\right)}^{2}}{n} = 5.29$

$\setminus \textrm{v a r} = \setminus \frac{\setminus {\sum}_{i}^{n} {\left(3 {x}_{i} + 7 - 19\right)}^{2}}{n} = \setminus \frac{\setminus {\sum}_{i}^{n} {\left(3 {x}_{i} - 12\right)}^{2}}{n}$

$= \setminus \frac{\setminus {\sum}_{i}^{n} {\left(3 \left({x}_{i} - 4\right)\right)}^{2}}{n} = \setminus \frac{\setminus {\sum}_{i}^{n} 9 {\left({x}_{i} - 4\right)}^{2}}{n} = 9 \setminus \frac{\setminus {\sum}_{i}^{n} {\left({x}_{i} - 4\right)}^{2}}{n}$

$\setminus \textrm{v a r} = 9 \cdot 5.29 = 47.61$