Standard deviation is calculated from mean $i.e$ arithmetic mean.

$\sqrt{\displaystyle\frac{72}{45}-\frac{144}{45\times 45}}\times 10$
$=12.36$

Assume $A=13;$ hence $d=A-13$

Use following formula:
$\sqrt{\displaystyle\frac{\displaystyle\sum d^2}{n}-\left[\displaystyle\frac{\displaystyle\sum d}{n}\right]^2}$
$\sqrt{\displaystyle\frac{72}{8}-\left[\displaystyle\frac{0}{8}\right]^2}$
$\sqrt{9}$
$=3$.

$^-{X}=\displaystyle\frac{1064}{76}=14$
$\sigma x=\sqrt{\displaystyle\frac{300}{76}}$
$=\sqrt{3.95}$
$=1.99$.

$\displaystyle\frac{2+5+7+8+13}{5}=7$
$\sqrt{\displaystyle\frac{4+25+49+64+169}{5}}-49=3.63$.

Here, $i=0.09=9\%$
$n=5$
$A _n=10,000$
Required present value $=\displaystyle\frac{A _n}{(1+i)^n}$
$=\displaystyle\frac{10,000}{(1+0.09)^5}$
$=Rs. 6499.42$.

$\sigma {x^2} = \frac{{\sum {{d^2}} }}{h}${here deviation are taken  from mean}

Given data values are $4,6,8,10$

The sum of the squared deviations from their their mean is the least value.

Given that variance of $5$ number is $10$

$Var(2X+3)=Var(2X)+Var(3)$

$Var(a+bX)=Var(a)+Var(bX)$

We know that variance is square of standard deviation.

$SD(X-Y)=SD(X)+SD(-Y)$

If each observation is multiplied by $2$, then mean$(\mu )$ will get doubled.

The correct formula for Chi square distribution is given in option D, where O means the observed number in the table and E means the corresponding expected number.

For a large sample, the sampling distribution of Chi square distribution may form a continuous curve. Chi square distribution helps in measuring how well the observed distribution of data fits with the distribution that is expected.

The correct formula for the estimation of Chi-Square test is mentioned in Option C, where O represents the frequency observed while E represents the frequency expected.

The mean of t distribution is always equal to 0 and also the variance is always greater than 1 and it is symmetrical like normal distribution.

Standard error can be calculated by applying two formulas mentioned in option A and option B that shows how a sample mean deviates from the mean of whole population.

Chi square distribution is a statistical test commonly used to compare observed data with data we would expect to obtain according to a specific hypothesis.

In the curve, with the large sample size, there are more samples with a mean around the middle and fewer with sample means at the extreme, the curve becomes more symmetrical. Larger samples tend to have lower standard errors

Both (a) and (b) statements regarding students t-distribution is correct i.e. its value varies from +infinity to-infinity and the variance of t distribution and the variance of normal distribution become closer and closer as the sample size increases and also the variance of t distribution is always greater than 1.

If the sample size of the population is small, then the curve would be skewed on one side and also as the sample size increases the sampling distribution of the mean gets narrower.

If the standard deviation is small, it signifies that data are closely distributed around the mean value instead of widely spread and here, we define a new variable as Students T-variable.

Standard deviation is the square root of the arithmetic mean of the squares of the deviations measured from the arithmetic mean of the data.

Coefficient of variation is the coefficient of dispersion based on the standard deviation of the statistical series. The coefficient of standard deviation is calculated by dividing the standard deviation of the series by its mean and then multiplying it by 100. It is regarded as the best measure of dispersion to compare two different series because it is expressed in percentage. 

Standard deviation is the square root of the arithmetic mean of the squares of the deviations measured from the arithmetic mean of the data. It is considered as the best and most commonly used measure of dispersion to ensure high degree of reliability as it is a measure of average of deviations from the average.

Standard deviation is the square root of the arithmetic mean of the squares of the deviations measured from the arithmetic mean of the data. So the deviations are affected by division and multiplication. Therefore, if each observation of the set id divided by 10 then the whole standard deviation also becomes 1/10 th of the prior standard deviation. 

A set of values from an observation is said to be relatively uniform if all the observations of the series have very little dispersion from each other. 

Variance is the mean of the squares of the deviations from the mean. Variance is not affected by the addition, subtraction, multiplication and division of the given value. Therefore, if each value of the series is multiplied by 15, the coefficient of variation will be unaltered.

Coefficient of variation is the coefficient of dispersion based on the standard deviation of the statistical series.

Coefficient of variation = ( standard deviation / mean )

=> 80 /100 = S.D / 40 

=> S.D = 32 

Variance is the mean of the squares of the deviations from the mean. Variance is not affected by the addition, subtraction, multiplication and division of the given value. Therefore, if each value of the series is multiplied by 15, the coefficient of variation will be unaltered.

Standard deviation is the square root of the arithmetic mean of the squares of the deviations measured from the arithmetic mean of the data. Standard deviation is not affected by the increase of decrease of observations in the series. 

Standard deviation is the square root of the arithmetic mean of the squares of the deviations measured from the arithmetic mean of the data.

Standard deviation = { (sum of the squares of the observations/ number of observations ) – (sum of observations/ number of observations ) } ^½

                               = { (1444) –(18.4) } ^½

                                = (1425.6) ^½

                                = 37.75 

Coefficient of variation is the coefficient of dispersion based on the standard deviation of the statistical series.

Coefficient of variation = ( standard deviation / mean )x 100 

                                     = ( 37.75/ 18.4)x100

                                     = 20.51 

In probability two events are said to be independent if the occurrence of one event is not affected by the occurrence of another event. Both the variable are completely independent of each other. These events are also known as mutually exclusive event as they have no common point. 

• Mean deviation is the arithmetic average of the deviations of the observed values from an  average of the observed values. Since Mean deviation do not include negative value of the deviations. Therefore, it does not follow the algebraic value of the deviations.
  
• Range is defined as the difference between the highest(or largest ) and lowest(or smallest) observed value in a series. It is the most simple and commonly understandable measures of dispersion. Since it is the most affected measures of dispersion by the extreme values of the series, it is regarded as the crudest measure. 
  
• Coefficient of variation is the coefficient of dispersion based on the standard deviation of the statistical series.Therefore, Coefficient of variation is a unit less or relative measure of dispersion as variation is the absolute measure of dispersion. 
  

Standard deviation is the square root of the arithmetic mean of the squares of the deviations measured from the arithmetic mean of the data.

Variance is the mean of the squares of the deviations from the mean. 

Standard deviation is the square root of variance or variance is the square of standard deviation. 

S.D = {Var(X)}^1/2 

or 

Var(X) = (S.D)²

Standard deviation is the square root of the arithmetic mean of the squares of the deviations measured from the arithmetic mean of the data. If standard deviation of a certain series is zero than it denotes that all the values of that series is equal to the mean of the series which made all the deviations zero and therefore standard deviation also zero. 

$\sigma^2=\dfrac{1}{n}\sum _{i=1}^{n}i^2-\left(\dfrac{1}{n}\sum _{i=1}^{n}i\right)^2$