Variance, Standard Deviation, and Measures of Dispersion

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

Standard deviation is defined mathematically as the square root of the variance, which is the mean of the squared deviations from the arithmetic mean.

Standard deviation can be calculated using the actual mean method, the assumed mean method, or the step-deviation method.

Standard deviation is independent of the change of origin (adding or subtracting a constant from all values does not change the spread).

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

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

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

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

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

$\sigma {x^2} = \frac{{\sum {{d^2}} }}{h}${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 formula for correlation coefficient r is r = Cov(x,y) / (SDx * SDy). Plugging in the values: 0.8 = 20 / (4 * SDy). Thus, 0.8 = 5 / SDy, so SDy = 5 / 0.8 = 6.25.

For a normal distribution, the relationship between standard deviation (SD) and mean deviation (MD) is approximately SD = 1.25 * MD. Here, 1.25 * 16 = 20.

Variance is the square of the standard deviation. Therefore, 3.24^2 = 10.4976, which rounds to 10.49.

Standard deviation is independent of the change of origin. Adding a constant to all items does not change the spread of the data.

Standard deviation = sqrt(Sum of squared deviations / N) = sqrt(1444 / 100) = sqrt(14.44) = 3.8. Coefficient of Variation (CV) = (SD / Mean) * 100 = (3.8 / 18.4) * 100 = 20.65.

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 

• 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

Given N=100, old mean=40, old SD=5.1. The sum of observations is S_old = 100 * 40 = 4000. Correct sum = 4000 - 50 + 40 = 3990, so correct mean = 3990 / 100 = 39.9. For the standard deviation, old sum of squares (SS) can be found using SD^2 = (SS/N) - mean^2, yielding SS_old = 100 * (5.1^2 + 40^2) = 162601. Correct SS = 162601 - 50^2 + 40^2 = 161701. Correct variance = (161701 / 100) - (39.9^2) = 1617.01 - 1592.01 = 25, giving correct SD = sqrt(25) = 5.

The formula for covariance is Cov(x,y) = r * SD(x) * SD(y). Plugging in the values: 35 = 0.7 * 5 * SD(y). Thus, 35 = 3.5 * SD(y), which means SD(y) = 35 / 3.5 = 10.

Standard deviation is a measure of dispersion. Adding a constant to every observation shifts the data but does not change the spread or distance between points, so the standard deviation remains unchanged.

The formula relating covariance, correlation coefficient (r), and standard deviations is Cov(X,Y) = r * SD(X) * SD(Y). Substituting the given values: 32 = 0.8 * SD(X) * 4. This simplifies to 32 = 3.2 * SD(X), which gives SD(X) = 32 / 3.2 = 10.

Coefficient of Variation (CV) = (SD / Mean) * 100. Given CV = 22.5% and Mean = 7.5, then 22.5 = (SD / 7.5) * 100. SD = (22.5 * 7.5) / 100 = 168.75 / 100 = 1.6875.

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. 

Assuming a normal distribution, 10% of students scored more than 137. This corresponds to the top 10% tail, where the z-score for 0.90 is approximately 1.28. Using the formula z = (x - mean) / SD, we get 1.28 = (137 - 134) / SD, so 1.28 = 3 / SD, SD = 3 / 1.28 = 2.34375. Option B is closest.

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

Data: 3, 5, 12, 3, 2. Mean = (3+5+12+3+2)/5 = 25/5 = 5. Variance = [(3-5)^2 + (5-5)^2 + (12-5)^2 + (3-5)^2 + (2-5)^2] / 5 = [4 + 0 + 49 + 4 + 9] / 5 = 66 / 5 = 13.2. SD = sqrt(13.2) = 3.63. If using sample SD (divide by n-1=4), Variance = 66/4 = 16.5, SD = sqrt(16.5) = 4.062.