Some Important Point Regarding Statistics - Practice Questions & MCQ

Some Important Points Regarding Statistics

1. The sum of the deviation of an observation from their mean is equal to zero. i.e. $\sum_{i=1}^n\left(x_i-\bar{x}\right)=0$
   2. The sum of the square of the deviation from the mean is minimum, i.e. $\sum_{i=1}^n\left(x_\bar {x}\right)^2$ is least.
   3. The mean is affected accordingly if the observations are given a mathematical treatment i.e. addition, subtraction multiplication by a constant term.
   4. If set of $\mathrm{n}_1$ observations has mean $\bar{x}_1$ and a set of $\mathrm{n}_2$ observations has mean $\bar{x}_2$, then their combined mean is $\frac{n_1 \bar{x}_1+n_2 \bar{x}_2}{n_1+n_2}$
   5. If set of $\mathrm{n}_1$ observations has mean $\bar{x}_1$ and set of $\mathrm{n}_2$ observations has mean $\bar{x}_2$ then their combined variance is given by

   $
   \sigma^2=\frac{n_1\left(\sigma_1^2+d_1^2\right)+n_2\left(\sigma_2^2+d_2^2\right)}{n_1+n_2}
   $

   where, $d_1=\bar{x}_1-\bar{x}$ and $d_2=\bar{x}_2-\bar{x}$

   $
   \begin{aligned}
   & \sigma_1^2=\frac{1}{n_1} \sum_{i=1}^{n_1}\left(x_{1 i}-\overline{x_1}\right)^2 \\
   & \sigma_2^2=\frac{1}{n_2} \sum_{j=1}^{n_2}\left(x_{2 j}-\overline{x_2}\right)^2
   \end{aligned}
   $

   $\bar{x}_1, \bar{x}_2$ are the means and $\sigma_1, \sigma_2$ are the standard deviations of two series.
   6. Z-Score Formula: It is a method to compare test results from a normal distribution.

   For a random variable (x) from a normal distribution with mean $(\mu)$ and standard deviation $(\sigma)$, the Z -score can be determined by subtracting the mean from x and then dividing it by the standard deviation.

   $
   \mathrm{z}=\frac{(\mathrm{x}-\mu)}{\sigma}
   $

Where x is a test value
$\mu$ is the mean and
$\sigma$ is the standard deviation (SD).
For the average of a sample of size $(\mathrm{n})$ from a population, the standard deviation is $\sigma$ and the mean is $\mu$.

Relation between Mean, Mode and Median 

If the distribution is symmetric, the values of mean, mode and median coincide. In symmetric distribution, frequencies are symmetrically distributed on both sides of the centre point of the frequency curve.

A distribution which is not symmetric is called a skewed distribution.

The empirical relation between mean, mode and median in such distributions is: $
\text { Mode }=3 \text { Median }-2 \text { Mean. }
$