Maths Statistics

Variance = $\frac{\sum {\left(x_i-\overline{x}\right)}^2}{n}=\frac{\sum \left({x_i}^2+{\overline{x}}^2-2\overline{x}xi\right)}{n}$  

$=\frac{\sum {x_i}^2+{\overline{x}}^2\sum 1-2\overline{x}\sum x_i}{n}$

$=\frac{\frac{n\left(n+1\right)\left(2n+1\right)}{6}+{\left(\frac{n\left(n+1\right)}{2n}\right)}^2.n-\frac{2\left(n+1\right)}{2n}.\frac{n\left(n+1\right)}{2}}{n}$ $=\frac{n^2-1}{12}$

Now, $\frac{n^2-1}{12}=14son=13$

$\sum X=200$

$\sum x^2=2125$

For new data

$\sum x=200-25+35=210$

$\overline{x}=10.5$

$\sum x^2=2725$

${\sigma }^2=\frac{2725}{20}-{\left(10.5\right)}^2=26$

$mean=\frac{n\left(n+1\right)/2}{n}$           

  $=\frac{n+1}{2}$            For n = 2k – 1, k   $\in N$

mean = k

mean deviation = avg. of deviations

$=\frac{2\left(\stackrel{0+}{1}+2+...\left(k-1\right)\right)}{2k-1}=\frac{\left(k-1\right)k}{2k-1}$

$=\frac{5\left(2k\right)}{k-1}\Rightarrow k=11\Rightarrow n=2k-1=21$            

               

Z distribution and Chi-Squared are some of the most popular distribution patterns of probability, and it is vital to recognise the variations between them and when to use the distribution pattern. A Z table is of no use when the operation revolves around a smaller sample size. On the other hand, the

Five types of sample statistics include sample mean, sample variance, sample standard deviation, sample proportion.

1. Sample mean is the average of all data points in a sample. It is calculated by summing all values in sample and then dividing by number of observations.
2. The sample variance measures the

Median = $\frac{2k+12}{2}=k+6$

Mean deviation = $\sum \frac{\left|x_i-M\right|}{n}=6$

$\Rightarrow \frac{\left(k+3\right)+\left(k+1\right)+\left(k-1\right)+\left(6-k\right)+\left(6-k\right)+\left(10-k\right)+\left(15-k\right)+\left(18-k\right)}{8}$

$\therefore \frac{58-2k}{8}=6$

k = 5

$Median=\frac{2\times 5+12}{2}=11$

Arithmetic mean is the measure of central tendency that is most affected by extreme items or outliers in the dataset. The reason behind this is since mean is calculated by summing up all the values and after that, it is divided by the number of values. Extreme values may disproprotionately impact th

Median is considered to be the most suitable average for qualitative measurement. It divides an entire frequency distribution into two haves. This is especially useful for ordinal data where values represent categories with meaningful order. However, it is not necessarily a linear scale. The median

$\overline{x}=\frac{{\displaystyle \sum _{i=1}^{10}{x}_i}}{10}=15\frac{{\displaystyle \sum _{i=1}^{10}{x_i}^2}}{10}-{\left(\overline{x}\right)}^2=15$

$\Rightarrow \Sigma x_i=150\Sigma {x_i}^2=2400$

Actual mean $\overline{x}=\frac{\Sigma x_i+15-25}{10}=\frac{140}{10}=14$

Actual variance = $\frac{\Sigma {x_i}^2+15^2-25^2}{10}-{\left(14\right)}^2$

$=\frac{2400-400}{10}-196$

$\begin{array}{l}{\sigma }^2=4\\ \sigma =2\end{array}$