Answer to Question #227000 in Statistics and Probability for Viraj Rambhia

Question

Calculate the Mean and Standard Deviation using Total MSMEs, Total Manufacturing MSMEs, and Total Service MSMEs. Write your comment if mean is affected by extreme observation.

Total MSMEs Total Manufacturing MSMEs Total Service MSMEs

755 246 509

2735 851 1884

36 17 19

1229 444 785

1581 668 913

578 229 349

1731 717 1014

409 181 228

4951 2848 2103

1014 563 451

1795 413 1382

148 42 106

Answer 1

If we have a dataset of size n given by $x_1, x_2,..., x_n,$ then,

$mean= \bar{x}=\frac{x_1+x_2+...+x_n}{n}.....(1)\\ sd= [\frac{(x_1-\bar{x})^2+(x_2-\bar{x})^2+...+(x_n-\bar{x})^2}{n-1}]^{\frac{1}{2}}......(2)$

For all the datasets, we calculate seperately as follows

2. Total Manufacturing MSMEs:

Here, $n = 12, x_1 = 246, x_2 = 851, ..., x_{12} = 42$

Thus, using (1), we get : 

mean of total manufacturing MSMEs.

$= \bar{x}_m \\ = 601.5833$  

Similarly, using (2), we get :

sd of total manufacturing MSMEs

$= sd_m\\ = 755.9.$

Theoretically, the mean is highly affected by extreme observations. For the given datasets, note that the 3rd and 9th data points can be considered as extreme observations, as they differ significantly from the rest of the observations.

Therefore, we discard the values 36, 4951 from the first dataset, the pair 17, 2848 from the second, and 19, 2103 from the third and calculate the new means using formula (1) with n = 10.

If we denote the new means for each of the datasets using ', we have:

$\bar{x}'_T=1197.5\\ \bar{x}'_M=435.4\\ \bar{x}'_S=762.1$

It is clear that means are highly affected by extreme observations.