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.

InternalAssignmentApplicableforSeptember2021Examination

State District Total Total  Total

MSMEs Manufacturing Service

MSMEs MSMEs

HIMACHAL PRADESH Chamba 755 246 509

HIMACHAL PRADESH Kangra 2735 851 1884

HIMACHAL PRADESH Lahul & Spiti 36 17 19

HIMACHAL PRASEDH Kullu 1229 444 785

HIMACHAL PRADESH Mandi 1581 668 913

HIMACHAL PRADESH Hamirpur 578 229 349

HIMACHAL PRADESH Una 1731 717 1014

HIMACHAL PRADESH Bilaspur 409 181 228

HIMACHAL PRADESH Solan 4951 2848 2103

HIMACHAL PRADESH Sirmaur 1014 561 451

HIMACHAL PRADESH Shimla 1795 413 1382

HIMACHAL PRADESH Kinnaur 148 42 106

Source: Data.Gov.in(MSMEsasper2011)

Answer 1

Let random variables $X, Y, Z$ denote Total MSMEs, Total Manufacturing MSMEs, Total Service MSMEs respectively.

$E(X)=(755+2735+36+1229+1581+578+1731+409+4951+1014+1795+148)/12=1414$

$E(X^2) = (755^2+2735^2+36^2+1229^2+1581^2+578^2+1731^2+409^2+4951^2+1014^2+1795^2+148^2)/12=3695316.67$

$Var(X)=E(X^2 )-E(X)^2=3695316.67-1414^2=1697334.42$

$\sigma(X)=\sqrt{Var(X)}=\sqrt{1697334.42}=1302.82$

One can note that mean is affected by extreme observation: $|X(Solan)-E(X)|/\sigma(X) >2.7$ . Without Solan the mean value will be $(1414\cdot 12-4951)/11=1092.45$ .

 

$E(Y)=(246+851+17+444+668+229+717+181+2848+563+413+42)/12=601.58$

$E(Y^2)=(246^2+851^2+17^2+444^2+668^2+229^2+717^2+181^2+2848^2+563^2+413^2+42^2)/12=885671.92$

$Var(Y)=E(Y^2 )-E(Y)^2=885671.92-601.58^2=523769.410$

$\sigma(Y)=\sqrt{Var(Y)}=\sqrt{885671.92}=941.1$

One can note that mean is affected by extreme observation $|Y(Solan)-E(Y)|/\sigma(Y) >2.387$ . Without Solan the mean value will be $(601.58\cdot 12-2848)/11=397.36$ .

 

$E(Z)=(509+1884+19+785+913+349+1014+228+2103+451+1382+106)/12=811.92$

$E(Z^2)=(509^2+1884^2+19^2+785^2+913^2+349^2+1014^2+228^2+2103^2+451^2+1382^2+106^2)/12=1083986.92$

$Var(Z)=E(Z^2 )-E(Z)^2=1083986.92-811.92^2=424778.24$

$\sigma(Y)=\sqrt{Var(Y)}=\sqrt{424778.24}=651.75$

One can note that mean is affected by extreme observation $|Z(Solan)-Z(Y)|/σ(Z) >1.98$ . Without Solan the mean value will be $(811.92\cdot 12-2848)/11=694.55$ .