How to Solve: Statistics(Range, Weighted Mean, Variance, SD)

Theory

Range

• Range of a set is the difference between the highest and lowest value of the set.
Example: Suppose the set is {-1,2,3,6,8} then the range will be
8 -(-1) = 9

Properties of Range

1. If all the numbers in the set are increased/decreased by the same number(k) then the range DOES NOT CHANGE!
Suppose the set is {a,b,c} (in increasing order)
Range = c-a
Now, lets increase all the numbers by k then the set will become {a+k, b+k, c+k}
New range = c+k -(a+k) = c-a = Old range

2. If all the numbers in the set are multiplied/divided by the same number(k) then the range also gets multiplied/divided by the same number(k)
Proof similar to that for mean.


Weighted Average

• Weighted Average = \(\frac{((Weight1∗Value1) + (Weight2∗Value2)…+ (WeightN∗ValueN))}{(Weight1 + Weight2 + ... WeightN)}\)

Q1. If an employee’s performance consists of 20% of component A, 30% of component B and 50% of component C and if he receives 10 in A, 20 in B and 10 in C, then find the overall performance of the employee

Ans 13 (Check Video for solution)

Variance

• Variance, V = Mean of (Square of difference of each number from the mean)

V = \(\frac{Sum of (Squares Of Difference Of Each Number From Mean) }{Total Number Of Numbers }\)

Q1 Find the Variance of the set { 1, 2, 3, 4, 5 }

Sol: Mean of this set is 3
Variance, V = \(\frac{((3-1)^2 + (3-2)^2 + (3-3)^2 + (3-4)^2 + (3-5)^2)}{ 5 }\)
= \(\frac{(4+1+0+1+4)}{5}\) = 2

Properties of Variance

1. If all the numbers in a set are increased/ decreased by the same number(k) then the variance DOES NOT change
Check Video For Explanation

2. If all the numbers in a set are multiplied/ divided by the same number(k) then the variance gets multiplied/divided by the square of the number (k2)
Check Video For Explanation


Standard Deviation(SD)

• SD is an indication of how spread the numbers are as compared to the Mean

• SD is equal to the Root Mean Square(RMS) of the distance of the values from the mean

• Standard Deviation =\( \sqrt{Variance}\), SD = \(\sqrt{V}\)

Q1 Find the SD of the set { 1, 2, 3, 4, 5 }

Sol: V = 2 (calculated above)
SD = \(\sqrt{V}\) = \(\sqrt{2}\)

Properties of SD

1. If all the numbers in the set are increased/decreased by the same number(k) then the Standard Deviation DOES NOT CHANGE!
(This happens because the mean also gets increased/decreased by the same number and the Variance or Standard Deviation are calculated by subtracting all the numbers by the mean and taking square of them and taking their average. )

2. If all the numbers in the set are multiplied/divided by the same number(k) then the Standard Deviation also gets multiplied by the same number.

Zero SD

• SD of a 1 element set
Check Video For Explanation

• SD of a set with all numbers equal
Check Video For Explanation

Recap of Properties



Hope it helps!