How to Solve: Statistics(Range, Weighted Mean, Variance, SD)
TheoryRange• 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 Range1. 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 Variance1. 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 SD1. 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 PropertiesHope it helps!