How to Solve: Statistics

[brushmyquant]

How to Solve: Statistics

Hi All,

I have recently uploaded a video on YouTube to discuss Statistics Basics in Detail:

https://www.youtube.com/watch?v=Cvy_HHw4KIs

Following is covered in the video

• Mean and Properties of Mean
• Median and Properties of Median
• Range and Properties of Range
• Mode
• Weighted Mean
• Variance and Properties of Variance
• Standard Deviation and Properties of SD

Theory

Mean /Average / Arithmetic Mean

• Mean is the average of the all the numbers in the set.
• Mean = Sum Of All The Numbers In The Set / Total Number Of Numbers In The Set

Suppose the set is {1,2,3,4,5}
Then, Mean = (1+2+3+4+5)/5 = 15 / 5 = 3

Properties of Mean

1. If all the numbers in the set are increased/decreased by the same number(k) then the mean also gets increased/decreased by the same number(k)

Suppose the set is {a,b,c,d,e}
then the Mean = (a+b+c+d+e)/5
Now, lets increase all the numbers by k. So, the new set is {a+k,b+k,c+k,d+k,e+k)
New Mean = (a+k +b+k +c+k +d+k + e+k)/5
= (a+b+c+d+e + 5k)/5 = (a+b+c+d+e)/5 + k = Old Mean + k

2. If all the numbers in the set are multiplied/divided by the same number(k) then the mean also gets multiplied/divided by the same number(k)
Proof same as above. In this case if we multiple all the numbers by k then
New Mean = k* (Old Mean)

SUGGESTION: Don't try remembering the points 1 and 2 above. It does not take much time to calculate them!


Median

• Median is the middle value of the set.
• In case of even number of numbers in the set: Median is the mean of the two middle numbers (after the numbers are arranged in the increasing / decreasing order)
Example: If the set is {5,1,4,6,3,2} then we will arrange the set as {1,2,3,4,5,6} and median will be mean of middle two terms. Middle two terms in this case are 3 and 4 so Median = (3+4)/2 = 3.5

• In case of odd number of numbers in the set: Median is the middle number (after the numbers are arranged in increasing/ decreasing order )
Example: If the set is {4,5,3,1,2} then we will arrange the set as {1,2,3,4,5} and the median will be the middle number which is 3

Properties of Median

1. If all the numbers in the set are increased/decreased by the same number(k) then the median also gets increased/decreased by the same number(k)
Proof same as for mean.

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

3. In Case of evenly spaced set
Mean = Median = Middle term (if the number of terms is odd)
= Mean of middle terms (if the number of terms is even)

4. In case of consecutive integers: IF the number of integers is even then then the Mean = Median ≠ Integer
Suppose the set is {1,2,3,4,5,6}
then Mean = Median = 3.5

SUGGESTION: Don't try remembering the points 1 and 2 above. It does not take much time to calculate them!


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.


Mode

• Mode is the number which has occurred the maximum number of times in the set.
Suppose the set is {1,1,2,2,3,3,3,3,4,5}
then the mode is 3, as 3 has occurred the maximum number of times in the set.

Weighted Average

• Weighted Average = ((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 = 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 = ((3-1)^2 + (3-2)^2 + (3-3)^2 + (3-4)^2 + (3-5)^2)/ 5
= (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 = √(Variance), SD = √(V)

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

Sol: V = 2 (calculated above)
SD = √(V) = √(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!