How to Solve: Statistics (Mean, Median, Mode)

Theory

Mean /Average / Arithmetic Mean

• Mean is the average of the all the numbers in the set.
• Mean = \(\frac{Sum Of All The Numbers In The Set }{ Total Number Of Numbers In The Set}\)

Suppose the set is {1,2,3,4,5}
Then, Mean = \(\frac{(1+2+3+4+5)}{5} \)=\(\frac{ 15 }{ 5} \)= 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 = \(\frac{(a+b+c+d+e)}{5}\)
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 = \(\frac{(a+k +b+k +c+k +d+k + e+k)}{5}\)
= \(\frac{(a+b+c+d+e + 5k)}{5}\) = \(\frac{(a+b+c+d+e)}{5}\) + 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!

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.

Next Check out Range, Weighted Average, Variance and SD

Hope it helps!