Statistics: Mean

Mean Math Or Mean Statistics
In Mathematics in the branch of Statistics, the expression for the mathematical mean of a statistical distribution is the mathematical average of all the terms in the data. To calculate this, we add up the values of the terms given and divide the sum by the number of terms in the data. This expression is also called the Arithmetic Mean.
Example: Find the Arithmetic Mean of the following data 5, 5,10,10,15,15,20,30
Solution: Arithmetic Mean = Regular average = sum of the values of the terms/number of terms
Sum of the values of the terms =5+5+10+10+15+15+20+30=110
Number of terms = 8
Mean = 110/8 = 13.75
Sample Mean
The sample mean in statistics branch of Mathematics is the sum of all observed outcomes from the sample divided by the total number of events. It is denoted by the symbol x with a (bar) above it. The formula used to compute the sample mean is as follows:
X (bar)= (1/n) (x1+x2+x3………xn)
 If we consider the sampling of some non-indigenous weed in a land of five acres in Springfield and came up with the counts of this weeds in that area as 38, 56, 84,105,116
We calculate sample mean for the above sampling by adding the weed counts and divide by the number of samples, 5
(38+56+84+105+116)/5= 79.8
So, we get the sample mean of non-indigenous weeds = 79.8
Weighted Mean
While computing Arithmetic Mean all the terms or values have equal importance. But at times we come across some situations where all the terms or values do not have equal importance. For instance, when we compute the average number of marks per subject, as per the different subjects like English, Science, Mathematics, Social Sciences; each subject has different levels of importance.
So, weighted mean is the arithmetic mean calculated by considering the relative importance (weight) of each term.
Each item is assigned a weight in proportion to its relative importance
Formula to compute Weighted Mean: xw (bar) = sigma(wx)/sigma(w)
(here x =value of the items  w= weight of the item)
Example:  A student scored 50, 70, 80 in English, Science and Math respectively and assume the weights of each subject to be 4,5,6  respectively. Find the weighted arithmetic mean of each subject.
Solution: let us tabulate the given data
Subjects              Marks obtained          weight              wx
English                50                          4                  150
Science                       60                          5                  300
Math    80                          6                  480
         Sigma(w)=15  sigma(wx)=930
Arithmetic Weighted mean = 930/15= 62 marks/subject
Short-cut Method
A short-cut method of calculating the arithmetic mean is based on the property of arithmetic average. In this method the deviations (D) of the items from an assumed mean are first calculated and then multiplied with their respective frequencies (f). Then, the total of these products [sigma(fD)] is divided by the total frequencies [sigma(f)]and added to the assumed mean(A). The figure we get is the actual arithmetic average or the Arithmetic Mean.
Formula used in the short-cut method of calculating the arithmetic mean:
X(bar) = A + sigma(fD)/sigma(f)