Properties of Arithmetic Mean || Lesson 5 || Probability & Statistics || Learning Monkey ||

Properties of Arithmetic Mean
In this class, We discuss Properties of Arithmetic Mean.
The reader should have a prior understanding of arithmetic mean. Click Here.
1) if X’ is the arithmetic mean of n observations x1,x2, . . . xn. then (x1 - x’)+(x2 - x’) + . . .+(xn - x’) =0
Sum of deviations of the set of values from their arithmetic mean.
Understand the property with an example
1, 2, 3, 4, 5
X’ = mean = (1 + 2+ 3 +4 +5)/5
X’ = 3
(1-3)+(2-3) + (3-3) + (4-3) + (5-3) = 0
(1-3) gives how much the data value 1 deviates from the mean value 3.
The sum of all the deviations of the data values from the mean is zero.
Proof:
X’ = (X1+X2+. . +Xn)/N
(X1+X2+. . +Xn) = X’N - Important to understand.
The sum of all the data values is equal to X’N.
(x1 - x’)+(x2 - x’) + . . .+(xn - x’) = (X1+X2+. . +Xn) - NX’
NX’ - NX’ = 0
2) Mean of composite series
Mean of heights of N1 girls is X1′
Mean of heights of N2 boys is X2′
X1′ = (xg1 + xg2 + . . . +xgn1)/N1
X2′ = (xb1 + xb2 + . . . +xbn2)/N2
Total Mean X’ is as follows.
X’ = ((xg1 + xg2 + . . . +xgn1) + (xb1 + xb2 + . . . +xbn2)) / (N1 + N2)
X’ = (N1X1′ + N2X2′)/(N1 + N2)
3) If each of the data values is increased or decreased by a fixed value then the mean of the new data is increased or decreased by the fixed value.
Example:
1, 2, 3, 4, 5
X ‘ = ( 1+ 2+ 3+ 4+ 5)/5 = 3
Each data value is increased by 5.
6,7,8,9,10
X’new = ( 6+7+8+9+10)/ 5 = 8
X’new = X’ + 5
4) Similar to property 3 with multiplication or division
Example:
1, 2, 3, 4, 5
X ‘ = ( 1+ 2+ 3+ 4+ 5)/5 = 3
Each data value is multiplied by 5
5,10,15,20,25
X’new = (5+10+15+20+25)/5 = 15
Example:
Each data value multiplied by b and added with a.
X’new = a + bX’
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[@FUTUREIITIAN-s2y]

Beautiful explanation

[@LearningMonkey]

Have a great learning in CSE

[@antisocialbarbie1587]

Thank you

[@akshayaa7388]

Bro, is Mean a positional average?

[@LearningMonkey]

What do you mean positional average.
Elaborate a little

[@akshayaa7388]

@@LearningMonkey Positional average is an average that will be picked up from a series of data, which will represent the whole series. And in some lectures I read, median is called as positional average. So, my doubt is : Is arithmetic mean a Positional average or is there any other term that represents the Arithmetic mean as some average? And I also see, in some tutorial videos, they call mean as ordinary average.