Mean all formulas and concepts.

 

Arithmetic mean

In statistics we use arithmetic mean before learning that see the fundamentals of statestics to understand this concept easily.

Here are the set of formulas to solve the measures of central tendency

Individual series

If the data is given in individual series we should use
         ∑x 
    x̅=------ ⇒∑x = nx̅.    
           n
Where n is no. Of observations given in data
 
If the values are in AP then  x̅= (a+l)/2
 For Ex:- what is the mean of first 10 multiples of 8
 Sol:- a is first multiple that is 8
         l is last multiple that is 80
 Now, mean =(a+l)/2. = (8+80)/2. = 88/2 = 44.

If they asked assumed mean for individual series we have to assume, that means select a number of your wish from the given data and subtract that assumed number from every number in the data. This set of subtracted numbers should be added and then divided by number of observations and add assumed number to that number which you get after adding.
                        ∑(x-A)
           x̅= A+  ------------
                           n
For example:- let the given data be 2,4,5,8
Now assumed mean =
      I assumes 5 as a assumed number from data. You can select your wish.
                          (2-5)+(4-5)+(5-5)+(8-5)
             x̅=    5+ —————-————----————
                           no. Of observations
           
                     -3+(-1)+0+3
        x̅=.   5+  ————————.   = 5-(1/4)=19/4 = 4.75
                                   4

Descrete series

When the data in question is given in descrete series we should use
                        ∑fx                    
           x̅=.   ————
                               N

 Let me give you an example for this

    X.         |        f.      |.   fx    (f×X)                 ————————————————

     5.       |        6       |.     30     (5×6)

    6         |         7      |       42.   (6×7)

    4         |         5      |       20.   (4×5)               —————————————————

                |     N=18  |     ∑fx=  92   |                              ————————————      

                             ∑fx.             

      Mean=    —————   =   92 /18. =5.1

                             N

Continuous series

If the data is given in continuous series 

                    ∑fx
  Mean.= ———.     which is same as                                     N       Descrete series formulae
    In descrete series we find x given directly in question but in continuous series the x will be midvalue.
                  Upper limit + lower limit         
   X   =     ———————————–——
                                  2
 Suppose let a class be 
        Class interval. |.    f.     |.  X. |. fx
     ————————————————       
             20-30.           |     5     | 25  |  125
             30-40            |     7     |  35 |  245
           40-50.         |    4     | 45 |. 180
       _______________________________
                                   16.   |. 105 | 550
  Mean= 550/16 = 34.37
                         

Combined mean

   Let me say, you done 2 mean sums and you got 2 different means for each. I asked you the means you said 1st mean is x¹ and 2nd mean is x². Ofcourse it's right answer. Now I asked combined mean of both questions. Then you must say a single value as x¹².
The formula for combining n2 different means of 2 different groups is
      X¹²=    (n¹x¹ + n²x²)/(n¹+n²)
 n¹ is number of observations in 1st data
n² is number of observations in 2nd data.
X¹ is mean of 1st data.
X² is mean of 2nd data.

Corrected arithmetic mean

                                                                  C– W
Corrected mean= existing mean + ———
                                                              n
C means correct number
W means wrong number
For example in individual series if we entered a number as 45 instead of 54 we can correct by using this formula

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