Question

Write the names of two methods used to determine the mean of grouped data
1) Direct Method
2) Step DeviationMethod

Answer 1

one is more assumed mean method

Answer 2

Hey.!!


Methods and Formulae

Mean of the grouped data can be calculated by using following two methods -

Direct Method:
Direct method can be used when the grouped data is given. If the given observations are denoted by xixi and their corresponding frequencies are fifi, 
then the formula for calculating mean using direct method is as follows:

x¯x¯= ∑ni=1fixi∑ni=1fi∑i=1nfixi∑i=1nfi

Where, 

x¯x¯ = Mean of the given data
∑ni=1fi∑i=1nfi = Sum of all the frequencies. This is also denoted by N.
∑ni=1fixi∑i=1nfixi = Sum of the product of frequencies and corresponding observations.Assumed Mean Method:
Assumed mean method is a method that uses an assumed mean in order to calculate actual mean. This method does cut short the calculations that are more in the direct method. 

The formula used for estimating mean of grouped data by using assumed mean method is given below:

x¯x¯ = A + ∑ni=1fidi∑ni=1fi∑i=1nfidi∑i=1nfi

Where, 

x¯x¯ = Mean of the given data.

A = The middle value that is assumed as the mean for calculation.

∑ni=1fi∑i=1nfi = Sum of the frequencies given, can be denoted by N.

∑ni=1fidi∑i=1nfidi = Sum of the product of frequencies and corresponding deviations.

di=xi−Adi=xi−A This deviation makes the calculation quite easier even if the data is too big.

How to Calculate

Direct Method
Let us suppose that the given observations are x1,x2,...,xnx1,x2,...,xn and the frequencies corresponding to them are given as f1,f2,...,fnf1,f2,...,fn.

In order to calculate mean by this method, we should follow the steps mentioned below.

Step 1: Construct a table in which there are three columns.

Step 2: In first column, write all the observations.

Step 3: In second column, write all the frequencies. Also, down below calculate the sum of all frequencies as ∑ni=1fi=f1+f2+...+fn∑i=1nfi=f1+f2+...+fn

Step 4: In third column calculate the product of each observation and corresponding frequencies, i.e. fixifixi and down calculate their sum, i.e. ∑ni=1fixi∑i=1nfixi.

Step 5: Now divide the value of ∑ni=1fixi∑i=1nfixiby the value of ∑ni=1fi∑i=1nfi. The required mean is obtained.Assumed Mean Method:
In order to calculate mean by assumed mean method, one should follow the following steps.

Step 1: Let us construct a table in which there are four columns.

Step 2: In first column, write all the observations. From these observations, one middle observation should be consider as assumed mean. This observation is to be denoted by A.

Step 3: In second column, we should write all the frequencies and below calculate the sum of all frequencies, i.e. ∑ni=1fi=f1+f2+...+fn∑i=1nfi=f1+f2+...+fn

Step 4: In third column, we should calculate the deviations, i.e. di=xi−Adi=xi−A.

Step 5: In fourth column, the product of each observation and corresponding deviation is to be calculated, i.e. fidifidi and below their sum is
to be calculated, i.e. ∑ni=1fidi∑i=1nfidi.

Step 6: Now, using the formula $\bar{x} = A + ∑ni=1fidi∑ni=1fi∑i=1nfidi∑i=1nfi, the required mean is calculated.

If the data is given in the form of class intervals; 30 - 40, 40 - 50 etc, then xixi would 

be calculated by xixi = lowerlimit+upperlimit2lowerlimit+upperlimit2

i.e. 30+40230+402 = 35

and 40+50240+502 = 45


I hope it's Help you....!!!


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