In Statistics or even in Maths, sometimes for the sake of convenience, Data are grouped within a class at the expense of not knowing individually each of these values.

In case one has to find them it is often assumed they are EVENLY DISTRIBUTED within the class. But no such formula to determine them exists in my statistics book

Therefore on my own, I have developed two methods so-called:Arithmetic Method&Graphical Method.

Both methods seem to work but givesdifferent answers,Which of the two Methods is the best if ever they are?

For example:

There are 5 terms within this class with lower boundary② and upper boundary⑩. What are these terms?

{ ②, __ , __ , __ , ⑩ } (frequency = 5)

I. Arithmetic Method:

With some arithmetic progression knowledge I have managed to invent this formula:

$\displaystyle {\color{DarkBlue} Term = A + \frac{W}{(f - 1)}\times (R - 1)}$

where,.

A= lower class boundary,W= class width,f= frequency,R= rank

Applying,

1st term : ② + $\displaystyle \frac{8}{5 - 1}\times (1 - 1) = $②

2nd term : ② + $\displaystyle \frac{8}{5 - 1}\times (2 - 1) = $4

3rd term : ② + $\displaystyle \frac{8}{5 - 1}\times (3 - 1) = $6

4th term : ② + $\displaystyle \frac{8}{5 - 1}\times (4 - 1) = $8

5th term : ② + $\displaystyle \frac{8}{5 - 1}\times (5 - 1) = $⑩

Giving the following 5 terms data set{②, 4, 6, 8,⑩}

II. Graphical Method:

Plotting that class②-⑩ (frequency = 5)as it would be normally drawn on a grouped frequency graph, assuming the values are evenly distributed:

But this gives the following 5 terms data set{3.6 , 5.2, 6.8, 8.4,⑩}in CONTRAST with{②, 4, 6, 8,⑩}fromarithmetic method.

Alternatively to using the graph, one can use this slightly modified formula also:

$\displaystyle {\color{DarkBlue} Term = A + \frac{W}{f }\times R} $

Applying,

1st term : ② + $\displaystyle \frac{8}{5}\times 1 = $3.6

2nd term : ② + $\displaystyle \frac{8}{5}\times 2 = $5.2

3rd term : ② + $\displaystyle \frac{8}{5}\times 3 = $6.8

4th term : ② + $\displaystyle \frac{8}{5}\times 4 = $8.4

5th term : ② + $\displaystyle \frac{8}{5}\times 5 = $⑩

Giving thesame5 terms data sets as on the graph{3.6 , 5.2, 6.8, 8.4,⑩}

So which Method (Arithmetic or Graphical) you think is best or you think is faulty or whether there is a better solution to determine the data evenly distributed within a class. All the Best and Thanks in advance.