Determining Individual Values EVENLY Spread within a Class - Best Method?

• Sep 6th 2013, 12:43 PM
zikcau25
Determining Individual Values EVENLY Spread within a Class - Best Method?
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 gives different answers, Which of the two Methods is the best if ever they are?

For example:

There are 5 terms within this class with lower boundary ② and upper boundary ⑩. 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 + $\displaystyle \frac{8}{5 - 1}\times (1 - 1) =$

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 + $\displaystyle \frac{8}{5 - 1}\times (5 - 1) =$

Giving the following 5 terms data set { , 4, 6, 8, } :)

II. Graphical Method:

Plotting that class ②- (frequency = 5) as it would be normally drawn on a grouped frequency graph, assuming the values are evenly distributed:

Attachment 29126

But this gives the following 5 terms data set {3.6 , 5.2, 6.8, 8.4, } in CONTRAST with { , 4, 6, 8, } from arithmetic 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 + $\displaystyle \frac{8}{5}\times 5 =$

Giving the same 5 terms data sets as on the graph {3.6 , 5.2, 6.8, 8.4, }

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.
• Sep 6th 2013, 05:42 PM
chiro
Re: Determining Individual Values EVENLY Spread within a Class - Best Method?
Hey zikcau25.

Can you please give a one or two sentence explanation of what you are trying to do? Name the data, its type, the analysis you want to do, the question you are trying to answer, the problem you have regarding statistical techniques, and what the nature of the transformation to the data is in simple words (not math).
• Sep 6th 2013, 10:52 PM
zikcau25
Re: Determining Individual Values EVENLY Spread within a Class - Best Method?
LOL. I am a bit desperate that no one is able to prove at least their basic statistics insight to this very basic statistics problem.
That problem is at the very beginning of statistics lessons and it should be easily understood.

It is a about grouping discrete data into classes while making a grouped frequency table.

Let take this example:
______________________________________________

$\displaystyle \begin{tabular}{ l c r } Score & 0-9 & 10-19 \\ Frequency & 14 & 9 \\ \end{tabular}$
______________________________________________

- For example, In CLASS 0-9, or if ever data was rounded-off (that don't pause problem) then instead you will take -0.5-9.5 and 9.5-19.5 etc. but you know there are 14 observations there, and you don't know what these values are exactly. Grouping the data into classes means losing some information. If you are told to write back the 14 values grouped in that class, provided they are evenly distributed, What Should you do??? - That the whole problem I was explaining. At the start, I gave a simple example of class 2-10 so that the whole attention may be directed directly into the problem. You don't have to mind about the nature of the data, whether it was round-off or not etc...

______________________________________________

$\displaystyle \begin{tabular}{ l c r } data & 2-10 & \\ Frequency & 5 \\ \end{tabular}$
______________________________________________

Thank you for showing at-least an enthusiasm.
• Sep 7th 2013, 04:53 PM
chiro
Re: Determining Individual Values EVENLY Spread within a Class - Best Method?
If everything is evenly distributed in a bin (which is what you call a class, but this is the standard term), then basically you have a uniform distribution.

What you can do is then aggregate these uniform distributions and describe a global distribution that has a "stair-case" design.

From this you can calculate moments, and probabilities as if you had a continuous distribution.

So as an example, lets say we had two bins: one goes from 0 to 2 and another goes from 2 to 5. Each value in the same bin has the same probability where bin 1 probability is 3/4 and other bin is 1/4. The continuous PDF under these assumptions would be:

P(0 < X <= 2) = 3/4 and P(X = x) = (3/4)/2 = 3/8 for x in [0,2)
P(2 < X <= 5) = 1/4 and P(X = x) = (1/4)/5 = 1/20 for x in [2,5)
• Sep 8th 2013, 09:52 PM
zikcau25
Re: Determining Individual Values EVENLY Spread within a Class - Best Method?
I found out the right name of the method I was looking for in another statistics book.
It is called "LINEAR INTERPOLATION" which matches with my Graphical Method : $\displaystyle {\color{DarkBlue} Term = A + \frac{W}{f }\times R}$
• Sep 8th 2013, 10:59 PM
chiro
Re: Determining Individual Values EVENLY Spread within a Class - Best Method?
If you are linearly interpolating the PDF (probability function), then linear interpolation will not do what you describing in your assumptions (where you stated that all probabilities were the same in some region).

Linear interpolation means you connect points by lines (straight lines), so if you want probabilities to hold the same properties in some region, you need to re-formulate your distribution.