Deriving the equation for oxygen estimation by Winkler’s method
[Concepts]


Normality, N: The normality is the relation between the sustance equivalence and the liters of a solution. The sustance equivalents are referred to the charges per mole of a sustance. The number of equivalents (NOE) of a sustance is obtained as follows:


Acid equal to number of hydrogens.
Hidroxide equal to number of hydroxils.
Salt equal to number of cations.


To determine the normality of a solution, these are the steps you must follow:


Weight Equivalent Gram, WEG:
\begin{equation}
    WEG=\frac{Molecular\ weight\ of\ solute}{NOE}
   \end{equation}
Weight Equivalent Number, WEN:
\begin{equation}
    WEN=\frac{Mass\ of\ solute}{WEG}
   \end{equation}
Normality, N:
\begin{equation}
    N = \frac{WEN}{L}
   \end{equation}
Doing some basic algebra, you find that:
\begin{equation}
   N=M\cdot NOE
  \end{equation}
Where M stands for Molarity.

Dilution factors, DF: It is employed when the solution concetration of your sample is too concentrate. In order to determine the concentration of the substance you want to find, you have to dilute the initial solution. This is done through the dilution factors. I.e. you have 25 ml of a solution 10 M and you are asked to dilute it 100 times. To do so, you have to take an aliquot (A portion of a large whole) of 0.250 ml, pour into a volumetric flask of 25 ml and then fill it with destilled water to capacity. Then, you have a diluted solution of 0.1 M. In general, a dilution factor is expressed as:
\begin{align}
   Dilution\ factor=\frac{Total\ volumen}{Sample\ volume}
  \end{align}
Dilution: Dilution is the process of decreasing the concentration of a solute in solution, usually simply by mixing with more solvent. To dilute a solution means to add more solvent without the addition of more solute. This can be represented as:
\begin{align}
   V_1\cdot M_1 &amp;= V_2\cdot M_2\\
   M_2&amp;=\frac{V_1}{V_2}\cdot M_1
  \end{align}


[Answer]

Now, if you put all together:


If you look at the first parenthesis at the right of the equa, having in mind that E is WEG and doing some basic algebra:
\begin{align*}
   V_1\cdot N\cdot E\cdot 1000&amp;=V_1(L)\cdot\frac{WEN}{L}\cdot WEG\cdot 1000\\
   &amp;=V_1(L)\cdot\frac{\frac{Mass\ of\ solute}{WEG}}{L}\cdot WEG\cdot 1000\\
   &amp;=V_1(L)\cdot\frac{Mass\ of\ solute}{WEG\cdot L}\cdot WEG\cdot 1000\\
   &amp;=V_1(L)\cdot\frac{Mass\ of\ solute}{L}\cdot 1000\\
   &amp;=V_1(mL)\cdot\frac{Mass\ of\ solute}{mL}\\
   &amp;=Mass\ of\ solute\cdot\frac{V_1(mL)}{mL}\\
   &amp;=Mass\ of\ solute
  \end{align*}
The square bracket is just a dilution factor.
If you put together the last two steps with the dilution section, you find a similarity.


[History]

If you are interested in the origins of this method, I recommend you the following paper:


Winkler, L. "Die Bestimmung des in Wasser Gelösten Sauerstoffes". Berichte der Deutschen Chemischen Gesellschaft. (1888) 21, 2, 2843–2855.


I cannot give you a brief handout of it, my german is basic. But, I want to point out that in page 2845, you will find an equation similar than the indicate in dilution section.

[How to understand the Winkler method]


  The Winkler method for the determination of dissolved $\ce{O2}$ involves adding an excess of ‘white’ $\ce{Mn(OH)2}$ which contains $\ce{Mn^{2+}}$ ions.





  For every molecule of $\ce{O2}$ two molecules of $\ce{I2}$ are formed. The quantity of iodine present can be found though titration of the sample with sodium thiosulphate.


[Bibliography]


Shashi,  B.P. A complete review on: effluent testing and treatment in pharmaceutical industry.