I'm currently learning about grouped frequencies and I was wondering how we could determine the Median and Mode from grouped frequencies. We don't know the precise values so we can only determine and estimate Median and Mode. I found these 2 formulas that help me do this.
However what is the intuition behind this formulas. I've tried searching but I can't seem to find anything. Is there a way to derive these formulas?
In both cases, we're calculating some fractional value $f$, and then finding the point within the appropriate bin that is a fraction $f$ along its width by calculating $L + fw$. So for example, if we have a bin starting at 10 and with width 5, then $f = 0.2$ gives $10+5\times0.2 = 11$.
Then it's just a matter of picking the right bin and a suitable fraction. For the median, it's a bit simpler - we know that the median has to be in the bin containing the $\frac{n}{2}$th value, and if there are $B$ values in the bins below it, then the median must be the $(\frac{n}{2} - B)$th value in the bin, which is a fraction of $f = \frac{1}{G}(\frac{n}{2} - B)$ of the count of all values in the bin. If we assume the values are uniformly distributed within the bin, then we can just apply that fraction to the formula above for the estimate of the median.
For the mode, define the first-order differences of consecutive groups as $\Delta_n = f_n - f_{n-1}$. Then notice that the fraction in question can be written as $\frac{\Delta_m}{\Delta_m - \Delta_{m+1}}$, which is the fraction that divides an interval in the ratio $|\Delta_m| : |\Delta_{m+1}|$. One way to view this is to look at the following diagram:
If we assume the two triangles are similar, i.e. the lines have the same slope (up to a negative sign), then their heights are $|\Delta_m|$ and $|\Delta_{m+1}|$ respectively, so the point where they meet divides the width of the bin in that ratio.