Why would $L \ + \frac{f_{m}-f_{m-1}}{(f_{m}-f_{m-1}) + f_{m}-f_{m+1}} \cdot w$ correspond to the mode of grouped frequencies?

Question

$\text{Mode} = L \ + \frac{f_{m}-f_{m-1}}{(f_{m}-f_{m-1}) + f_{m}-f_{m+1}} \cdot w$

where,
       • L is the lower class boundary of the modal group.
       • fm-1 is the frequency of the group before the modal group.
       • fm is the frequency of the modal group.
       • fm+1 is the frequency of the group after the modal group.
       • w is the group width.

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Mode is the most frequently occurring value in a data set - I don't understand why this formula would give the mode of grouped frequencies. Why would the mode even be in the modal group? It could be inside any other group, right!?

Answer 1

You are correct that the modal interval of a histogram may or may not contain the mode of the sample. In the following data, the mode is 2 (the only tie), but in my histogram the modal interval is $(4.5, 6.5).$

x = c(1:10, 2, 4.9, 5.1); sort(x)
##  1.0  2.0  2.0  3.0  4.0  4.9  5.0  5.1  6.0  7.0  8.0  9.0 10.0
hist(x, br=seq(.5, 10.5, by=2), col="skyblue2")

For continuous data, the sample will not ordinarily have a mode, because ties result only from rounding. Often histograms are made in an attempt to find the approximate location of the mode of the population distribution. A better way to estimate the mode of the distribution is to use a 'kernel density estimator' (KDE), which attempts to make a smooth curve that approximates the population density. [You can google 'kernel density estimator' or look at the help page in R statistical software for density if you want technical details.]

The sample illustrated below, from $\mathsf{Norm}(\mu = 100, \sigma = 15),$ is large enough $(n=1000)$ that the histogram and the KDE (red curve) are roughly suggestive of the population density (blue dashed curve). The mode of the population is at $y=100,$ the modal histogram interval is just above 100, and the KDE (although not a perfect match for the population density) correctly locates the population mode as very near 100. (There is no sample mode because continuous values are expressed to enough decimal places to avoid ties altogether.)

Notes: Many elementary statistics texts give formulas that attempt to approximate mean, median, and mode from grouped data. In my experience, only the formula $\bar X \approx \sum_j m_jr_j$ for approximating the sample mean from centers $m_j$ of histogram bins and bin relative frequencies $r_j = f_j/n$ is useful in practice. For the other two measures of centrality, it usually suffices to identify the median and modal intervals.

I have not seen your formula before; it seems to be an attempt to interpolate the location of the 'mode' within the modal interval. However, lacking context, I have no idea whether that is an attempt to approximate the mode of the sample, to estimate the mode of the population, or to define a new kind of 'mode' for histograms.