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We know that formula of finding mode of grouped data is

Mode = $l+\frac{(f_1-f_0)}{(2f_1-f_0-f_2)}\cdot h$

Where, $f_0$ is frequency of the class preceding the modal class and $f_2$ is frequency of the class succeeding the modal class. But how to calculate mode when there is no class preceding or succeeding the modal class.

Sumit Kumar
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  • The traditional formula you propose using may not be up to this task. Depending on the amount and nature of your data, a kernel density estimator (KDE) [roughly a smoothed histogram] may work better. – BruceET Mar 27 '20 at 22:34
  • Thanks , didnt think someone would have had this same late night doubt as me :) – Substitute_Y Mar 20 '23 at 19:40

2 Answers2

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we can take their value as 0. The frequency of the succeeding model class is taken as 0 if model class is the last observation.

You can also check it from the equation as-

$l =$ lower limit of the modal class,

$h =$ size of the class interval (assuming all class sizes to be equal),

$f_1 =$ frequency of the modal class,

$f_0 =$ frequency of the class preceding the modal class,

$f_2 =$ frequency of the class succeeding the modal class.

Even if $f_2$ is $0$, the mode can be easily found by using the above expression.

Especially Lime
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ANANT
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Here is an elementary example of the use of a density estimator in R.

First we generate a thousand observations from the gamma distribution $\mathsf{Gamma}(\mathsf{shape}=\alpha=2, \mathsf{rate} = \lambda = 1/3)$ and plot their histogram in such a way that the 'modal bin' includes the smallest values.

set.seed(327)
x = rgamma(1000, 2, 1/3)
hist(x, prob=T, br=7, col="skyblue2")

enter image description here

Then we find the default density estimator in R. It consists of 512 points. Plotting them imitates a smooth curve.

den.est = density(x)
hist(x, prob=T, br=7, ylim=c(0,.15),col="skyblue2")
lines(den.est, type="l", lwd=2, col="red")

enter image description here

Here is a summary of the $(x,y)$ points of the density estimator. We can use these points to find where the estimated density curve is at its highest points. Thus, we can locate the 'mode' of the data, as defined by the density estimator. For our simulated data its about $3.65.$ (We take the 'mean' of x's with highest y-value because there may be ties.)

den.est

Call:
        density.default(x = x)

Data: x (1000 obs.);    Bandwidth 'bw' = 0.8625

       x                y            
 Min.   :-2.480   Min.   :6.260e-06  
 1st Qu.: 4.383   1st Qu.:2.507e-03  
 Median :11.247   Median :1.828e-02  
 Mean   :11.247   Mean   :3.639e-02  
 3rd Qu.:18.110   3rd Qu.:6.596e-02  
 Max.   :24.974   Max.   :1.203e-01  

mean(den.est$x[den.est$y == max(den.est$y)])
[1] 3.644313

Usually the point of finding the mode of a histogram is to estimate the mode of the population distribution. We did pretty well in this example: The gamma distribution $\mathsf{Gamma}(\alpha=2,\lambda=1/3),$ from which we simulated the data has its mode at $(\alpha-1)/\lambda = 1/(1/3) = 3.$

Note: By way of full disclosure: (1) With as many as $n = 1000$ observations, we might have used more bins in our original histogram so that the traditional formula could be used. Here is a frequency histogram of the data with more bins. (I will leave it to you to see what value the traditional method gives.)

hist(x, ylim=c(0,260), labels=T, col="skyblue2")

enter image description here

(2) Also, if the population distribution has its mode at one end of its support, a modification of the default kernel density estimator in R may be required for a good estimate of the mode. (An exponential distribution, with its 'mode' at $0,$ would be an example.)

BruceET
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