Week 6 ( Doubt ): Knowing Mode From Histogram

Dear,

Premise:
We can not find mode just by looking at the histogram if the bin size is more than 1.

Background:
In One Video from Week 6 on Knowing Measure of Central Tendency From Distribution, Professor shows that for left-skewed distribution with bin size more than one ( visible from diagram ) the mode is the value from the tallest bar ( also shown in a red vertical line )

I am adding a snapshot from the video.

My Question:
Can the mode of the histogram showing above necessarily be lying in the tallest bar?

According to the premise, it should not be the case as in diagram the bin size is greater than 1.

Please, clarify this to me.

Thanks
Nimit

In the above shown diagram, the bin size (or interval size) is clearly not 1.
I guess it’s 20 (assuming a discrete random variable X for the x-axis).
This means that each bin represents the total frequency of the 20 discrete values (x-axis).
That is, from the diagram we only know the total count of an interval range, not each value’s count.

Hence, the tallest bar in the histogram does not tell you the mode value when bin_size>1. Why?
Let me give an example:

Assume the mode value is present somewhere in the [260, 280) interval with bin_size=20 and frequency=20 (approximated from the histogram for explanation). Think of a possibility where each discrete value of X {260, 261, 262, … 279} has frequency=1, giving rise to total_freq=20.

Now let us consider the interval [240, 260), with frequency=15 (approx). Think of a possibility, where the count of X=250 is equal to 15, and other counts as 0. So now, can we suddenly tell the mode as X=250? No! We don’t know until we see the count of each values of the histogram (which is, making the bin size as 1). There can even be multiple modes.

So why does the histogram show the mode in the tallest bar in the diagram?
It just happened to be the case for the data used to plot the histogram.

But not always true.

Note:
The following fact always holds true though:
The median of a left-skewed data always lies between the mean (in the left) and the mode (in the right)

1 Like

So what I understood is:

Median will always lie between mean and mode for left-skewed distribution as well as right-skewed distribution.

Thanks :slight_smile:

@GokulNC
I think @NIMIT’s doubt is very valid.
For bin size more than 1, we cannot say for sure that Mode will be the largest (for left skewed) or smallest (for right skewed).

So I believe Mean < Median < Mode (for left skewed) or Mean > Median > Mode (for right skewed) does not hold good if bin size is greater than 1.

I think the example in the presentation need to be changed.