Finding the mode (descriptive statistics)

I have 5 classes :

• [0,4[ containing 4 people (density 1)
• [4,6[ containing 8 people (density 4)
• [6,7[ containing 2 people (density 2)
• [7,10[ containing 9 people (density 3)
• [10,12[ containing 2 people (density 1)

The class with highest density is [4,6[.

Think of the histogram. I read somewhere that the mode is the point of intersection of the lines (AC) and (BD) with A=(4,1), B=(4,4), C=(6,4) and D=(6,2) (the first coordinate is the abscissa of a class, the second coordinate is the density of a class). Why is it so ? What are the hypothesis behind ?

The mode is given by the formula : $m_0=x_i+(x_{i+1}-x_{i})\frac{d_i-d_{i-1}}{(d_i-d_{i-1})+(d_i-d_{i+1})}=4+(6-4)\frac{4-1}{(4-1)+(4-2)}$

Why is it so ?

Where can I find a document on finding the mode ?

The mode is the observed value that occurs most often (i.e. with highest frequncy) in a sample or a population. In your data, I don't see how you can tell what the mode is, because you don't know what the actual values are -- you only know what intervals these values fall in.

The mode is not unique -- some collections have two or more modal values. That being the case, I don't see how there could possibly be a simple arithmetic formula for the modal value.

Disclaimer: I am not a statistician. I used to teach elementary statistics, but that was 40 years ago. I'm sure some things have changed, but I'd be surprised if the definition of "mode" has changed.

Added after reading the cited paper The concept of "mode" has been extended to "modal interval". If values are organised into intervals (of equal width, presumably), then the interval(s) that contain the most values are said to be "modal intervals". There may be more than one modal interval, in some cases. Then, we would like to designate a specific "modal value" within each modal interval. So, we need some formula that will give us a value within the given modal interval. The formula should give us a value that is "skewed" to follow the frequencies in the intervals to the left and right of the modal one.

Ths picture shows the technique: You draw lines using the tops of the histogram columns to the left and right of the modal interval, as shown in the picture, and the modal value is the given by the intersection of these lines. You can see that as the column on the left gets higher, the intersection point will get moved to the left.

The formula given by the OP is just a formula for the $x$-coordinate of the intersection of the two lines. This formula gives you a reasonable value for the mode within a given modal interval. It works nicely because it biasses the value to reflect the heights of the histogram bars on adjacent intervals.

• Thank you very much. There is a formula because you can make assumptions on the distributions in each interval. – user63008 Mar 8 '13 at 14:46
• What do you assume? Uniform distributions? – bubba Mar 8 '13 at 15:10
• I found this document explaining the method but I don't understand it : conte.u-bordeaux4.fr/Enseig/SE1-stat/Mode/MODE.html . Look at the diagram. – user63008 Mar 8 '13 at 15:49