Histogram normalisation: a question about the correct terminology

Question

Given a histogram and the following quantities

v_i = the bin value.
c_i = the number of elements in the bin.
w_i = the width of the bin.
N   = the number of elements in the input data. 

Is the following terminology correct in Statistics?

1. By normalizing the histogram by v_i = c_i / N, I would get the "Relative Frequency Histogram"

2. By normalizing the histogram by v_i = c_i / N, I would get an "empirical estimate of the Probability Mass Function"

3. By normalizing the histogram by v_i = c_i / N, I would get an "empirical discrete probability distribution"

4. By normalizing the histogram by v_i = c_i / (N * w_i), I would get an "empirical estimate of the Probability Density Function"

Answer 1

Choice of terminology can reflect personal taste as well as each writer's idea of technical correctness, and what is considered correct should depend on the details of the application.

I doubt that there could be strong objections to #1, relative frequency histogram. Many would be happy to say fraction, proportion or even probability instead of "relative frequency", with the understanding that what is shown may be empirical estimates.

I don't find the terms normalize or normalization (whether z or s) attractive, if only because they can mean so many different things. In my readings in statistical and data science they imply typically scaling in one of various different ways, but occasionally some kind of transformation intended to bring a distribution closer to Gaussian (normal) in shape. Conversely, either could be a fine term to use if people in your field use that term often and (as here) you always explain with a formula or other recipe precisely what normalization means in each application.

The terminology discrete and probability mass function I would commend if and only if the underlying variable was discrete, in which case bin width would usually reflect discreteness, most commonly by being 1 if the distinct values of the variable were integers. These comments qualify your usages #2 and #3.

Conversely, the term probability density function applies most comfortably to continuous variables. The term probability mass function was introduced precisely to stress the difference between probability distributions for continuous and discrete variables. But many writers, including several with a strong sense of mathematical correctness, are happy to use density generally, each version of density being calculated given some measure, which could be counting measure as well as some continuous measure. So, many statistical people would be happy with the idea of a histogram (for any kind of variable) showing an empirical binned estimate of probability density in this wider sense.

I think it's germane to underline that the term density was in general use long before it became standard in probability and statistics. Across physics, chemistry, demography, ecology, hydrology, and other sciences, there are uses for density as amount of stuff in some space, regardless of whether the space is one, two or three dimensional, or the stuff is continuous or discrete. Thus in physical science density is mass/volume, in population studies it is number of people or other organisms/area, in hydrology it can be drainage density (stream length/area), and so on.