# The frequency in histograms

My question is trivial, but the problem is that I found contradictory answers on internet. So I have the following sample data:

31/08/2015, Monday, 'Seat Ibiza', 3
31/08/2015, Monday, 'Ford Focus', 2
01/09/2015, Tuesday, 'Seat Ibiza', 3
02/09/2015, Wednesday, 'Toyota', 1
07/09/2015, Monday, 'Honda', 2
08/09/2015, Tuesday, 'Volkswagen Golf', 1
08/09/2015, Tuesday, 'Seat Ibiza', 1


1) Now I want to build a classical histogram. I wonder if the frequencies should be absolute or relative numbers??

For instance, I would build the following frequency table, but I'm not sure if it's a must to use relative numbers (i.e. divide each number by total - 7):

Monday, 3
Tuesday, 3
Wednesday, 1


2) Another question is: can I call such graphic as a histogram, though discrete values are used? I read that the term 'histogram' refers only to continuous space.

3) Final question is: how the probability distribution would be estimated for these data?

You have a discrete variable with, in your example, very small frequencies. It would be fine to show relative frequencies so long as you made clear the small sample size. I suggest that it would be nevertheless be simpler to show sample frequencies directly:

In this case we hope that people are able to look at the graph and think (e.g.) 3 out of 7 were on Monday, and so forth. Small children are often encouraged to do this, which presumably explains why it is sometimes thought too challenging for business people.

I am happy, personally, with the idea that this is a histogram, as it shows the frequency distribution of a numeric variable. The fact that the variable is discrete and not continuous does not worry me. As a convention to underline discreteness many people like to leave some space between the bars. Some even like to show spikes. I have often done either. Here there is no signal what the data are, beyond showing car makes. Suppose that the data are daily sales or numbers of cars serviced. If there is an idea that some business is open for part of the day only, then spaces between bars convey a fair picture. Conversely, if the days were touching 24 hour periods, then I think the bars are better shown as touching.

Do not be surprised if some people prefer to call this a bar chart, not a histogram. Graphically a histogram is just a bar chart, but in many statistical discussions pedagogic tradition and minor snobbery join forces to insist that a histogram showing a frequency distribution is not a bar chart at all!

All that said, I imagine that your real data set is larger than your example, so it might well be in order to show relative frequencies as well as frequencies (counts).

It is hard to get people to agree on whether the numeric information should be given (a) on the graph (b) in a table (c) in accompanying text, and indeed any or even all could make sense, depending on the data, the readership and the goals of he analysis.

Without more said, the underlying probabilities are to be estimated as the sample proportions.

1. I would show absolute counts, not relative frequencies. Absolute counts make the total weight of the evidence (which is not much!) clear.

2. It's a "bar chart" you want, not a "histogram". Although many people say "histogram" for both discrete and continuous data, you may as well head-off any complaints by using the correct terminology.

3. If you have categorical data (e.g. counts for days of the week or types of car) and want to estimate the underlying probabilities then, well, "it depends". You can use the observed frequencies, e.g. $(\frac{3}{7},\frac{3}{7},\frac{1}{7})$, as a point estimate; or, to allow the possibility of currently-unseen values, add a small flattening pseudo-observation of size $\epsilon$ to everything to get say $(\frac{3+\epsilon}{7+4\epsilon},\frac{3+\epsilon}{7+4\epsilon},\frac{1+\epsilon}{7+4\epsilon},\texttt{other=}\frac{\epsilon}{7+4\epsilon})$; and you might want to put some sort of confidence interval around your estimates. It all comes down to your "loss function", or the penalty you'd pay for being wrong. It's a bit of a black art, to be honest. How about starting with the observed frequencies and clarifying your question if you want a more problem-specific answer?