9 Relative and absolute frequencies: Marvelous dice once again.

Slide 0

I hope you’re having a wonderful day. Today we will talk about two important terms in statistics, specifically absolute frequency and relative frequency. And along the way we will come to a third key term, and we’ll look at different ways of representing data.

Incidentally, we will look at some experiments again that we have worked with in previous episodes. Of course, you can also do precisely that in school: Use a good idea more than once.

Slide 1

Let’s roll a die. We take a die and roll it 10 times, 100 times, and 1,000 times.

What you see here in the table are the results. And these results are the numbers, the absolute quantity of times a particular number between 1 and 6 was rolled. Instead of “quantity” we say “frequency” and speak of the absolute frequency for the results 1, 2, 3, 4, 5, or 6.

Slide 2

And if you now want to determine the proportions of the individual results, it clearly makes sense to divide the absolute frequencies by the number of rolls. What you end up with is the relative frequency.

For 10 rolls the results still look somewhat varied, but for 1,000 rolls these values are very close to each other at 0.16, 0.17, and 0.18, although here we rounded to two decimal places. Calculate for yourself: 1/6 = 0.166666 and that really is between 0.16 and 0.17.

Slide 3

Perhaps you remember this survey of how students get to school. The table shows the absolute numbers or – expressed another way – the absolute frequencies.

Slide 4

If you add up the numbers, you see that 1,073 students participated in the survey. If you divide the absolute frequencies by this number, you obtain the relative frequencies. I have written them here as percentages and rounded to one decimal place. This representation is familiar, and at a glance you can assess whether many or few children get to school using a particular means of transportation.

Slide 5

Of course, you can also represent these numbers in a bar chart. It is surely not surprising that the height of the bar shows the number of observed results. In this representation as well, you can quickly see which means of transportation are used more or less frequently. Well, we call that a frequency distribution. Make a note of this term; it will also occur in somewhat more complicated contexts.

Slide 6

Let’s discuss another example. Once again, it deals with a basis of statistics, the suitable collection of data. The numbers are somewhat larger this time, but the principle observations remain the same. 

We’re looking at how many new infections of Covid-19 there were in the week of 3/23/2021 in a few selected countries, namely in Chile, Germany, Colombia, Mexico, Peru, and Spain. Collecting this data ourselves is less suitable in this case; rather, we will rely on data from the Johns Hopkins University.

Slide 7

In this case as well, not only are the absolute numbers interesting, but especially the relative proportion of infected persons. And again we divide, this time the number of new infections during that week by the population of a country. You see the result in the middle column of the table. For example, Chile comes to 0.00224 newly infected people and Mexico to 0.00023, which means that 0.00224 and 0.00023 new infections occur per inhabitant or 0.224% and 0.023% of the population became infected during the week in question.

Although these figures are rounded, you seldom make friends with them outside of mathematics – and maybe even not within mathematics. That’s why we don’t indicate them in this form, rather translate them to 100,000 population. In a nutshell, you multiply by 100,000. You remember that this means moving the decimal point five places to the right. So, 0.00224 becomes 224 and 0.00023 becomes 23, and these are incidences that we’re all familiar with in the meantime.

Slide 8

Let’s look at these six countries again. This time we’re interested in births. In this case as well, collecting our own data is not the means of choice; these figures can be found on the Internet. By the way, some date back to 2018 and others to 2019. It is almost always difficult to obtain data that are based on the same reference date. You should take this uncertainty into account when evaluating the results.

In principle, we proceed exactly as we did in the previous example. We have to know what the population of a country is in a specific year and now many births there were in that year. Then we can calculate the relative frequency from the absolute frequency by dividing. And because these relative figures – thus about 0.0094 for Germany and 0.0179 for Peru – are not everyone’s cup of tea, we extrapolate again, this time to 1,000 population. 

So, 0.0094 • 1,000 = 9.4 and 0.0179 • 1,000 = 17.9. In Germany there were 9.4 births per 1,000 population in 2019, and in Peru there were 17.9, significantly more. By the way, the figures for Peru are for 2018. 

Slide 9

There are many examples that are suitable and interesting for school. You can plan to collect your own data on favorite subjects, the number of siblings, or leisure activities. And you can also continue to work with professionally acquired data, such as data found in statistical yearbooks. The only requirement is that the data should be somewhat interesting to your specific class.

Slide 10

Let’s summarize.

We conduct a random experiment n times and determine that a certain event occurs k times. Then k is the absolute frequency of this event.

Let’s assume that an event occurs k times when we conduct the random experiment n times. Then k/n is the relative frequency of this event.

 

If you assign the absolute frequency to each event of a random experiment, that’s called a frequency distribution.

Slide 11

Do you remember when we rolled two dice? We added up the number of spots and considered the theoretical possibilities. This resulted in the table you see at the right.

For the sums of 2 and 12, there is exactly one combination because both dice show a 1 or a 6. The most combinations exist for the sum of 7: 1 and 6, 2 and 5, and 3 and 4. You see the theoretical relative frequencies as percentages at the far right in the table.

Slide 12

We’ll look at this again, and for this purpose I did not just roll 1,000 times, but I also noted the intermediate results after every 100 rolls.

Slide 13

What you see here are the frequency distributions for N = 100, N = 400, N = 600, and N = 1,000. You see very nicely how these absolute numbers approach our theoretical ideal step by step. For N = 100 the chart still looks a bit uneven, for instance for the sums of 5, 6, and 10. For N = 400 the sum of 7 slips out of the pattern, but later on everything makes a good impression.

Slide 14

And this good impression is verified when you determine the relative frequencies. I have given them here in the form of percentages; then you can easily compare them with the theoretical frequencies that you see in the bottom row. It takes a little time to read the table and you’ll probably need to pause the image.

Allow me to single out two results. The sums of 2 and 12 should theoretically occur in 2.8% of the cases; here they occur 3.5% and 3.0%, thus quite close. And the sum of 7 meets all expectations: 16.7% is both the empirically obtained proportion and the theoretical proportion of this sum of spots.

Slide 15

If you transfer the results to a line chart, it looks like this. The curve for N = 100 shows the trend, but it’s still somewhat unsteady; the curve for N = 1,000 visually confirms the closer approximation to the theoretical values.

Slide 16

Let’s take a look back.

In this episode, we looked at the terms absolute frequency and relative frequency of an event. We selected different ways of representing the results, namely a pie chart, a bar chart, a table, and a line chart.

In addition, using a very comprehensible example, we introduced the term frequency distribution, which plays an important role in stochastics.

Finally, we saw once more that there’s nothing better than doing your own experiment. And that applies especially to stochastics in class. Experiment yourself and allow your students to experiment. In this way, they’ll more likely have fun with this field of mathematics.

Slide 17

That’s all for today. Thank you for being here and hopefully until next time.