Data and tables

Whenever we make measurements, whether that means quantitative measurements (numbers) or qualitative ones (things like categories), we need to be able to organize and present our data. Most of the time, that means organizing into tables or charts of various kinds. The idea is always to communicate something about the data, and to do so powerfully. That is, we'd like a reader to be able to learn something meaningful from our work with as little effort as possible. We also want to strive to present data and our analytical results as honestly as we can.

Kinds of data

There are basically two kinds of data that we can organize into tables and charts: categorical data and quantitative or numerical data.

Tables

We use tables to organize data in preparation for analysis and to present it so that trends and conclusions are obvious. When organizing data, it's worthwhile to try a few table layouts so that you can judge which best highlights the trends in the data, and which are easier for a reader to decipher.

The table below shows data taken from a sample of men and women, asking them which, of baseball, basketball or football, is their favorite sport. This is categorical data organized into what's known as a two-way table. There are two sets of categories. Category 1 includes males and females. The second set of categories are the sports.

We can do some easy calculations with a table like this. These include so-called marginal distributions, which consider differences among all individuals in the table or dataset, and conditional distributions, which consider differences among one classification of individuals (like males and females).

Marginal distributions

Marginal distributions involve all individuals represented in the table, 168 in this case. We might then ask, what is the marginal distribution of the two genders shown, male (♂) and female (♀)? Because there are 84 in each group, the distribution is {50%, 50%}.

Practice 1 – Tables

A picture is worth 1,000 words

Very often in science and data presentation, a chart or "graph" can be more powerful than a wordy explanation of trends in the data. At the very least, an appropriate chart can be complementary to the text, helping a reader to catch the meaning of a description or discussion. Books (see this link, for example) have been written about presentation of data in charts.

In this section we'll look at just a few of the main chart types used to present data. You'll want to be very familiar with these chart types and how to interpret them.

Pie charts

Pie charts – they look like a sliced pie – are a common way of showing proportion among categories. Here is a pie chart, using data from 2022, that shows the broad categories of the federal budget of the United States. About 62% of all federal funds are spoken for each year including this one. These are things like salaries, social security, medicare, etc. 8% of the budget has to go to "debt service," that is, interest on borrowed money, and the rest, 30%, can be reallocated by congress and other agencies with authority to do so.

The pie chart is a nice way to display this data. It's understood that the full circle represents 100% of the budget, and we can see right away that mandatory spending takes up the largest portion of it, followed by discretionary spending and interest payments. Not only do we get a good look at which categories are bigger and smaller, but how much bigger or smaller.

There are plenty of options for pie charts. For example, we can pull a pie wedge out for emphasis, as in the chart below.

Bar charts

Bar charts are ubiquitous in nearly all fields that involve data. The reader of a bar chart compares the relative heights of the bars against a scale to make judgements about the data. They can be used just like pie charts to show proportion, but more often they're used for comparison and sometimes to show growth. Here's an example using data on the sales of personal computers in 2021.

It is a little easier to see the ordering of total sales (most to least, right-to-left) in the bar chart, but to get a sense of how much of the "pie" each complany owns, nothing really beats a pie chart. You can roll over or tap the image to see the percentages of total personal computer sales.

Histograms

A histogram is a special kind of bar chart. It represents all of the data from a given observation or experiment, breaking the independent variable into "bins," usually of equal size. The heights of the bars are proportional to the frequency of observations in each bin. Here's an example.

Here are 41 measurements — of what doesn't really matter for this example:

Here is a sample histogram from that data. It's divided into eight bins of width 0.5 units each. This is a pretty informative histogram. It shows us that the data is distributed approximately in a bell-shaped or normal curve.

Making a histogram on the TI-84

Dot plots

A quick and easy way to visualize distributions

A dot plot (sometimes "dotplot") is related to a histogram, but simpler to construct — they can just be jotted down with paper and pencil if needed. To construct a dot plot, we simply make a number line that includes the full range of the data, then begin stacking dots above it as data points accumulate.

The dot plot at right shows the results of 44 independent measurements of the length of a room. At a glance we can tell that the distribution of lengths is approximately bell-shaped or "normal, " with a mean of about 14 or so units.

We can also tell that there are some measurements that seem like outliers that we might want to do some follow-up work on. For little effort, a dot plot allows us to get a good idea of the kind and quality of the data it represents.

Stem plots

or Stem & leaf plots

Stem plots are another quick way to visualize the shape of a distribution of data and to quickly calculate a few simple statistics. Let's show how it works by example. Here's a small data set of integers:

{18, 18, 19, 23, 24, 24, 24, 25, 25, 26, 26, 26, 26, 27, 28, 29, 29, 30, 31, 31, 33, 33, 35, 39, 40}

This set has been sorted for convenience, but its not necessary. In this case, the first (tens) digit will form our "stems." These are simply 1, 2, 3 and 4, and we'll write them vertically like this

            1|
            2|
            3|
            4|
        

Now we'll use the second (ones) digits to form the "leaves,". The leaves that go with the stem of 1, for example, are 8, 8, 9. Here's how that part looks:

            1|889             Key: 1|9 = 19
            2|34445566667899
            3|0113359
            4|0