This article is Chapter I from the author’s book Statistics and Probability Flashcards.

Definitions

Individuals and variables

In a dataset, the individuals are the items with one or more properties, called variables. Individuals can be events, cases, objects, people, etc.

Student (individuals)Height (variables)
John190 cm
Ali175 cm
Paul165 cm
Clara160 cm

Table 1.1. example of a data set with items and variables.

Individuals and variables are called data. Table 1.1 is called a data table.

Here’s another example of a data table containing other variables:

StudentHeightWeightLikes football
John190 cm100 kgYes
Ali175 cm90 kgNo
Paul165 cm60 kgNo
Clara160 cm63 kgYes

Table 1.2. example of a data set with items and more than 1 variable category.

Variables can be categorical or quantitative. In table 1.1 there’s one quantitative variable: the height whereas in table 1.2 there are two quantitative variables (height and weight), and one categorical variable (likes football).

Quantitative variables are numerical variables: counts, percents, or numbers.

Categorical variables are non-numerical variables. Their values aren’t represented with numbers: words, not numbers.

This data set presented in table 1.1 and table 1.2 is called one-way data because we have just a single individual (item) that has one or many properties attached to it.

How to build a data table?

When you build a data table, it is important to think about whether you have more individuals or more variables.

In tables 1.1 and 1.2 the number of individuals listed was greater than the number of variables. If we have many variables but only a few individuals, it is advisable to list the individuals across the top and the variables down the left side.

JohnAli
Height190 cm175 cm
Weight90 kg75 kg
Likes footballYesNo
Likes pizzaYesYes

Table 1.3. Since the number of variables is bigger than individuals, listing the variables vertically would make the data table more appropriate than if we had tried to list all the variables horizontally.

Data visualization

Bar graphs and pie charts

Two of the simplest ways to summarize and graphically represent data are bar graphs and pie charts.

Bar graphs apply a series of rectangular bars to show absolute values or proportions for each of the data categories whereas pie charts show how substantial each data category represents as a part or proportion of the whole, by using a circular format with different-sized “slices” for different percentages of the total.

RankCountryOil production (bbl/day)
01USA15,043,000
02Saudi Arabia (OPEC)12,000,000
03Russia10,800,000
04Iraq (OPEC)4,451,516
05Iran (OPEC)3,990,956
06China3,980,650
07Canada3,662,694
08United Arab Emirates (OPEC)3,106,077
09Kuwait (OPEC)2,923,825
10Brazil2,515,459

Table 1.4. Top 10 world Oil producers (“Production of Crude Oil including Lease Condensate 2019” U.S. Energy Information Administration)

Figure 1. Bar chart – Top 10 world Oil producers (“Production of Crude Oil including Lease Condensate 2019” U.S. Energy Information Administration)

Notice that we have a list of the Oil producers (countries) across the bottom of the bar graph, with the count of the Oil production (bbl/day) up the left side.

The countries are the individuals, and the count is a quantitative variable because it represents the numeric property of each of the individuals. The bar graph is one of the best ways to represent this data because it is possible to get quickly an overview of which countries produce the most oil.

Figure 2. Pie chart – Top 10 world Oil producers (“Production of Crude Oil including Lease Condensate 2019” U.S. Energy Information Administration)

Now we can quickly see that the United States produces the most of the total oil daily, biggest than any other country, Saudi Arabia occupies second place, and Brazil is the 10th world’s biggest oil producer.

Venn diagrams

A Venn diagram is a diagram that shows all possible logical relations between a finite collection of different sets from a two-way table.

GoodCheapFastTotal
Expensive1001020
Low quality0101020
Slow delivery1010020
Best choice10101030
Other20202060
Total505050150

Table 1.5. two-way data table

Figure 3. Venn diagram

Box-and-whisker plots

Box-and-whisker plots (also called box plots) are a great method for graphically depicting groups of numerical data through their quartiles. It is very useful when you want to show the median and spread of the data (see chapter IV) at the same time.

Assuming that we have the following data set: [1, 2,2, 2, 3, 3, 4, 6, 8,8, 10, 11, 11, 16]:

Figure 4. Box-and-whisker chart

The horizontal line in the center of the box is the median of the data set, so the median of the data set represented in the chart above is 5.

The dot at the end of the bottom whisker is the minimum of the data set, and the dot at the top of the right whisker is the maximum of the data set. So in this plot, we can say that the minimum is 1, that the maximum is 16, so the range would be 16 − 1 = 15.

The IQR (interquartile range) is given by the ends of the box. Since the box above extends from 2 to 10.25, the IQR is 10.25 − 2 = 8.25.

We can summarize the information given by the Box-and-whisker chart above in the following table:

MinQ1MedianQ3Max
12510.2516