Tables are often used to communicate data. Tables can be filled with numerical data or written data. They can be used to compare data or when a single item has several data points associated with it.

A table is made up of a series of rows and columns and has a title that describes what is shown in the table. Rows run horizontally (left to right), while columns run vertically (top to bottom). Each row and column has a title that describes what is contained in each cell of the table.

The title of Table 1 is, “The weight of certain dogs according to their breed”; this lets us know what information is contained in the table. The row titled “Weight (lb.)” indicates the weight of the dog in pounds. The columns are titled with the name of the dog and its breed. This is a simple table that can be read quite easily. For example, Lincoln is a Dachshund that weighs [latex]18[\latex] lb., while Bailey is a Shetland Sheepdog that weighs [latex]25[\latex] lb.

    The information in the table could be written as a set of data pairs in the form (Breed, weight in pounds). So the same information could be written as {(Bassett Hound, [latex]52[/latex]), (Border Collie, [latex]37[/latex]), (Dachshund, [latex]18[/latex]), (Shetland Sheepdog, [latex]25[/latex]), (Labrador Retriever, [latex]68[/latex])}. Each data pair contains two pieces of information: the breed of the dog and its weight. The weight of the dog is dependent upon the breed. Consequently, “breed” is called the independent variable, while “weight” is called the dependent variable.

Table 2 is a bit more complex, and because of that, the data is not quite as accessible to the reader. But the data is complex as well, and if it’s going to be displayed for ease of review, this seems like a decent choice. Table 2 shows the results of a survey where participants were asked to score eight speakers (a male and a female of different English Language dialects) according to their professionalism, intelligence, education, friendliness, and sociability on a scale of 1 – 10. A score of 1 indicates the lowest rating and a score of 10, the highest rating. An average was calculated for each dialect from the responses.

To read this table, we have to be aware of exactly which row and column any given cell is in. For example, a female who is speaking standard American English is rated [latex]5.83[/latex] for sounding professional. This compares to a male who is speaking standard American English who is rated [latex]6.92[/latex]. Since [latex]6.92 > 5.83[/latex], a male who is speaking standard American English is perceived as being more professional than a woman who is speaking standard American English.

Another example from Table 2 is that a woman speaking British English is rated [latex]7.33[/latex] on the perception of being educated. This compares to a women speaking southern American English who is rated [latex]5.00[/latex] on the perception of being educated. Since [latex]7.33 > 5.00[/latex], the women speaking British English is seen as being more educated than the women speaking southern American English.

Tables help us manage more complex sets of data. A table can be used if you’re looking to display individual values, if values are being compared, or if data is going to be shown and then summarized. Tables can convey a large amount of information in an easy-to-understand way.

Frequency Tables

A table that shows the frequency of each data value in a sample is called a frequency table. It is used when each value in the data set is not unique so that the frequency of the value can be shown.

Table 3 shows the grades of thirty students after the first test in a math class. If a C or better is a passing grade, then [latex]5+9+11=25[/latex] of the 30 students were passing after the first test, and [latex]3+2=5[/latex] students were not passing. A frequency table gives an idea of how the grades are distributed. More students earned a C than any other grade. Five students earned an A; the same number as those who did not pass. A table like this is useful for teachers to get a beat on how much the students are learning.

Notice that table 3 does not give individual scores on the exam. Grade is a categorical variable, so this table only tells us the number of students in each grade category.

Grouped Frequency Tables

A table that shows the frequency of grouped data values in a sample is called a grouped frequency table. It is used when individual values in the data set are unknown but the frequency of the grouped values can be shown.

Table 4 shows the results from a question on the Spring 2021 UVU Student Survey, that asked, “How many hours per week are you currently employed?” To find the total number of students who answered this question we can add up the number of responses in each group: [latex]155+60+196+269+206+143=1029[/latex]. We can also see that working 21 to 30 hours per week is most common as that is the group with the highest frequency.

Notice that the number of hours worked is a continuous quantitative variable. The hours per week are grouped and each group does not intersect with any other group. This is very important for a grouped frequency table. The intervals of hours worked (e.g., 11-20) is called a class. No matter the endpoints of the class, there will always be a gap between one class and the next. For example, there is a gap between 10 and 11. One question that often comes up is, “What if someone works 10.4 hours per week or 30.7 hours per week? Where does that fit into the table?”. Technically, the halfway point between adjacent class endpoints is called a boundary value. For example, halfway between 10 and 11 is 10.5, so 10.5 is a boundary value. Likewise, 0.5, 20.5, 30.5, and 40.5 are all boundary values. The boundary value belongs to the interval following it. So, 10.5 belongs to the 11-20 class, and 30.5 belongs to the 31-40 class. This means that if we have someone who works 10.4 hours per week, which is less than the boundary value 10.5, that person would count in the 1-10 class. Similarly, 30.7 hours per week is greater than or equal to the boundary value 30.5, so it would be counted in the 31-40 class.  This is equivalent to rounding the number of hours worked to the nearest whole number: 10.4 rounds to 10; 30.7 rounds to 31. In summary, if a value falls between classes, we round to the number of decimal points used in the class endpoints.

Relative Frequency Tables

Often seeing raw numbers like these is not as helpful as seeing the percent of students in each group. By finding the percent of students in each group, we are determining the number of students out of 100 that would fall in each group. To find the percent of students in each group, we divide the number of students in a group by the total number of students surveyed, For example, [latex]60[/latex] students work between 1-10 hours per week. This translates to [latex]\frac{60}{1029}=0.056[/latex] when divided and rounded to 3 decimal places. To turn any number into a percent, we multiply by 100 and add a percent sign, so [latex]0.056[/latex] is equivalent to [latex]5.6%[/latex]. This means that for every 100 students, 5.6 work between 1 – 10 hours per week. Obviously we can’t have 0.6 of a student!! Percents are often decimals so we usually round to a number that makes the most sense. Since we are talking about whole numbers of students, we can round [latex]5.6%[/latex] to [latex]6%[/latex].

Table 5 shows the same data as Table 4 with the percent of total students shown for each group. When we give the percent of the total in a frequency table, it is referred to as a relative frequency table. Any time data is displayed in a relative frequency table, the total number of data points must be given.

Notice that if we add up all of the relative frequencies, we get 100%. This makes perfect sense, since 100% (=1) of the total frequencies is the total number of students.