Mathematics Study Guide for the TABE Test

Measurement, Data, and Probability: Data and Measurement—Part 1

The TABE 13&14 has combined the topics of measurement, data analysis, statistics, and probability into one section called “Measurement, Data, and Probability.”

Measuring is a mathematical method of assigning a number or value to the quantity or size of a particular object based on standards, calculations, formulas, and units. The distance between two points is a measure of length. The size of a flat surface or plane is measured by determining its area. The capacity of or space occupied by a three-dimensional object is measured by calculating its volume. Measurement can be done on almost anything that can be quantified, including time, weight, mass, force, speed, angles, density, temperature, and much more.

The word data can refer to a set of measurements or statistics, such as a list of heights of each student in a classroom or the amount of rain that fell each day of the month. Although in written work you will often see data as a plural (“The data are clear.”), it’s also commonly used as a singular word (“Our data is incomplete.”). Probability is a method for predicting future events based on known data.

Measuring Length

Though it’s probably obvious, length is just another word for distance, usually used to mean the longest dimension of an object. The units used are chosen depending on the size of the item being measured. Larger units like miles or kilometers are used for massive objects or long distances, and smaller units like inches or centimeters are used when the objects or distances are more compact.

Measuring Objects with Tools

Objects of all sizes can be measured, from the atoms that make up all matter to mountains and planets. In your day-to-day life, you may need to measure all kinds of things, such as a table to see if it fits in your dining room or a piece of wrapping paper to see if it’s big enough for a gift. There’s not always just one unit that you can use. For instance, when measuring a table, you could use inches or feet. So long as you are consistent with the unit you use, you should get an accurate measurement.

Technically, we can use any object as a unit to measure another. We could say that a table is \(11\) playing cards long if we lay laying the cards end to end. However, it is more accurate to use standard units of measurement that are the same everywhere.

We use different tools for measuring various objects. One of the common tools is a simple ruler, which is one foot or \(12\) inches long. You can also use a yardstick (one yard is equal to three feet) or a tape measure, which come in lengths from as short as a few feet to as long as hundreds of feet. Based on the situation, one or more measuring tools may be appropriate.

Estimating and Comparing Length

If you don’t have a tool readily available, you can still estimate lengths, either by eyeballing it or by making a comparison against something else. For instance, a US quarter is roughly an inch across, and a dollar bill is slightly longer than six inches. In situations where absolute accuracy isn’t necessary, those may be close enough.

At other times, you will need to compare lengths of objects in different units. For instance, you may need to measure an object using two separate units. Or you may be asked to determine which of two objects is larger based on their measurements given in different units. Though you don’t need to have these memorized for the TABE, it may be helpful to have a basic familiarity with some of the most common length/distance conversions:

• \(1\) foot = \(12\) inches
• \(1\) yard = \(3\) feet
• \(1\) inch = \(2.54\) centimeters
• \(1\) mile = \(5\text{,}280\) feet

Common Unit Measurements

There are two main systems of measurement: the metric system (e.g., meters, grams, liters) and the US standard system (e.g., feet, pounds, gallons). It is important to be familiar with the process of converting one unit to another. You will encounter various common measurement units on the TABE (and their abbreviations), including:

• distance/length—feet (ft), inches (in), centimeters (cm), meters (m), kilometers (km)

• area—square meters (sq m), square inches (sq in), square feet (sq ft)

• volume/capacity—cubic feet (cu ft), cubic meters (cu m), liters (L), pints ({pt), gallons (gal)

Converting Measurements

A unit can be converted to another unit within its system (from feet to inches) or to another unit in the other system (from meters to inches). The important thing to remember when solving measurement problems or math questions involving measurement units is to convert the different units into a common unit.

For example, if you are asked the total length of four poles, and the lengths are given as \(3\) m, \(6\) ft, \(40\) in, and \(5’10’’\), you should decide on one common unit for all four measurements before proceeding to addition. If the answer choices are given in feet, for instance, common sense dictates that you convert all the values to feet.

There is a process for converting measurements from one unit to another. It involves conversion factors, which are ratios like the unit rates you learned about earlier in this guide. For instance, we know there are \(12\) inches in a foot, so, depending on which unit we were trying to cancel out, we could set up a conversation factor like this:

\[\frac{12\text{ in}}{1 \text{ ft}} \quad \text{or} \quad \frac{1 \text{ ft}}{12 \text{ in}}\]

We’ll practice by converting \(45\) meters per second (m/s) to miles per hour (mph). Since the two units in the answer (miles and hours) are different from the two units that we’re starting with (meters and seconds), we will need two conversion factors. The first step is writing the value to be converted as a fraction:

\[\frac {45\text{ m}}{\text{ s}}\]

Then, we’ll multiply by the relevant conversion factors, also written as fractions:

\[\frac {45\text{ m}}{\text{ s}} \cdot \frac {1\text{ mi}}{1\text{,}609\text{ m}} \cdot \frac{3\text{,}600\text{ s}}{1\text{ hr}} =\]

Note: The conversion factors are written in such a way that the same units can be canceled. For instance, the m in the numerator will cancel out with the m in the denominator of the conversion factor.

To get the answer, multiply all numerators, and multiply all denominators, then divide the resulting numerator by the resulting denominator and affix the units that have not been canceled:

\[\require{cancel} \frac{45\cancel{\text{ m}}}{\cancel{\text{s}}} \cdot \frac{1\text{ mi}}{1\text{,}609\cancel{\text{ m}}} \cdot \frac{3\text{,}600\cancel{\text{ s}}}{1\text{ hr}} = 100.68\text{ mph}\]

Representing Data

When we measure objects or collect information about something, what we’re doing in a broad sense is collecting data. We can collect data on a wide range of things, from how tall the kids in a class are to the number of earthquakes that occur at the North Pole. Collecting data is essential to the scientific process, as is sharing and interpreting that data.

Once data has been collected, it can be organized and represented in different ways. A good first step is to put it in a table, like the one below, with labels of all the relevant categories and values:

Retrieved from: https://openstax.org/books/introductory-business-statistics example 2.7.

For a simple data set like this one, the table itself is quite easy to interpret. More complicated sets need other ways to show the data, the most common way being graphs.

Types of Data Display

You have learned about graphs already, but there are several types of graphs that are especially useful for displaying data. These types of graphs contain information displayed visually so that it’s easier to read and interpret the data.

Graphs often compare values of a specific category or measurement over another dimension, such as a period of time. To interpret a graph, read the title first, because it will summarize what the graph represents. Next, read the vertical and horizontal labels and, if there is one, the legend, which defines the symbols and values in the graph.

In these next few sections, we’ll review some of the common graphs used for displaying data.

Dot Plot

Dot plots are one of the simplest types of data displays. These types of graphs look like this:

In the above dot plot, each dot represents one watermelon (the title tells us this). The graph reveals that three watermelons weighed eight pounds, two watermelons weighed eight and a half pounds, one watermelon weighed eight and three-quarters pounds, and so on.

Note: Based on this dot plot, we know that the measured watermelons have weights that roughly cluster around nine pounds. The idea of data clusters will be discussed more thoroughly later.

Line Graph

Line graphs (or line plots) are another simple type of data display that can make comparisons quick and easy. You can also use them to predict additional data points not on the given graph.

Line graphs work a bit like coordinate grids, except instead of coordinate points, you rely on the scale of the two axes to tell you the distance between the grid boxes. Consider this example below that shows the change in temperature over a period of time split into half-hour segments:

The line increases steadily, meaning temperature is changing at a constant rate. We can assume the pattern also continues backward beyond the range of the graph. That will allow us to determine the temperature at an earlier time, such as \(5{:}30\) a.m.

Since there is a temperature rise of \(10^{\circ}\)F for every one hour, to find the temperature at \(5{:}30\) a.m., we move backward along the line. Since it’s not a full hour and the line is at a constant rate, this point falls halfway between \(0\) and \(-10\) on the temperature scale. So, the temperature at \(5{:}30\) a.m. must have been \(-5^\circ\)C.

Note: In the real world, temperatures don’t really follow a perfectly linear path, but we’re just using this as an illustrative example.

Bar Graph

Then there are bar graphs, which are another useful graph type for making comparisons within a data set. Below is an example of a bar graph with seven categories, showing the ethnicity of students in a school:

Retrieved from: https://openstax.org/books/introductory-business-statistics Figure 1.7.

Notice how easy it is to make comparisons between the groups represented in each bar? This type of graph is among the most commonly used for displaying data for the general public.

Histogram

Histograms are a special kind of bar graph that show how often things fall into different categories. Histograms look like this:

Retrieved from: https://openstax.org/books/statistics/pages/2-2-histograms-frequency-polygons-and-time-series-graphs. Figure 2.6

In this histogram above, we can see how many students in a room have read how many books. This graph introduces the concept of frequency, which means the number of times a specific data point appears. In this case, it’s students who read a specific number of books. The tallest bar, which is for reading \(2.5\) to \(3.5\) books, goes up to \(16\), meaning \(16\) students have read between \(2.5\) and \(3.5\) books. That is the number that more students have read than any other number. On the other hand, it shows that only two students have read from \(5.5\) to \(6.5\) books.

Pie Graph

A pie graph is unique among the data displays because it does not have vertical and horizontal labels. Instead, it compares values relative to other values using percent, fraction, or ratio. The pie chart below represents the data from the student ethnicity bar graph above, but in a very different visual representation:

Retrieved from: https://openstax.org/books/introductory-business-statistics Figure 1.10

Pictograph

Data can be represented in a more visually interesting way using a pictograph (or picture graph). This type of graph uses pictures to represent quantities. Pictographs can be of two types. The simplest pictograph uses one picture for every item in the data set. Here is an example of a simple pictograph:

The image shows the number of each type of pet that the children owned:

• dogs: \(5\)
• cats: \(6\)
• fish: \(3\)
• lizards: \(2\)

When larger numbers are involved, you can also use one picture to represent more than one item in the data set. We call this a scaled pictograph. Below, the same number of the same pictures is used, but notice the title and the explanation on the left side:

This graph may look a lot like the graph before it, but it does not communicate the same information at all. Obviously, there will be many more animals across a county than there are in students’ homes. Based on the legend on the side, the totals represented in each category of this graph are:

• dogs: \(100\)
• cats: \(120\)
• fish: \(60\)
• lizards: \(40\)

You can see how important it is to read every part of a graph, including all captions and other information. If you don’t, you may not come away with accurate information.

Scatter Plot

Whereas other graph types are useful for comparing data, scatter plots are especially useful in revealing trends in the data, which can help us predict future data points. The trend, also known as a pattern or tendency, is generally represented by a line going up or down.

Each dot in the scatter plot below represents one person who took the third exam and the final exam in a course:

Adapted from: https://openstax.org/books/statistics/pages/12-2-the-regression-equation. Figure 12.5

As an example, one person scored \(75\) on the third exam and \(200\) on the final. The graph gives you a rough idea of the relationship, if any, between the scores. The fact that the dots tend to go up as they go to the right suggests a trend, which is that the higher a student scored on the third exam the higher they scored on the final exam.

This graph below shows a similar scatter plot with a line added. The line is called a best-fit line because it has been drawn to come as close to each point as possible. The line represents the trend of the points a bit more clearly than just the points themselves.

Adapted from: https://openstax.org/books/statistics/pages/12-2-the-regression-equation Figure 12.7

Is the line a good fit? That’s a bit of a judgment call, but it actually hits some of the points, and is close to the others. That would make it a pretty good fit. Rarely, if ever, with actual measured data will a best-fit line hit all the points. A decent best-fit line can often be drawn by hand with a straight edge.

Note: Notice that there is one dot at \(65\) that is also fairly high on the final exam score. That goes against the trend of the other dots, so we would call that an outlier. You’ll learn more about that concept shortly.

Two-Way Table

Two-way tables are another good way to show patterns between two categories. They are sometimes called two-way frequency tables. Suppose a study was done to see who gets the flu most often in the general population. The study used men, women, boys, and girls, with one hundred members of each group. The table below shows the results:

Group	Flu
Men	17
Women	15
Boys	23
Girls	19

Based on this data, boys are more likely to get the flu than men, women, and girls. Now, after that initial study, the researchers did a follow-up study of who gets the common cold. The results were added to the original table:

Group	Flu	Cold
Men	17	20
Women	15	19
Boys	23	34
Girls	19	27

This is an example of a two-way table because it is comparing two different categories to the same subjects. Remember, frequency means how often this particular event happened. This table would be good for answering the question, “Are people likely to get colds more often than the flu?” It looks like the answer is yes. Each group got more colds than the flu.

Bivariate Data

Two-way tables are one way of displaying bivariate data, which is just another way of saying we’re comparing two types of variables, such as gender and age. Another common way to graph bivariate data is with a scatter plot, as you’ve seen.

Solving Data Problems

The bar graph below shows the numbers of students at Albert Einstein High School who bought their lunch in a certain week:

Notice that the actual number of students is \(100\) times larger than the scale number shown on the vertical axis. That is, \(2\) means \(200\), and so on. This is a type of scaled graph. It has small numbers representing much larger numbers, which can be very useful when the data comes in larger sets.

On the TABE, you will be presented with graphs like this and expected to answer questions that require interpreting the data. For instance, based on the given bar graph, approximately how many more lunches were bought on Tuesday than on Thursday?

To answer that question, look at the two bars labeled “Tuesday” and “Thursday.” On Tuesday, the bar goes up to \(3\) on the lefthand scale. On Thursday, the bar goes up to halfway between \(1\) and \(2\), so we can say \(1.5\). We just need to subtract Thursday’s value from Tuesday’s. Those aren’t the total numbers, though. We have to multiply those values by \(100\) and then we can do the subtraction:

\[300 - 150 = 150\]

Therefore, \(150\) more lunches were sold on Tuesday compared to Thursday.

Note: If the exact total isn’t obvious on the graph, you can estimate. The answer choices will likely let you know if you’re in the right ballpark.

Interpreting Data

You should be able to interpret a data table like the one below:

Retrieved from: https://openstax.org/books/introductory-business-statistics Table 1.2.

These are examples of the type of questions you may be expected to answer based on the above graph:

Are there more full-time or part-time students?

Answer: Part-time

How many more part-time students than full-time are there?

Answer: \(13\text{,}296-9\text{,}200=4\text{,}096\)

Is the ratio of full-time to part-time students greater than or less than one?

Answer: Less than one, because it is \(9\text{,}200/13\text{,}296\).

Graphing Negative Values

The graph examples we provided all deal with real-world situations where the values are positive. There are, however, some situations in which values can be negative. You can still use graphs to show the data. Suppose this graph represents average weekly temperatures over a \(12\)-week period:

How many weeks had a negative average temperature? Three. How many weeks had an average temperature over $$$30? Six. Nothing has changed about how you use the graph, you just need to be comfortable doing basic math operations with negative numbers.

Now, consider another example. Lake Mead is in Arizona and Nevada and has been shrinking for a number of years now. The graph above shows its depth (in feet) compared to its approximate median depth from the years \(2000\) to \(2020\):

With a graph like this, you could be asked, “Roughly how much did the water level change from \(2000\) to \(2016\)?”

Looking at the graph, we can see that the level went from \(60\) feet above the median to a little more than \(40\) feet below it. That’s a change of approximately \(60 - (-40) = 100\) feet.