Mathematics: Levels E, M, D, and A Study Guide for the TABE Test

Statistics and Probability: Part 2

Percentage of Test Level Specifically Assessing Statistics and Probability (— = Assumed)

L	E	M	D	A
not tested	not tested	5%	22%	16%

Concepts in Data Display

In answering a statistical question, there is always the question of how to show the data in a way that will be useful. There are values that can be calculated to help with understanding what the data are showing.

Distribution

Suppose you measure the nose length of one thousand randomly chosen people. Some people will have longer noses and some will have shorter ones, of course. A few will have very long noses and a few will have very short noses. Most peoples’ nose length will be found near the average of all nose lengths.

If you graph the numbers of people vs. their nose lengths, you will get a graph a lot like the one below. It’s called a bell curve. The dashed line in the center represents the average nose length. The fact that the graph is so high there tells us that more people have that nose length than any other length. The shape of the graph shows that nose lengths farther from the center are less common. The graph being very low on both ends shows that very few people have really long noses or really short noses.

The point of this section is to show that measurements of a large number of a randomly distributed property, like nose length, will show a distribution across a range of values forming a bell curve when graphed.

Measures of Center

Sometimes it’s nice to be able to boil a whole data set down to one central number to represent the whole set. This is useful when you want to compare different data sets. Suppose you weigh 100 horses and 100 cows. Does a typical cow weigh more or less than a typical horse? Rather than trying to figure this out by looking at all 200 weights, it would be useful to just find the average of each animal’s weight and compare those. Each average is one number that represents the center of the whole set of weights. Other than the average, there are other ways to come up with a number that represents the central tendency of a data set.

Mean, Median, and Mode

The TABE Math test includes statistical concepts, such as finding the central value of a set of data. Computing for the central value is usually done by solving for the mean, median, or mode.

To solve for the mean of a list of numbers, add all the numbers then divide the sum by how many numbers there are in the list. The mean of \(40, \,56, \,38,\, 67,\, 50, \,40, \,55 \, 70\) is \(52\) because:

\[Mean = \frac{40 + 56 + 38 + 67 + 50 + 40 + 55 + 70}{8} = \frac {416}{8} = 52\]

To find the median, arrange the numbers in ascending or descending order. Find the middle number (or numbers) of the sorted list. That middle number is the median. If there are two numbers in the middle, as in this case where there is an even number of elements in the list, add the middle pair and divide by \(2\). The result is the median. The median of this list is:

Sorted list in ascending order: \(38, \,40,\, 40,\, 50, \,55, \,56, \,67, \,70\)
Middle pair: \(50 \text{ and } 55\)
Median: \((50 + 55) \div 2 = 52.5\)

The mode of a set of numbers is the number that occurs most frequently. In this set, \(40\) occurred twice while the rest occurred only once each. Therefore, the mode is \(40\).

Variability

Suppose a group of 10 Yorkshire Terriers are weighed and their weights are graphed as below. It would be surprising if they all weighed the same and, as you can see, they don’t. You would expect them to be fairly close to each other, and they are, ranging from 5 pounds up to 10 pounds. This idea of being close to each other or not is called variability, and there is definitely some variability in this data set. If the dogs were all the same weight, we would call that zero variability. That would be a truly rare occurrence. We can get some useful information from this graph, namely that the terriers tend to weigh from 5 to 10 pounds.

Below is a graph of the weights of small mixed-breed dogs. You can see that there is a lot more variability in these weights, seeing that they range from 1 to 20 pounds. We can get some information from this graph too: a range of dog breeds can have a big range of weights, which is not surprising.

The point of this section is to show what variability is. Sometimes it’s called the spread of the data. In general, the more variability there is, the less certain any conclusion based on those numbers will be.

Shape and Spread

Imagine seeing this graph. What’s up with dog number ten? Its weight is so far away from the other weights that something weird must have happened. Maybe the scale malfunctioned. Maybe that dog was just a puppy. This kind of weird result is called an outlier. Whatever the reason for it, we probably can’t trust it much. It adds variability (or spreads the data out more). Most likely we would just throw it out of our data set. If we left it in, it would drag the average down below what it likely should be.

Clusters

When there is an enormous population to study, researchers sometimes group this large number into smaller groups and sample those groups during a study. These groups are called clusters.

Association: Positive and Negative

Think about skin wrinkles and how we get more as we get older. The association between age and wrinkles is called a positive association because they both change in the same way. As age increases, wrinkles increase.

A negative association, as you might guess, is the opposite. For example, think of your car’s gas gauge as you drive. As your miles increase, the gas in your tank decreases.

Linear and Nonlinear Association

It’s easy to see the different kinds of association between two variables if you can see their graph. A graph of a** linear association’s data will be a straight line, and a **nonlinear association’s graph will not be straight. That’s it.

An example of a linear association is distance and time if you’re moving at a constant speed. Every mile takes the same amount of time. On the other hand, you may know that if you drop something, it actually picks up speed as it falls. Each foot that it falls takes a little less time than the one before it. That will make a curved graph.

Correlation and Causation

Just like we talked about positive and negative association, we could just as well have said positive or negative correlation. They mean essentially the same thing. The variables being studied change in ways that are connected (correlated) to each other. Causation means that a change in one quantity causes a change in the other.

The take-away from this section is that even a strong correlation doesn’t necessarily mean that one thing causes the other.

A well-known example of this is ice cream sales. They go up in the summer. You know what else goes up in the summer? Murder rates. If you graph ice cream sales against the murder rate, you get a very strong positive correlation. They both go up. Does eating ice cream cause murders? Of course not. Even though there is a strong correlation, there is no causation.

Summarizing Numerical Data

By looking at the various tools for understanding that we have been talking about, you should be able to come up with a summary of the important characteristics of your data set. You will want to include things like the number of observations that were made, their units, and the nature of the things being studied.

Describe the measures of central tendency, such as the mean, the median, and the mode, and report any data points that seem not to fit the overall pattern, like outliers. It’s also good to note the spread of the data. Does it appear to be closely grouped around a central tendency, or widely scattered?

Bivariate Measurement Data

Bivariate means that there are two variables being compared, such as ice cream sales and outdoor temperatures. We’ve seen scatter plots in an earlier section (“Types of Graphs”). Be able to make a scatter plot graph and interpret it. Look for any pattern or tendency as you would with any statistical graph. Is there a pattern such as going upward from left to right, or going downward? Is there any repetitive pattern? If you see any general tendency, do the data points cluster closely around that tendency, or are they widely scattered?

Probability

Probability is a mathematical concept of predicting or calculating the chance that an event will happen or occur. It is often given by a decimal number between \(0\) and \(1\), where \(0\) means the event won’t happen, and \(1\) means it definitely will happen. It can also be given as a ratio, such as \(\frac{3}{5}\), or three out of five.

It can be calculated using the following formula:

\[\text{probability} = \frac {\text{number of ways a certain event can happen}} {\text{total number of possible outcomes}}\]

A clown has prepared \(10\) tricks for the party, including pulling a bunny out of a hat. If the time allows him to perform only one trick, what is the likelihood that he will perform that bunny-from-the-hat trick?

Probability = \(\frac{1}{10}\)

This number and counting concept is expanded to include combinations and permutations.

Combination is a concept of counting the possible number of ways a thing can be done or the pairing of things where order of elements does not matter. Combinations can be calculated by using this formula:

\[C = \frac {n!}{r!(n-r)!}\]

where n is the number of things to choose from r is the number of choices

Example:

A salad bar has \(9\) kinds of fruits. Customers may mix \(4\) kinds of fruits per serving. How many possible fruit combinations can customers come up with?

\[C = \frac{9!}{4!(9-4)!} = \frac {9!}{4!5!} = \frac {9 \cdot 8 \cdot 7 \cdot 6}{4 \cdot 3 \cdot 2 \cdot 1} = \frac {3024}{24} = 126\]

Permutation is a similar concept except that the order of the things matters. Permutation is computed using these formulas:

\(P = n^r\) (Repetition allowed)

\(P = \frac{n!}{n-r)!}\) (Repetition not allowed)

Example: How many three-letter combinations, without repetition, can be formed from the letters a, b, c, d, e?

Here order matters; abc is different from bac. Also, we can’t repeat letters such as aab.

We will use the second formula.

\[P = \frac{5!}{3!} = \frac{5 \cdot 4 \cdot 3 \cdot 2 \cdot 1}{2 \cdot 1} = 60\]

Probability Model

Be able to come up with a reasonable idea of what the probability is of a certain simple event happening. For example, suppose there are three red marbles, four green marbles, and nine blue marbles in a cloth bag. If you reach in and randomly pull a marble out, what is the probability that it will be green?

Recall that probability is:

\[\frac {\text{number of favorable outcomes}} {\text{total number of possible outcomes}}\]

The bag has \(16\) marbles in it, so there are only \(16\) outcomes possible when you pull one out. Green is the favorable outcome and of the \(16\) possible outcomes, there are \(4\) that will be green. That gives us a reasonable guess for the probability.

\[\frac{4}{16} = \frac{1}{4} = 0.25\]

If we were to actually try this with marbles, would we get one green marble every time we picked four marbles? If not, can you think of any reason why not?

For one thing, probability only tells us what is most likely to happen. It tells us that in the very long run we will get really close to getting a green for every four marbles we pick. In the short run, it’s unpredictable.

For another thing, there might be something weird happening, like maybe green marbles are lighter than the other ones so they tend to rise to the top. Far fetched? Maybe, but try to think of things like this to try to explain why real observations don’t agree with a calculated probability.

Probability of Compound Events

A compound event means an event that is made of two or more simple events. First, here is an example of a simple event. If you spin a game spinner that has the numbers \(1, 2,\) and \(3\), what is the probability that you will get a \(2\)? There are three possible numbers that could come up, and only one of them is a \(2\), so the probability is \(\frac{1}{3}\).

Now, if you spin the spinner twice, what are the odds that you will get a \(2\) both times? Each time you spin, the probability of getting a \(2\) is \(\frac{1}{3}\). Since you rolled twice, it turns out that the probability of getting two \(2\)s is the product of the individual probabilities for each roll.

\[\frac{1}{3} \cdot \frac{1}{3} = \frac{1}{9}\]

The diagram below is called a tree diagram and shows this in another way.

This shows every possible way two spins could happen. There are nine different complete paths and each path shows one result of spinning twice. The top path shows a \(1\) on the first spin and a \(1\) on the second spin: \(1,1\). The second path down shows \(1,2\). The third path shows \(1,3\), and so on. The other six paths show \(2,1; 2,2; 2,3; 3,1; 3,2; 3,3\). Out of nine possible paths, only one gives us the number \(2\) twice. That backs up our first method of getting a probability of \(\frac{1}{9}\).

Probabilities can get complicated very quickly and this is just a small sample.

Plotting on the Real Number Line

In some cases, graphs of real situations can go into negative territory. 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.

Lake Mead is in Arizona and Nevada and has been shrinking for a number of years now. The graph above shows its depth compared to its approximate median depth from the year \(2000\) to \(2020\). You need to be able to interpret graphs like this that involve negative numbers. Typical questions could be like these:

What year was the water level about \(10\) feet above the median? \(2004\).

Roughly how much did the water level change from \(2000\) to \(2016\)?

The level went from \(60\) ft above the median to a little more than \(40\) ft below it. That’s a change of a little over \(100\) ft.