Subtest II: Mathematics Study Guide for the CSET Multiple Subjects Test

Statistics, Data Analysis, and Probability

Collecting, interpreting, analyzing, and representing data are the operations of statistics. Probability can be used to predict the likelihood of some event occurring.

Collecting and Representing Data

Anywhere you find things being observed, counted, and recorded, those counts or measures makeup what are known as data. The population in California, for example, has been recorded for decades. This set of data can be represented in different ways for different purposes that we will go into below.

Representing Data

Data is commonly set forth in a table and from that it can be represented in different ways. To help visualize trends, some sort of a graph (sometimes called a chart) is usually drawn. Histograms, pie graphs, bar graphs, scatter plots, and line graphs are good ones to know.

Data Concepts

Know how to find the following.

mean— The mean is what is commonly called the average. It’s the sum of all the data values divided by the number of values.

median—The median is the value that is in the center of a set of values that are arranged in numerical order. If there are an even number of values, there will be two values in the middle. The mean of those two is the median.

mode—The mode is the value that shows up most often. There can be more than one mode if there are ties for the most often used value.

range—The range of data values is the difference between the highest value and the lowest. You can think of it as how much the data is spread out.

Surveys

Many times, data is collected by surveys, and it’s good to know something about survey design. Things to consider are sample size (how many took the survey), having unbiased questions, and randomizing those being surveyed. The bigger the sample size, the better. Randomizing helps to insure that a wide cross-section of the population is included, which reduces bias. Designing the questions themselves so as to not induce bias is a whole topic of its own and way beyond the scope of this guide.

Analyzing Data

Collecting, organizing and, maybe, graphing data is a first step, but interpreting it all is just as important.

Interpretation

Know how to interpret graphs or tables, noticing upward, downward or cyclic trends. Just what do all the numbers and lines mean?

Finding Patterns

While noticing trends in the data, look for details. Does the graph show a straight line pattern, or maybe one that curves up or down? How steep is it? In a scatter plot, does the line seem to closely follow the majority of points (goodness of fit)? In a frequency plot, do you see an asymmetric curve indicating a left or right skewing of the data?

Drawing Conclusions

Drawing conclusions involves evaluating all of the above, and there are specific statistical calculations to help decide the likely correctness of your conclusion, though they are beyond the scope of this guide. Be able to identify potential sources and effects of bias, such as small sample size, poor randomizing, and poorly worded questions.

Probability

In a number of equally likely events (outcomes), some of them will satisfy a certain requirement. If we call those winners, probability is the ratio of winners to the number of possible events. The set of all possible events is called the sample space. Suppose you want to draw an ace from a deck of 52 cards. There are 52 distinct events that could occur when you draw one card, but only 4 of them will be aces.

The probability of drawing an ace in this scenario is

\(\frac{4}{52}\) or \(\frac{1}{13}\)

Events

Suppose you have a 4-sided die numbered 1, 2, 3, and 4. The event you want is to roll a 3.

The probability of that happening is \(\frac{1}{4}\). What is the probability of not rolling a 3? That event would be rolling a 1, 2, or 4, i.e. \(\frac{3}{4}\).

Those are two complementary events. Notice two things here:

1. First, the two probabilities add up to 1.

2. Second, they are mutually exclusive, meaning there is no overlap of events. If you roll a 3, you can’t have rolled a 1, 2, or 4..

Now, let’s say you have 2 blue and 2 red marbles in a bag and you pull out two, one after the other. What is the probability of getting a blue marble on the second draw?

If you pull out a blue marble first, you will have 1 blue and 2 reds left.
The probability of picking out a blue then will be \(\frac{1}{3}\).

If you pull out a red marble first, you will have 2 blues and one red left.
The probability of picking out a blue then will be \(\frac{2}{3}\).

Notice how the probability of drawing a blue changed, depending on which marble you pulled out first.That makes getting a 2 on the second draw a dependent event. Events where this isn’t true are called independent. (In this example, the events would be independent if the drawn marble was replaced in the bag after each draw.)

Expressing Probabilities

Probabilities can be written in a variety of ways, including fractions (ratios), proportions, decimals, and percents. For example:

\[\frac{1}{13}= 0.077 = 7.7\%\]

Compound Events

A compound event consists of more than one simple event. Imagine we have two four-sided dice, one red and one green. We will roll the red one, then the green one. What is the probability of rolling a sum of 5? We can use an outcome table to find out. Here is a table showing all possible outcomes of the two rolls, red across the top and green on the left side. Each outcome is a complex event.

\[\begin{array}{|c|c|c|c|c|} \hline &1 & 2 & 3 & 4 \\ \hline 1& (1,1)&(1,2)&(1,3)&(1,4)\\ \hline 2& (2,1)&(2,2)&(2,3)&(2,4) \\ \hline 3 & (3,1)&(3,2)&(3,3)&(3,4) \\ \hline 4 & (4,1)&(4,2)&(4,3)&(4,4) \\ \hline \end{array}\]

You can see there are 16 possible outcomes, so that is the sample space. How many of those will give us a sum of 5? These four: (4,1), (3,2), (2,3), and (1,4). This gives us a probability of
\(\frac{4}{16} = \frac{1}{4} = 0.25\).

It would also be good to know about using a tree diagram to do a similar thing. The tree diagram below represents two spins of a spinner that has only numbers 1, 2, and 3 on it. If you spin it twice, what is the probability of the spins adding up to 4?

Count up all the paths. From the top down, it shows a path from 1 to1, then 1 to 2, 1 to 3, 2 to 1, 2 to 2, 2 to 3, 3 to 1, 3 to 2, and last, 3 to 3. The number of paths is 9 and only 3 of them add up to 4: 1 to 3, 2 to 2, and 3 to 1.

That gives a probability of \(\frac {3}{9}\) or \(\frac{1}{3}\).