Math Skills Review

Below are examples of the kinds of problems and mathematical operations you will encounter during this course. Please familiarize your self with them. This page gives you an overview; you are responsible for reviewing basic math and algebra operations.

Operations

When you do mathematical operations, remember this pneumonic device: Please Excuse My Dear Aunt Sally. This will tell you the order in which to do operations -- Parentheses, Exponentiate, Multiplication, Division, Addition, and Subtraction. For most of the operations that you will do in class, parentheses will indicate which operations you should do first.

Try some examples below. The carrot symbol (^) represents exponentiation (raising to the xth power):

(1).  ((5-4)^2)/4

(2). (((100/2)/2)/5)

(3). 3^3*2

(4). 25+192-(3+5)2

Solutions:

(1). ((1*2)^2)/4 (2^2)/4 1/4 = .25

(2). (50/2)/5 25/5 = 5

(3). 27*2 = 54

(4). 25 + 192 - (8)^2 25 + 192 - 64 217 - 64 =153

Converting Proper Fractions into Decimals

To convert a fraction into a decimal, you simply divide the bottom number into the top number (or in math speak, divide the denominator into the numerator). Figure 1 demonstrates this operation.

Figure 1. Converting 1/2 into a decimal.

You must always divide the denominator into the numerator, but if the denominator is a factor of 10 (100, 1000, 10,000 etc.), you can simply move the numerator's decimal point over to the left the same number of places that there are zeros in the denominator. For example, to convert 3/10 into a fraction, you move the decimal of the numerator's number over one place to the left, or .3 (see figure 2).

Figure 2. Converting 3/10 and 3/100 into a decimal.

Consider the following examples:

(1) 1/10

(2) 4/10

(3) 8/100

(4) 77/1000

Solutions:

(1) .1

(2) .4

(3) .08

(4) .077

Finding Proportions and Percents

To find a proportion, you must divide the whole into the part. A proportion is simply a fraction of some portion of the whole. For example, if I asked you to find the proportion of men who major in criminal justice at UMSL, you would need to know how many men (or women) there are in the program and the total number of undergraduates enrolled. So, if there are 400 criminal justice majors and 230 are men, the proportion would be 230 (part) divided by 400 (the whole) or 230/400, which equals .575. Proportions can never be lower than zero or greater than one.

To get the percent, simply multiply the proportion by 100, and you get 57.5 percent. You can find the percent though this formula:

Consider the same example: 230/400 = %/100. You would cross multiply 230 x 100 and then divide by 400, which yields 57.5%. You may also use this formula if you know the percent and whole amount to find the part. If x represents the part, the equation looks like this x/400 = 57.5%/100. Now you would cross multiply 57.5 x 400 and divide by 100. And we get 230. Mathematically it looks like this: x = (57.5 * 400) / 100.

Here are some examples. Try them on your own and check your answers.

Solutions:

(1) 57/130 = x/100 step 1: x = (57*100/130) step 2: x = 5700/130 step 3: x = 43.846%

(2) x/1080 = 27%/100 step 1: x = (27*1080)/100 step 2: x = (29160)/100 step 3: x = 291.6 or 292

(3) 27/x = 5%/100 step 1: x = (27*100)/5 step 2: x = 2700/5 step 3: x = 540

Summation

Many statistical operations will ask you to compute the sum of variables in your data. We will use x to represent the variable we are interested in and the Greek letter sigma to represent summation (think of it as a command to add all the values of your variable). Consider the class year of 5 students in enrolled in a course (4 = senior, 3=junior, 2= sophomore, and 1 = freshman). Thus is x has the following values for five students: 4 3 2 3 1 2; then the sum would be 4 + 3 + 2 + 3 + 1 + 2 = 15. In addition, you need to do other operations in addition to summing:

In the first example, you would square each value of X and then add the resulting values: 4^2 + 3^2 + 2^2 + 3^2 + 1^2 + 2^2 = 16 + 9 + 4 + 9 + 1 + 4, which equals 43.

In the second example, you would add each value of X and then square the total; thus, 4 + 3 + 2 + 3 + 1 + 2 = 15. 15^2 = 225. Easy Peasy. Just like the rules for other operations, follow the PEMDAS paradigm when you come across the summation sign.

George's Final Thoughts

When you are asked to compute values with your data, it will help you to draw a table of the data. Of course this would be burdensome if you had a variable with a large number of cases. For class, however, you will be given variables with relatively low number of cases. Make a table for your problems like this (using the example in the summation).

If you are asked to compute some value of X, you can add this as a column in your table or compute the value at the bottom. For example, if you are asked to compute the sum of X^2 you should add the following column:

Drawing these tables will help you double-check your work and help you see how you derived the final product. Don't worry, you'll see a lot more of this in class.