This basic course in Statistics (SOC 231) is not particularly mathematical, even though you will encounter many numbers to work with and numerous formulae to use. The major emphasis is on understanding the role of statistics in research and the logic by which we attempt to answer research questions scientifically. Yet, there is some arithmetic that you simply cannot avoid if you want to do some empirical research. This appendix is designed to give a review of basic arithmetic operations which you may have learnt sometime in the past and which you may have forgotten since then. A pocket calculator will be very useful to save time and effort and to minimize the tedium and hassle of mere computing.

1. Basic Operations: You are familiar with the four basic mathematical operations of addition, subtraction, multiplication and division, and the associated symbols of operation (namely, +, -, x, ÷). Some of you may be confused with the different ways the symbols for multiplication and division are written sometimes. For example, the multiplication of two numbers a and b can also be expressed as follows:   a x b,   ab,    a.b,   a*b,      a(b),     (a)(b).

The easiest symbol is the second one (called "adjacent symbols"), and the fourth one is used mostly on calculators and computers. The others are used mostly for clarity when an expression involves many operations. In a similar manner, the operation of division is also indicated by many symbols, the most common ones being: a ÷ b,    a/b,      a
                                                                                     b

There are a few other symbols used in conjunction with the above basic ones that you should be familiar with. The following table presents these symbols and their meaning along with an example for each.

                          Symbols                       Meaning                        Example

                               +                               Addition                          7 + 5 = 12
                               -                                Subtraction                      7 - 5 = 2
                            x, or  ( )                        Multiplication                   3 x 9 = 27, 3(9) = 27
                            ÷, or  /                          Division                           15 ÷ 3 = 5, 15/3 = 5,
                               >                                greater than                     20 > 10
                               <                                less than                          10 < 20
                              =/                                not equal to                      3 =/  6
 

2. Squares, Square Roots and Powers: Several formulae in basic statistics require us to find the square or square root of a number. To find the square of a, simply multiply the number by itself, that is a x a = a². The square is symbolized by writing 2 as a superscript. If a = 4, then a² = 4 x 4 = 16. The square root of a number is the value which, when multiplied by itself, will result in the original number. Thus, the square root of 16 is 4. The square root is symbolized as   Ö16 = 4, or in general, the square root of a is written as  Öa.

In general, we can find the n-th power of a number a, written as aⁿ implying the same number multiplied by itself n times. When n = 2, we are finding the square of the number, when n = 3, the cube of the number, and so on.

There are two specific points which you should know in this context. Any number raised to the first power equals itself. Thus, 5¹ = 5. And, any number (except zero) raised to the zero power equals 1. Thus, 5⁰ = 1.
 

3. Summation: The summation symbol has already been used in the text. And, there are some examples for operations on summation in the tutorial section below. Pay special attention to the difference between the two operations on summation that are the most confusing to students of this course, namely,  Sx_i²   and ( Sx_i )²   -  the first is the sum of squares while the second is square of the sum).
 

4. Negative Numbers: A number can be either positive or negative. Positive numbers are greater than zero, and negative numbers are less than zero. Sometimes the expression "non-negative" is used. This means that a number cannot be negative, but can take the value of zero. It is important to keep track of signs (+ or -) associated with positive and negative numbers because they affect the final outcome of every mathematical operation. The following rules are very important to remember

        1. Adding a negative number is the same as subtraction.
                e.g. 3 + (-2) + 7 is the same as 3 - 2 + 7 = 8.
        2. Subtraction changes the sign of a negative number.
                e.g. 3 - (-2) + 7 is the same as 3 + 2 + 7 = 12.
                       3 - (-2) - 7 is the same as 3 + 2 - 7 = -2.
        3. A negative number times a positive number results in a negative value.
                e.g. 3 (-4) = -12, and (3) (-4) (5) = -60.
        4. A negative number times a negative number becomes a positive value.
                e.g. (-3) (-4) = 12, and (-3) (-4) (5) = 60, but (-3) (-4) (-5) = -60.
In other  words,  an odd number of negatives stay negative, while an even number  of  negatives become positive.
        5. Division follows the same patterns as for multiplication. For example, if there is a single negative number in the calculations, the answer will be negative. If both numbers are negative, the answer will be positive.
                e.g. (-3)/(4) = -0.75,     (3)/(-4) = -0.75,      while (-3)/(-4) = 0.75.
 

5. Order of operations: Any formula is a set of directions stated in symbols. Normally, a formula involves a series of operations or calculations. Even the most complex formula can be rendered manageable if it is broken down into smaller steps. When you do a series of calculations as implied in a formula, what you need to remember in particular is the order in which the calculations are to be performed. Consider, for example, a simple computation of: 2 + 3 (4). This involves two basic operations, one is addition and the other is multiplication. If you do the addition first, then you will get the answer as : 5 (4) = 20. But, if you do the multiplication first, you will get the answer: 2 + 12 = 14. A basic question then is: Which order should be followed?

The rules of precedence say that squares and square roots should be done first, then all multiplications and divisions, and finally additions and subtractions. [You may be able to recall here how closely this rule follows the hierarchy of numbers seen in an earlier chapter.] According to this rule, the expression 7 + 3 x 3²/4 will be evaluated as: 7 + 3 x 9/4   = 7 + 27/4   = 7 + 6.75 = 13.75.

Sometimes, for convenience, parentheses are used to clarify correct procedures. The rules of precedence are overridden when an expression contains parentheses. In this context, do all calculations within parentheses first before applying the rules of precedence. For example, consider the expression  (7 + 3) -4 x 3² /(12-5).   Resolving the parenthetical expressions first, we get: 10 -4 x 3² /7. Then, applying the rules, we proceed step by step: 10 - 4 (9)/7 = 10 - 36/7 = 10 - 5.14 = 4.86. If there were no parentheses in this example, the expression would be evaluated as (by applying the rules): 7+3 - 4(9)/12 - 5   = 7 + 3 - 36/12 - 5    = 7 + 3 - 3 - 5 = 10 - 8 = 2.
 

6. Accuracy and Rounding Off: Different people work at different levels of accuracy and precision. You should keep in mind that different levels of accuracy can result in different results. If your answer to a problem is different from your friend's answer, first check the accuracy level that both of you have used in your computations. When you are working with calculators, it is not necessary to round off the numbers even with multiple operations, because your calculator normally keeps nine to thirteen digits within memory. However, when you note down a result that will be used in turn for some other calculations, then it is better to have some convention of working with as much accuracy as possible. There are two issues here: When to round off and how to round off.

When to round off is easy to answer: If a set of calculations, particularly with multiplications and divisions, becomes lengthy and requires writing down of intermediate results, then round the number to two or three decimal places.

As for how to round off, look at the digit immediately to the right of the last digit you would like to retain. Thus, if you decide to round off to 100ths (two places beyond the decimal point), look at the third digit beyond the decimal point. If that digit is greater than 5, round up. If that digit is less than 5, do not make any rounding. If that digit is 5, round up if the digit immediately to the left is even and do nothing if the digit is odd.

In the above example, we had to divide 36 by 7. This gives the result as 5.142857... I decided to round this to 100ths. Since the third digit beyond the decimal point is 2, I just retained 5.14 as the answer. Had I decided to round the number to 1000ths instead of 100ths, then I would look at the fourth digit beyond the decimal point and follow the convention of rounding off. In this case, the fourth digit happens to be 8 that is greater than 5. So, I round it off to 5.143.

As more examples, check the following rounded numbers, and say why they have been rounded so:         56.459067 = 56.46;         56.453067 = 56.45;             67.455126 = 67.45;
             67.465126 = 67.47.
 

7. Fractions, Decimals and Percentages: A proportion is a part of a whole and can be expressed as a fraction, a decimal or a percentage. For example, in a class of 60 students, 35 are women. The proportion female in the class can be expressed as a fraction: 35/60, or as a decimal value = 0.58 or as a percentage 58%. A fraction is simply a concise way of stating a proportion. "Three out of four" is equivalent to 3/4. To convert a fraction to a decimal, you divide the numerator by the denominator. To convert a decimal into a percentage, simply multiply by 100 and place a percent sign (%) after the answer.

In a fraction, the denominator indicates the number of equal pieces into which the whole is split. If the denominator has a large value, then each piece of the whole is smaller. A larger denominator therefore indicates a smaller fraction.

An equivalent fraction is one which has the same proportional value. Thus, 1/2 has the same proportional value as 2/4 or 4/8 or 10/20 or 50/100. To create equivalent fractions, multiply (or divide) both the numerator and the denominator by the same value. For example, 4/10 =12/30 because both the numerator and denominator have been multiplied by 3. And, 40/100 = 2/5 because both the numerator and the denominator have been divided by 20.

We can use the above rule to find specific equivalent fractions. For example, find the fraction that has a denominator of 200 and is equivalent to 3/4. That is, 3/4 = X/200. Since the denominator has been multiplied by 50, we multiply the numerator also by 50 to get the value of X = 150.

To multiply two fractions, first multiply the numerators and then multiply the denominators. For example, 3/4 x 5/7 = (3 x 5)/(4 x 7) = 15/28.

To divide one fraction by another, invert the second fraction and then multiply. Thus, 3/2 ÷ 2/4 = 3/2 x 4/2 = (3 x 4)/(2 x 2) = 12/4 = 3.

To add or subtract two fractions, they must have the same denominator. If two fractions already have a common denominator, you simply add or subtract only the values in the numerators. Thus, 2/5 + 1/5 = (2 + 1)/5 = 3/5.

If the two fractions do not have a common denominator, first find equivalent fractions with a common denominator before you add or subtract. A simple trick here is that the product of the two denominators will always work as a common denominator although it may not be the lowest common denominator. For example, 2/3 + 1/10 = ? The common denominator here is 3 x 10 = 30. Thus, the equivalent fraction of each is 2/3 = 20/30 and 1/10 = 3/30. Now the two fractions can be added as follows: (20 + 3)/30 = 23/30.

Like a fraction, a decimal represents part of the whole. The first decimal place to the right of the decimal point indicates how many tenths are used. The next decimal place indicates how many hundred-ths, the next how many thousand-ths, and so on. For example, 0.52 = 52/100, 0.6233 = 6233/10,000, 0.05 = 5/100, 0.0015 = 15/10,000, etc.

To add and subtract decimals, the only rule is that you must keep the decimal points in a straight vertical line. For example,
            0.57                         5.595
         + 1.226                     - 0.87
        ----------                   ----------
            1.796                        4.725

To multiply two decimal values, first multiply the two numbers ignoring the decimal points. Then you position the decimal point in the answer so that the number of digits to the right of the decimal point is equal to the total number of decimal places in the two numbers. For example, consider 4.53 x 0.257. Ignore the decimals and simply multiply 453 and 257. We get 116421. The total number of decimal points in the two numbers being multiplied is 5. Thus, place a decimal point such that after that point there will be five digits. Thus, we get the required answer as 1.16421.

For dividing two decimals, simply consider it as a fraction, then multiply both the numerator and the denominator of the fraction by 10, 100, 1000 or whatever number is necessary to remove the decimal places. Remember that doing so we are in effect creating an equivalent fraction. For example, 0.25 ÷ 1.6   = (0.25 x 100)/(1.6 x 100)   =   25/160    = 0.15625.
 

8. Solving Equations: An equation is a mathematical statement that shows the identity of two quantities. For example, 12 = 8 + 4. Often, an equation will contain an unknown quantity (or variable); for example, 12 = 8 + X. In this case, the task is to find the unknown value (X) so that the equation is balanced. In the example, X = 4 will make the equation "true" or balanced. Finding the value of X is usually called solving the equation.

To solve an equation, two points should be kept in mind:
1) Isolate the unknown value (X) on one side of the equation and all other quantities on the other   side.
2) The equation remains balanced when both sides are treated in the same way. For example, adding, subtracting, multiplying or dividing both sides of the equation by the same value, the equation is unchanged.

There are four basic types of equations. We shall see the operations involved in each case for solving the equation.
a) When X has a value added to it, as in X + 3 = 15. To isolate the X to one side, remove 3 on    the left side. This means that you subtract 3 from both sides of the equation. That is, X+3 - 3= 15 - 3. This keeps the equation unchanged. Therefore, X = 12.

b) When X has a value subtracted from it, as in X - 3 = 15. To isolate X on the left side of the equation, now add 3 to both sides. Thus, we get, X - 3 + 3 = 15 + 3, or X = 18.

c) When X is multiplied by a value, as in 4X = 24. In this case, we need to remove 4 from the left side, so we divide both sides of the equation by 4. Thus, we get X = 24/4 = 6.

d) When X is divided by a value, as in X /3 = 5. To isolate X and to balance the equation, we multiply both sides by 3. Thus, we get X = 15.

For more complex equations, we use a combination of the preceding simple operations. The main idea is always the same: Isolate X on one side of the equation. Look at this equation: (X + 3)/4 = 4. What is X then? We first multiply both sides by 4 to get (X + 3) = 16. Then, we get X = 13.
 
 

9. Tutorial:  Do the following test and check how far you have understood these basic concepts.
1.   Convert 7/25 to a decimal.
2.   Express 7/45 as a percentage.
3.   Convert 0.68 to a fraction.
4.   Express 0.0156 as a fraction.
5.   Next to each set of fractions, write "true" if they are equivalent and "false" if they are not.
           a) 41/1000 = 22/100 _____           b) 5/6 = 52/62 ______        c) 1/8 = 7/56 _____
6.   Perform the following calculations:
      a) 4/5 x 2/3 = ?           b) 7/8 ÷ 2/3 = ?         c) 3/9 + 1/6 = ?          d) 5/28 - 1/6 = ?
7.   2.51 x 0.0173 = ?
8.   3.8825 x 0.0002 = ?
9.   3.17 + 17.0132 = ?
10.  5.25 + 11.7 + 0.711 + 3.33 + 0.0313 = ?
11.  2.04 + 0.288 = ?
12.  0.365 + 0.4 = ?
13.  5 + 6 - 6 - 4 + 8 = ?
14.  9 - (-2) - 177 + 32 - (-45) + 50 = ?
15.  58 + 33 - (-89) - (-12) + (-36) - 4 + 100 = ?
16.   8 x (-37) = ?
17.  -22 ÷ (-2) = ?
18.  -2(-41) x (-3) = ?
19.   84 ÷ (-4) = ?
20.   -5² = ?
21.   -5³ = ?
22.    If a = 4 and b = 3, then a² + b⁴ = ?
23.    If a = -1 and b = 4, then (a + b)² = ?
24.    If a = -1 and b= 5, then ab² = ?
25.    18/ Ö4 = ?
26.    Ö20/5 = ?
 

Solve the following equations for X:
27.    X - 5 = -2
28.   19 = X + 3
29.   X/24 = 11
30.   -3 = 5X/3
31.    (X + 32)/5 = 2
32.    (X + 116) /3 = -8
33.    2X + 3 = -11