Statistical Methods

Unit 1: Review of Basic Mathematics

by Adam J. McKee

Using

F. J. Gravetter and L. B. Wallnau's Essentials of Statistics for the Behavioral Sciences (4th Ed.).

Symbols and Notation

Addition

Symbol: +

Example: 5 + 4 = 9

Subtraction

Symbol: -

Example: 8 – 4 = 4

Multiplication

Multiplication has several symbols:

x
( )
 ·

Examples:

3 x 9 = 27
3(9) = 27
3·3 = 27

Division

Division has several symbols:

÷
/

Examples:

15 ÷ 3 = 5
15 / 3 = 5

Remember that a fraction is also division

Greater Than

Symbol: >

Example:

20 > 10

Less Than

Symbol: <

Example

7 > 11

Not Equal to

Symbol: ≠

Example:

Order of Operations Rules

  1. Any calculation contained within parentheses is done first.
  2. Squaring (or raising to another exponent) is done second
  3. Multiplying or Dividing is done third – sets of these are done left to right
  4. Adding or Subtracting is done fourth

An Example

Evaluate the expression:

(3 + 1)2 – 4 x 7/2

Do the stuff in parentheses first to get:

(4)2 – 4 x 7/2

Do the exponents next to get:

16 – 4 x 7/2

Perform the multiplication and division to get:

16 – 14

Do the addition and subtraction to get:

16 – 14 = 2

Sequences of Operations

When you have several multiplication and divisions within an expression—work left to right.

If you don’t work left to right, you will end up with the wrong answer!

Show your answer for each stage in the four steps in the Order of Operation Rules—like the preceding example.

Proportions: Fractions, Decimals, and Percentages

A proportion is a part of a whole and can be expressed as

  • Fraction
  • Decimal
  • Percentage
    •

    Example

    Say that out of a class of 40, 3 people fail an exam.

    The proportion of the class that failed can be expressed as a fraction:

    3/40

    Or a a decimal value:

    0.075

    Or as a percentage:

    7.5%

    Denominator

    The denominator is the bottom number in a fraction. For example, in the fraction

    ¾    

    Four is the denominator because it is the bottom number.

    The denominator tells how many equal sized pieces into which the whole is split.

    The Numerator

    The value of the top number in a fraction tells us how many of the equal pieces specified in the bottom number we are using. The figure to the right illustrates the fraction ¾.

    To convert a fraction to a decimal, divide the numerator by the denominator.

    ¾ = 3 ÷ 4 = .75

    Converting a Decimal to a Percentage

    To convert a decimal to a percentage, multiply the decimal by 100 and place a percent sign after the result.

    0.75 x 100 = 75%

    Percentages as a Special Fraction

    Remember that all a percentage is a special fraction where the bottom number is always 100.

    That is why percentages are so universally used—we know that the percentage represents how many pieces of "pie" we are considering out of 100.

    67% = 67/100

    Math With Percentages

    Some calculators let you deal with fractions, but most don’t.

    The easiest way is often to convert the percentage to a decimal and plug it into a calculator.

    32% x 50 = ?

    .32 x 50 = 16

    Adding Negative Numbers

    When adding numbers that have negative signs, look at the negative sign as subtraction.

    3 + (-2) + 5 = ?

    3 – 2 + 5 = 6

    Hint: When adding a long string of numbers, it is easier to

    add up all the positive values

    add up all of the negative values

    subtract the negative sum from the positive sum

    Subtracting Negative Numbers

    Change the negative number to a positive number and add.

    4 – (-3) = ?

    4 + 3 = 7

    Multiplying and Dividing Negative Numbers

    When two numbers being multiplied or divided have the same sign, then the result is always positive.

    When two numbers have different signs, the result is always negative.

    Multiplication and Division Example

    3 x (-2) = -6

    -2 x -2 = 4

    -6 ÷ 2 = -3

    -6 / -2 = 3

    Basic Algebra: Solving Equations

    An equation is a math statement that indicates that two things are identical—they have the same value.

    We are all familiar with the general form:

    12 = 4 + 8

    1 + 1 = 2

    Equations with Variables

    Some equations will have an unknown quantity that is identified with a letter or symbol that stands in—like a question mark—for what we do not know.

    Algebra teachers love to use the letter X, but any letter or symbol will work.

    12 = 8 + x

    Equations as a Balance

    The equal sign is like the center point of a balance. The idea is to make sure each side stays the same.

    The equation is still "true" if you make sure to do the exact same thing to both sides—it stays balanced.

    Finding the Value of X

    Finding the value of the unknown quantity is called solving the equation.

    When solving for x, keep the following 2 things in mind:

    Your goal is to have the unknown value by itself on one side of the equal sign. You do this by removing all the other symbols on that side of the equal sign.

    The equation must be made to stay in balance by treating both sides exactly the same. For example, you could add 10 points to both sides of the equation and not change the value of X.

    Exponents

    Whenever a number is multiplied by itself 2 or more times, a simplified notation can be used: A number—called an exponent—is raised above the number that is multiplied by itself.

    The number to be multiplied by itself is called the base.

    The exponent tells us how many times the base is to be multiplied times itself.

    52 = 5 x 5 = 25 the second power is said to be "squared"

    23 = 2 x 2 x 2 = 8 the third power is said to be "cubed"

    24 = 2 x 2 x 2 x 2 = 16 read "2 to the fourth power"

    Special Exponent Rules

    Any number raised to the first power equals itself

    The exponent applies only to the base that is just in front of it

    If a negative number is raised to an exponent, the result will be positive for exponents that are even and negative for exponents that are odd

    If an exponent appears outside of parentheses, then any operation inside the parentheses is done first then the result is raised to the power of the exponent.

    Square Roots

    The square root of a value is the number that when multiplied by itself gives the original value.

    The square root is the reverse of an exponent.

    If we square a number, we can "undo" it by taking the square root

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