Many people approach statistics with a sense of dread because, as they explain,
I'm no good at math. Well if you're not good at math, I have good news for you.
The mathematics of statistics are very basic. All you have to do is add, subtract,
multiply, divide, square, and square root. If you know how to use those
mathematical techniques, you can do statistics. The secret of feeling
comfortable with statistics is knowing that stats is more about logic than it
is about math. Now you will need some practice to know when to use what
technique, and the order by which you will approach the calculations, but to do
the math you will have lots of tools available to help you. I encourage my
students to use a calculator, or an abacus, or a computer to do the math. I
will show you how to use formulas in Excel to simplify the math. And of course,
we'll have all the statistical software that we need to do the heavy lifting for
us. When we are working by hand however, it is important to know what math to do
when. For that we need to know the order of operations. The order of operations
can be remembered with the mnemonic PEMDAS, or the phrase, "please excuse my dear
Aunt Sally". When the formula requires a string of arithmetic computations, use
PEMDAS that stands for parentheses, exponents, multiply and divide, add, and
subtract. Always work from left to right in the equation or formula. But because
you begin solving the parentheses first, you may not actually begin at the left
edge of the formula. Start with the parentheses, then solve for exponents
(numbers that are squared) then do multiplication and division, again
working from left to right, and finally do addition and subtraction working from
left to right. If you are solving a ratio you must solve the numerator and
denominator separately before you can divide. We will look at an example of
this in a moment. When you get to your final answer, you should round. Our
general rule will be to round to two decimal points for a final answer. The
rules for rounding are five rounds up and four rounds down. What you should do
is look at the number in the third decimal place. If the number in the third
decimal place is five or greater, round up the number in the second decimal
place by adding one to it. If the number in the third decimal place is four or
less, round down, by not changing the number in the second decimal place. And
you only do this rounding at the end. Be cautious about rounding during your
calculations, because rounding introduces error, so round at the last step. An easy
way to remember the rounding rule is four or less, let it rest. Five or more,
raise the score. Along the way, I'm going to teach you some formulas to calculate
statistics by hand. The formulas will look odd at first, but the secret is just
to plug in numbers. If I gave you the formula B equals AX plus K, then told you
that a equals 4, x equals 1, and K equals 3, you could plug in those numbers to the
formula. You would get B equals four, times 11, plus 3. You would work from left
to right, doing multiplication and division first, then addition and
subtraction. Four times 11 is 44, plus 3 is 47 so B equals 47.
Sometimes the formula will have this symbol that looks like a crooked
capital E. That is the Greek letter Sigma, and it means add up these numbers. It is
read as the sum of Sigma X is the sum of X, and it means add up all of the X
values to get a total. If 4, 11, and 3 were X values, then the sum of X would be
4 plus 11 plus 3 or 18. If we added up the squared X values, that would be the
sum of x squared. Often, what we will do is calculate values, such as by summing
them up then plug them in to a formula. For example, this is the formula for
standard deviation. We will learn about this formula when we study variability.
Notice the Sigma, the x's, the exponents and the letter n. To get the values, like
the sum of X, we would add up a series of X values. To get the sum of x squared,
first we would square each of the previous X values, then add up the
squared values. The letter n stands for sample size or the total number of X
values. Once we have calculated the needed values, we would plug them into
the blank formula. The sum of X plugs in as 146. The sum of x squared is 1484 and
n is 15. Using our PEMDAS rule, we would start with the parentheses. We do have a
146 in parentheses, but we would not need to do anything more with it. Then we
would move to exponents. We have a square. We would square the 146 to get 21316,
then we move to division, still trying to clean up the numerator 21316
divided by 15 is 1421.06, then we do
subtraction. We might as well clean up the denominator. 15 minus 1 is 14. In the
numerator, 1484 minus 1421.06 is 62.93 (with rounding). Now we are ready to
divide the ratio. Remember, we can't do this division until we clean up both the
numerator and the denominator. The value is 4.4952. Take the square root, and
the final answer, rounded to two decimal places, is 2.12. When we use formulas, we
do summation to get the values that we need, then we plug them into the formula.
Then we use PEMDAS to solve the formula. Most disciplines in the health
and human services field use APA style for write-ups. As a statistics student, it
is important that you not only know how to do the statistics, but that you also
know how to write up your results. It is important for you to remember that SPSS
does not give you output hat is improper APA style. Never turn in raw
SPSS output as your homework or copy it into a paper. You must always format the
output first. As we learn each new statistical technique, I will show you
exactly what you need to know to write up your results in APA style for publication.