The Basic Mathematics of Statistics – What Maths are Essential when you do Statistics? (1-7)

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.