3.2: Math Refresher

Statistics is not math. It does, however, use math as a tool. This section will review some of the tools that you'll use in statistical equations, and the next section will discuss rounding decimal points and scientific notation. You learned much of this in junior high or high school, but that was a long time ago for most of us!

Order of Operations

Example \(\PageIndex{1}\)

What does “Please Excuse My Dear Aunt Sally” or “PEMDAS” mean?

Solution

It's the order in which you should do each mathematical operation.

“Please Excuse My Dear Aunt Sally” or PEMDAS tells you what kind of calculations must be completed before other calculations. However, this can still be overwhelming. Although you must follow the PEMDAS order, you can start at the top left, and go right and down, just like how you read. In fact, PEMDAS is sorta like a sentence structure but with equations.

Note

• P- You must complete the calculations within parentheses before you can use those numbers in the rest of the equation.
• E- You must complete the exponents (which will always just be squaring a number in statistics) before you can use that number in the rest of the equation.
• M- You must complete the multiplication before you can add or subtract it with other numbers in the equation. I actually think that you're supposed to divide before you multiply?
• D- You must complete the division before you can add or subtract it with other numbers in the equation. I actually think that you're supposed to divide before you multiply?
• A- Once all of the other mathematical calculations are completed, you can now add or subtract. I actually think that you're supposed to subtract before you add?
• S- Once all of the other mathematical calculations are completed, you can now add or subtract. I actually think that you're supposed to subtract before you add?

What might throw you off in actual equations that we'll use is square rooting. Square rooting is actually the last operation because you have to get down to one number.

Historical Context

The earliest evidence of written mathematics dates back to the ancient Sumerians, who built the earliest civilization in Mesopotamia, from 3000 BCE.  From around 2500 BCE onward, the Sumerians wrote multiplication tables on clay tablets and dealt with geometrical exercises and division problems.  Knowledge of Babylonian mathematics is derived from more than 400 clay tablets that were inscribed whilst the clay was moist, and baked hard in an oven or by the heat of the sun; some of these appear to be graded homework (Wikipedia)!

Figure \(\PageIndex{1}\): Babylonian mathematical tablet Plimpton 322 by author unknown from University of British Columbia. Public Domain

Notation & Symbols

Table \(\PageIndex{1}\) will cover some of the major mathematical symbols used in statistical equations. The column on the right is where you can write explanations or examples in your own words.

Table \(\PageIndex{1}\)- Common Mathematical Symbols

Symbol	Read As	Example or Description
>	"greater than"
<	"less than"
* or XY	"multiply" or "times"
()^2	"square" or "times itself"
√	"square root" or "find the number that, multiplied by itself, equals the number in the square root sign"
|#|	“absolute value” or "make it a positive number" (the lines, not the #-sign)

Summing (Sigma or \(\Sigma\))

Many statistical formulas involve summing numbers. Fortunately there is a convenient notation for expressing summation. This section covers the basics of this summation notation. Let's say we have a variable X that represents the weights (in grams) of four grapes show in Table \(\PageIndex{2}\):

Table \(\PageIndex{2}\)- Grape Weights in Grams

Grape	X
A	4.6
B	5.1
C	4.9
D	4.4

The following formula means to sum up the weights of the four grapes:

\[\displaystyle\sum X \nonumber\]

The Greek letter \(\Sigma\) indicates summation. The X is a placeholder for all of the scores. Therefore,

\[\displaystyle\sum X=X_{A}+X_{B}+X_{C}+X_{D}=4.6+5.1+4.9+4.4=19 \nonumber \]

Sums of Squares

Many formulas involve squaring numbers before they are summed. This is indicated as

\[\displaystyle\begin{array}{l}{\left(\sum X\right)^{2}= 4.6^{2}+5.1^{2}+4.9^{2}+4.4^{2}} \\ {\quad \quad=21.16+26.01+24.01+19.36=90.54}\end{array} \nonumber \]

Notice that:

\[\left(\sum \mathrm{X} \right)^{2} \neq \sum \left(\mathrm{X}^{2}\right) \nonumber \]

because the expression on the left means to sum up all the values of \(\mathrm{X}\) and then square the sum (19² = 361), whereas the expression on the right means to square the numbers and then sum the squares (90.54, as shown).

Multiplication (Products)

Some formulas involve the sum of cross products. Below are the data for variables \(\mathrm{X}\) and \(\mathrm{Y}\). The cross products (\(\mathrm{XY}\), or \(X \times Y\)) are shown in the third column. The sum of the cross products is 3 + 4 + 21 = 28.

Table \(\PageIndex{2}\)- Table of Scores and Products

Participant	\(\mathrm{X}\)	\(\mathrm{Y}\)	\(\mathrm{XY}\)
A	1	3	3
B	2	2	4
C	3	7	21
\(\sum\)	\(\sum = 1 + 2 + 3 = 6\)	\(\sum = 3 + 2 + 7 = 12\)	\(\sum = 3 + 4 + 21 = 28\)

Practice with Equations

Let's practice some simple equations to see how the order of operations and the symbols actually work.

If you were given this formula: \[\dfrac{\sum X}{N} \nonumber \]

You would need to figure out what \[\sum \nonumber \] means, and then decide if you would do what \[\sum \nonumber \] is telling you to do first or after the division.

Since \[\sum \nonumber \] means to sum, you would add up each and very number in your set. PEMDAS says that you'd do Division before Addition, but there's nothing to divide until you add everything up! Let's practice with the numbers in each column of Table \(\PageIndex{2}\).

\[\dfrac{\sum X}{N} \nonumber \]

For the \(\mathrm{X}\) column,

1. Adding them all up is: \(\sum = 1 + 2 + 3 = 6 \nonumber \)
2. Then divide by 3 since there are three numbers in the \(\mathrm{X}\) column.

\[\dfrac{\sum = 1 + 2 + 3 = 6}{3} = \dfrac{6}{3} = 2 \nonumber\]

Example \(\PageIndex{2}\)

How would we apply this information on the numbers in the \(\mathrm{Y} \nonumber \) column?

Solution

\[\dfrac{\sum = 3 + 2 + 7 = 12}{3} = \dfrac{12}{3} = 4 \nonumber\]

You can try the last column on your own!

Exercise \(\PageIndex{1}\)

How would we apply this information on the numbers in the \(\mathrm{XY} \nonumber\) column?

Answer

\[\dfrac{\sum = 3 + 4 + 21 = 28}{3} = \dfrac{28}{3} = 9.3\overline{3} \nonumber\]

The line over the second 3 is telling you that that number keeps repeating indefinitetly. Check out the next section to learn how to round decimals numbers.

The next section will refresh you on rounding decimals and scientific notation. This section is adapted from the Support Course for Elementary Statistics, if you'd like more details or help in other areas!