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1.8 – Math Refresher

The math that is required in this textbook is not too complicated. I would argue that as long as you can do the following, you will be okay:

  • Addition
  • Subtraction
  • Multiplication
  • Division
  • Squaring
  • Square Root

Basically, you will learn different formulas and you simply need to plug in the correct numbers in the correct places, and then do the calculations in the correct order. Making sure you insert the correct numbers in the correct places will rely on your memory and understanding of the symbols (see the previous section: 1.7 – Notation). Making sure you perform the calculations in the correct order will rely on your use of the mathematical order of operations.

PEMDAS – Order of Operations

When performing mathematical calculations, there is an order in which the calculations should occur in order to get to the correct answer. Most people learned this order of operations as the acronym, PEMDAS, or as the acrostic, “Please Excuse My Dear Aunt Sally.” The order in which calculations should take place is:

  1. P – Parentheses – Calculate anything inside parentheses first (using the following rules)
  2. E – Exponent or Square Root – Calculate any exponents
  3. M – Multiplication – Calculate any multiplication
  4. D – Division – Calculate any division
  5. A – Addition – Calculate any addition
  6. S – Subtraction – Calculate any subtraction

For example, when calculating the following formula:

[latex](2*3+4)^2 + \sqrt{(\frac{4}{2}-1)}[/latex]

We would first do everything within any parentheses. And when doing what is inside the parentheses, we follow the order of operations, so we would start with any exponents, but there aren’t any, so we next move to multiplication, which means we will multiply 2 times 3 inside the parentheses on the left…

[latex](2*3+4)^2 + \sqrt{(\frac{10}{2}-1)} = (6+4)^2 + \sqrt{(\frac{10}{2}-1)}[/latex]

Then, we would move to division, which means we will divide 10 by 2 inside the parentheses on the right…

[latex](6+4)^2 + \sqrt{(\frac{10}{2}-1)} = (6+4)^2 + \sqrt{(5-1)}[/latex]

Then we move to addition, which means we will add 6 and 4 inside the left parentheses…

[latex](6+4)^2 + \sqrt{(5-1)} = (10)^2 + \sqrt{(5-1)}[/latex]

Then we move to subtraction, which means we will subtract 5 by 1 inside the right parentheses…

[latex](10)^2 + \sqrt{(5-1)} = (10)^2 + \sqrt{(4)}[/latex]

Because there are no longer any calculations inside any parentheses, we now move to step 2, and do any exponents or square roots. We have an exponent of squaring, so we need to calculate 10-squared…

[latex](10)^2 + \sqrt{(4)} = 100 + \sqrt{(4)}[/latex]

and we have a square root, so we need to take the square root of 4…

[latex]100 + \sqrt{(4)} = 100 + 2[/latex]

Now, we are done with Exponents, so we move to Multiplication, but there are none, so we move to Division, which also has none, so then we move to addition, which means we add 100 plus 2…

[latex]100 + 2 = 102[/latex]

And because we don’t have any calculations left, we have finished!

Summation

The final mathematical refresher involves the concept of summation. Oftentimes in a mathematical calculation, we want to insert a set of scores into an equation one at a time. For example, let’s say that I want to add up the total number of years of experience that a group of workers has at their job. To do this, I want to add up (or “sum”) each person’s years of experience. So let’s say I have these results:

Person Years of Experience
A 12
B 17
C 9
D 14
E 2
F 23

So, to figure out the total years of experience for this group of workers, I simply want to sum up all their years of experience:

12 + 17 + 9 + 14 + 2 + 23 = 77

Math, though, is all about finding shortcuts by creating formulas that can be applied in any similar situation. The formula that we just used could be expressed as:

[latex]\sum\limits_{i=1}^n X_{i}[/latex]

Let’s go over the different symbols in this equation:

  • [latex]\Sigma[/latex] – Summation Symbol – It means to “repeatedly add” or “sum.” In this case, it instructs us to repeatedly add each individual’s score to one another.
  • X – Individual Score – An individual’s result (or measurement). In this case, a person’s years of experience.
  • i – Iteration – This specifies which individual score to use.
  • n – Sample Size – The number of individuals in our sample. In this case, 6 people.

So, what is it telling us to do? The only variable or calculation that is connected to the [latex]\Sigma[/latex] is the [latex]X_i[/latex], which is an individual’s score. Notice that it has a subscript of “i,” which means iteration. At the bottom of the [latex]\Sigma[/latex], we see “i = 1,” which means that we are going to start with the first iteration of x, which is [latex]X_i[/latex]. This means that we are going to cycle through all the iterations of our measurement. This means that we can think of our set of scores like this:

Person Score
Years of Experience
A X1 12
B X2 17
C X3 9
D X4 14
E X5 2
F X6 23

start with the first person’s score, which is person A, who has a score or measurement of 12. Because there is nothing else besides the [latex]x_i[/latex], we have now completed entering the first iteration of our scores, so we then include an addition, like this…

[latex]\sum\limits_{i=1}^n X_{i} = X_1 + ... = 12 + ...[/latex]

Now, we plug in the second iteration, which is the second person’s score, person B, who has a score of measurement of 17. And, again, because there is nothing else besides the [latex]x_i[/latex] connected to the [latex]\Sigma[/latex], we have now completed entering the first iteration of our scores, so we then include an addition, like this…

[latex]\sum\limits_{i=1}^n X_{i} = X_1 + X_2 + ... = 12 + 17 + ...[/latex]

And now we move to the third, fourth, fifth, etc. iterations until we reach the nth iteration, which is why there is an n on the top of the [latex]\Sigma[/latex], because it tells us when to stop. Because we have a sample size of n = 6 people, will end at the 6th iteration…

[latex]\sum\limits_{i=1}^n X_{i} = X_1 + X_2 + X_3 + X_4 + X_5 + X_6 = 12 + 17 + 9 + 14 + 2 + 23[/latex]

At this point, you might be thinking, “that seems like an extremely complicated way to add up a bunch of numbers.” And you would be mostly correct.

However, using summation when working with groups of scores allows us to write formulas that are much smaller because we don’t have to enter every iteration. Again, it’s providing us with a shortcut formula we can use for many different scenarios.

For example, let’s say we own a subscription-based home dinner catering service, where our customers pay a monthly subscription fee of $25 per week to have access to our catering, and then they can choose as many of our dinner options they would like during the week, with each dinner costing $20. Suppose that we have 5 customers, and they ordered the following dinners last week:

Person Score
Number of Dinners
Amaya X1 5
Ishita X2 4
Carlos X3 3
Pat X4 7
Keyshawn X5 2

Then, if we wanted to know how much money we took in that week, we could just tally everything up, but we could also create a formula that we could then use every week from now on. That formula would be:

[latex]\sum\limits_{i=1}^n (20X_{i}+25)[/latex]

This formula means that we will take each person, one at a time, and multiply the number of dinners they ordered that week (X) by $20, which will tell us how much they spent on dinners that week, and then we will add $25 for their weekly subscription fee. This will tell us how much we earned from each customer. Then the summation will simply add it all up to tell us the total amount of money we took in that week. It would look like this:

[latex]= (20X_1 + 25) + (20X_2 + 25) + (20X_3 + 25) + (20X_4 + 25) + (20X_5 + 25)[/latex]

Which would calculate out, after plugging in each of the [latex]X_i[/latex] values and using order of operations, as…

[latex]= (20*5 + 25) + (20*4 + 25) + (20*3 + 25) + (20*7 + 25) + (20*2 + 25)[/latex]

[latex]= (100 + 25) + (80 + 25) + (60 + 25) + (140+ 25) + (40 + 25)[/latex]

[latex]= (125) + (105) + (85) + (165) + (65)[/latex]

[latex]= 545[/latex]

How Will We Use Summation?

In this textbook, we will simplify our expression of summation by hiding the iteration indicators on the top and bottom of the Σ symbol. We are always going to do every iteration from 1 to n, so we don’t need to make that explicit. For example, if we wanted to write out the formula for our catering business, it would look like this in the simplified form:

[latex]\sum (20X+25)[/latex]

Additionally, most of our summation formulas will be pretty basic, where we are simply adding up all the scores,

[latex]\sum X[/latex]

or adding up all of the scores after they have each been squared:

[latex]\sum X^2[/latex]

Finally, we use our order of operations with summation. Above, we saw it when we were completing all of the calculations. However, we will also follow the order of operations when the summation isn’t connected to all of the calculations, such as:

[latex]\sum X + \frac{18}{6}[/latex]

Notice that the Σ symbol is only connected to the X, and then there is a mathematical calculation. This is different than our catering business formula, where the Σ is connected to the parentheses, which means that we are going to do everything involving the parentheses first before we do the summation. So now we are going to treat the summation as a separate calculation and follow PEMDAS. It can then be useful to rewrite the formula like this as a way to highlight the summation as a component:

[latex](\sum X) + \frac{18}{6}[/latex]

Order of operations starts with parentheses, so now we can compute the summation. Imagine that we have 4 scores: 28, 72, 41, and 6. We can then fill out the summation part of our formula:

[latex](\sum X) + \frac{18}{6} = (28+72+41+6) + \frac{18}{6}[/latex]

And then complete any calculations involving the parentheses, which means we will add up the score inside the parentheses:

[latex](28+72+41+6) + \frac{18}{6} = 147 + \frac{18}{6}[/latex]

After parentheses, we then move to exponents, but there are none, so we move to multiplication, but there are none, so we move to division. Thus, we divide 18 by 6:

[latex]147 + \frac{18}{6} = 147 + 3[/latex]

Then we move to addition:

[latex]147 + 3 = 150[/latex]

And we have our answer!

Note:  If summation feels a little bit difficult for you, it will be important to practice until you feel more comfortable because there will be a number of formulas in the later chapters that will use summation.
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Introduction to Statistics and Statistical Thinking Copyright © 2022 by Eric Haas is licensed under a Creative Commons Attribution 4.0 International License, except where otherwise noted.

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