Probability and Statistics > Order of Operations (PEMDAS)

In order to succeed in statistics, you’re going to have to be familiar with

**Order of Operations**. The “operations” are addition, multiplication, division, subtraction, exponents, and grouping. They can be performed in multiple ways, and if you don’t follow the “order”, you’re going to get the problem wrong. For example:

4 + 2 × 3

can equal 18 or 10, depending on whether you do the multiplication first or last:

4 + 2 × 3

6 × 3

18 (Wrong!)

4 + 2 × 3

4 + 6

10 (Right!)

This article shows you how to get it right every time with an acronym: **PEMDAS**.

## Order of Operations: What Does PEMDAS Mean?

The acronym PEMDAS actually stands for **P**arentheses, **E**xponents, **M**ultiplication, **D**ivision, **A**ddition, and **S**ubtraction, but you may easily remember it as **“Please Excuse My Dear Aunt Sally”. **

## Order of Operations: Example

Let’s say we have a question that asks you to solve the following equation:

1.96 + z where

You are given:

= 5,

μ= 3 × 7,

s = 5^{2},

n = 100.

**Step 1:** *Insert the numbers into the equation:*

1.96 + z = 1.96 + (**(5 – (3 × 7)) / (5 ^{2} / √100)***)

or

1.96 + z = 1.96 + ((5 – (3 × 7)) / 5

^{2}/

**10**))**

**Step 2:** *Perform the first operation: **P**arentheses*. Work from the inside out if there are multiple parentheses. The inside parentheses is 3 × 7 = 21, so:

1.96 + z = 1.96 + (**(5 – 21)** / (5^{2} / 10))

and there’s only one operation left inside that first set of parentheses, so let’s go ahead and do it:

1.96 + z = 1.96 + (**(-16)** / (5^{2} / 10))

Next, we tackle the second set of parentheses.

**Step 3:** *Perform the second operation: **E**xponents*.

5^{2} = 25, so:

1.96 + z = 1.96 + (-16 / (**25** / 10))

**Step 4:** *Perform the third operation. There’s no Multiplication in this equation, so let’s do the next one: Division*. Notice there are two division signs. You may be wondering which one you should do first. Remember that we have to evaluate everything inside parentheses

*first*, so we can’t do the first division sign on the left until we evaluate (25 / 10), so let’s do that:

25 / 10 = 2.5

1.96 + z = 1.96 + (-16/(

**2.5**))

Now we can do the second division: -16 / 2.5 = -6.4 (thanks for the correction, Dean!)

**Step 5:** 1.96 + (-6.4) = **-4.44**

**Step 6:** *Do the Subtraction.* In this case there is nothing more to do, so we’re already done!

This should hopefully jog your memory–**PEMDAS** and Order of Operations is covered in basic math classes and you should be thoroughly familiar with the concept in order to succeed with statistics.

*Notes:*

* Notice that I placed parentheses around the top and bottom parts of the equation to separate them. If I hadn’t, the equation would look like this:

z = 5 – 3 × 7 / 5^{2} / √100

which could easily be interpreted (incorrectly) as this:

z = (5 – 3 × 7 / 5^{2}) / √100

so use parentheses liberally, as a reminder to yourself which parts of the equation go together.

** √100 = 10. You don’t need PEMDAS to evaluate simple square roots like this one.

You can find another example of Order of Operations at the Ask Dr. Math FAQ.

If you prefer an online interactive environment to learn R and statistics, this free R Tutorial by Datacamp is a great way to get started. If you're are somewhat comfortable with R and are interested in going deeper into Statistics, try this Statistics with R track.

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You said, “25/10=2.5 1.96 + z= -16/(2.5)

Now we can do the second division: -16/2.5= 6.4”

This value should be -6.4 as opposed to positive shouldn’t it?

Further, moving to step 5 … 1.96 + z = – 6.4

1.96 + – 1.96 + z = – 6.4 + – 1.96

therefore, z should = – 8.36

Is this correct or am I missing something?

thank you

Hi Dean,

Thanks for catching the missing -.

As for Step 5, you’re right if you interpreted what I wrote literally with 1.96 + z = [the equation]. What I really meant was 1.96 + z = 1.96 + [the equation], which is now reflected in the post. I hope that’s cleared it up!

Stephanie

I loved doing these problems! and memorizing pemdas!, the explanation was helpful and with these typs of problems you have to do the order of operations correctly because one little mistake and your answer is gonna be way off.

I found this really helpful, just because if I haven’t worked any math for a while, I do forget the steps in which you are suppose to follow. So, it was very nice to work through these type problems to get a refresher!

It was great, very clear, thank you for your help, LUZ

nice

thanks for youre help!!!!!!!

it really helps me… :)