$$\List B: 5, 10, 15, 20, 25$$

Quantity A |
$\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;$ | Quantity B |

$\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;$ | ||

The standard deviation of the numbers in list $A$ | $\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;$ | The standard deviation of the numbers in list $B$ |

- Quantity A is greater.
- Quantity B is greater.
- The two quantities are equal.
- The relationship cannot be determined from the information given.

So, you were trying to be a good test taker and practice for the GRE with PowerPrep online. Buuuut then you had some questions about the quant section—specifically question 1 of Section 6 of Practice Test 1. Those questions testing our knowledge of **Numerical Methods for Describing Data** can be kind of tricky, but never fear, PrepScholar has got your back!

## Survey the Question

Let’s search the problem for clues as to what it will be testing, as this will help shift our minds to think about what type of math knowledge we’ll use to solve this question. Pay attention to any words that sound math-specific and anything special about what the numbers look like, and mark them on our paper.

Looks like we’re comparing the **standard deviations** between between two sets of numbers, which we remember is covered in the **Numerical Methods for Describing Data** math skill for the GRE. Let’s keep what we’ve learned about this skill at the tip of our minds as we approach this question.

## What Do We Know?

Let’s carefully read through the question and make a list of the things that we know.

- We have lists $A$ and $B$
- Each list contains five numbers
- We want to compare the standard deviations between the two lists

## Develop a Plan

We remember learning that **standard deviations** are a measure of *dispersion* or spread in a data set. Basically, it’s how far the numbers tend to be spread out from each other. For example, the list $(8, 9, 10, 11, 12)$ has a smaller standard deviation than the list $(4, 7, 10, 14, 17)$, because the latter list has values that are spread further away from the average value of $10$.

Standard deviations actually have quite the complex equation to calculate them, as part of it requires finding the square root of the sum of the squares of differences between each number in a list and the average value of a list of numbers. Yeah, definitely sounds complicated. Luckily for us, the GRE *won’t* expect us to do this type of calculation. However, we will be expected to have a general understanding about standard deviations, particularly a couple of interesting properties pertaining to them.

One of the most commonly tested properties of standard deviations is understanding that shifting an entire set of numbers up or down, all by the same amount, does NOT change the standard deviation. For example, the standard deviation of the list $(8, 9, 10, 11, 12)$ is the EXACT same as the standard deviation of the list $(18, 19, 20, 21, 22)$. This former list had $10$ added to each value to create the latter list. But we should note that the numbers aren’t spread apart any more or less; they’re just shifted up $10$. Likewise, the list $(28, 29, 30, 31, 32)$ would also have the same exact standard deviation. So the important lesson here is **shifting a list of numbers up or down by the same value does NOT change the standard deviation**.

Let’s keep this in mind as we solve this question.

## Solve the Question

Let’s look at the two lists first.

$$\List A: 0, 5, 10, 15, 20$$

$$\List B: 5, 10, 15, 20, 25$$

Comparing the two lists, let’s look for any *coincidental numerical relationships* that we can exploit to help us answer this question. Coincidentally, it looks like list $B$ is the same as list $A$, except it has $5$ added to each of the values. Ah ha!

Even though list $B$ is shifted up $5$ from list $A$, the spread in the numbers hasn’t changed. So their standard deviations MUST be the same! **The correct answer is C, the two quantities are equal**.

## What Did We Learn

This is a common way to test knowledge about standard deviations on the GRE. It’s the kind of question where before doing it, the question might seem confusing and difficult. However, now that we’ve done it and learned that shifting a set of numbers does NOT affect the standard deviation, we’ll be ready the next time we’re tested on the relationship between standard deviation and shifting a data set up or down a certain amount.

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