GRE Special Quantitative Topic: Standard Deviation
While not frequently tested on the GRE, standard deviation is one of those scary sounding math concepts – like probability or combinations/permutations – that always seems to get GRE test-takers’ hearts beating fast. “Wait a minute,” you may say to yourself when presented with a standard deviation problem. “I thought all of the math on the GRE is basic algebra, geometry, and arithmetic. Doesn’t standard deviation have something to do with statistics?”
Fear not, brave GRE test-taker! As scary as they may sound, standard deviation problems are actually pretty easy to wrap your head around once you take the time to sit down, understand what they’re all about, and try a few practice problems. In fact, the main reason these questions tend to throw GRE test-takers for a loop is precisely because they show up so rarely on the test. Developing even a small measure of comfort with these question types will help ease the anxiety you may have about seeing “gotcha” questions on GRE test day.
First of all, let’s talk briefly about what standard deviation means. Essentially, standard deviation measures how far from the mean different numbers are within a set. In other words, in a set of numbers, how far do all of the unique values in the set “deviate” from the average? The farther away from each other the different numbers tend to be, the larger the standard deviation; the closer they are to each other, the smaller the standard deviation. Below, I’ve listed two sets of numbers. According to the basic definition above, which set would you say has the greater standard deviation?
Set 1: 4, 17, 28, 53, 88
Set 2: 15, 23, 28, 32, 57
Just by looking at them, you can probably tell that Set 1 has a larger standard deviation, because the numbers in that set are more “spread out” than the numbers in Set 2. That’s the gist of standard deviation.
On the test, standard deviation questions are sometimes, but not always, simply a matter of comparing two sets. In addition, it’s possible that you’ll be asked to find a more specific value for the standard deviation of a set. Here’s how we can do that.
- Find the mean of the set of the numbers.
- For each individual number in the set, calculate the difference between that number and the mean.
- Square each of those differences.
- Average together those new squared values.
- Calculate the square root of the average of the squared values.
Let’s try this out with the values from Set 2 above. We’ll go through step by step to find the standard deviation of that set.
- The mean, or average, of the values in Set 2 is 31.
- The difference between 31 and each value in the set is, respectively: 16, 8, 3, 1, and 26.
- Squaring each of those values results in: 256, 64, 9, 1, and 676.
- Add those values together to get 1,006, then divide by 5: the average is 201.2.
- Calculating the square root of that value results in the standard deviation of the set: 14.18.
Now, while it’s good to know how to find the precise standard deviation of a set, the GRE doesn’t always ask you to do so. Often, you will be asked to simply compare two sets of numbers and determine which has a larger or smaller standard deviation. However, knowing the steps to find the exact standard deviation of a set will occasionally be necessary to answer a high difficulty question.
Ready for a little practice? Here’s another set of numbers (we’ll call it Set 3): 34, 36, 43, 66, 71. Using those values, answer these two questions:
- Which has a larger standard deviation, Set 2 or Set 3?
- What IS the standard deviation of Set 3?
Can you answer Question 1 by just looking at the two sets? Give it a guess, then check to see if you’re right by finding the answer to Question 2, then comparing that value to the standard deviation of Set 2. Leave your answers in the comments!