# GMAT Statistics: Standard Deviation

Many people find statistics to be a complicated subject and shy away from all the numbers and data related questions. One of the biggest causes of concern is the complicated language that is associated with statistics. Words such as 'variance' and 'standard deviation' immediately strike fear because they sound far too complex to comprehend.

Fortunately, they sound much more complicated than what they mean. Standard deviation is actually a very simple concept when it is explained in the right manner. In addition, the GMAT will rarely (if ever at all) ask you to work out any complex statistical calculations. More likely, it will test your general understanding of the concepts instead. For that reason, we can start with an overview of standard deviation and its usefulness.

**What is Standard Deviation?**

Simply put, standard deviation is a measure of how spread out are numbers in the set. It is often given the Greek letter sigma *σ*; however, on the GMAT, they will always refer to it by name rather than by letter.

A large standard deviation implies a large spread of numbers (or range) in the set. A smaller standard deviation, in contrast, implies that the numbers in the set are closely clustered.

**Why is Standard Deviation Useful?**

Standard deviation adds an extra dimension to our calculation of the mean. When we calculate the mean, it is simply an average of all the numbers in a set, but it does not tell us much about the actual numbers involved.

Consider the following example. There are two groups of 6 people who are asked to record how much exercise they did this morning. Here is their data:

Group 1:

10 min, 0 min, 10 min, 10 min, 0 min, 0 min

Group 2:

4 min, 6 min, 6 min, 4 min, 4 min, 6 min

Those are two very different sets of results. In group one, we have people who did 10 minutes of exercise mixed with people who did nothing. In group two, people spent a similar amount of time exercising, and everybody did something.

If we were to work out the mean of each set, then the calculation would simply be:

Group 1 mean:

Group 2 mean:

Both sets of data, although very different have the same mean. So, although the mean tells us the average of each group, it gives us no indication of the range of values in each set.

That is where the standard deviation plays its part. Standard deviation involves the variance (or distance) of each number from the mean. The larger the distance, the larger the standard deviation. When the numbers are closer together as in group two, then we would expect a smaller standard deviation.

In the GMAT exam, the standard deviation will be given to you, and you will not need to calculate it.

In the examples above:

Group one has a standard deviation of 5.

Group two has a standard deviation of 1.

We can see that group two's data values were much closer together and therefore gave us a much lower standard deviation. In group one the range of number was much further apart and thus gave us a higher standard deviation.

We can infer that data sets with higher standard deviation have more values at the extremes, whereas a lower standard deviation implies that the values are rather similar.

The GMAT will often give us questions that mention values being one or two standard deviations above the mean. This simply means that we need to add one or two standard deviations to the mean. So in group one above the standard deviation is 5 and coincidently the mean is also 5. One standard deviation above the mean would therefore be:

*One standard deviation above mean=mean+standard deviation*

5+5=10

One standard deviation below the mean, for the same set would be:

*One standard deviation below mean=mean-standard deviation*

5-5=0

Let us work through a quick example to test your basic understanding.

**The standard deviation of set {22, 30, 78, 45, 16, 56, 54, 35} is 18.3. How many of the values are at least one standard deviation below the mean?**

As with most standard deviation questions on the GMAT, we already have the value of standard deviation and do not need to calculate it. What we do not have is the mean, so that is what we need to calculate.

The mean is:

To calculate one standard deviation below the mean we simple use this equation:

*One standard deviation below mean=mean-standard deviation*

*One standard deviation below mean*=42-18.3=23.7

Now we simply need to count how many values in the set are less than 23.7. Values below 23.7 are 22 and 16. So there are two values that are at least one standard deviation below the mean. Quite easy and straightforward when you understand how standard deviation works.

The calculation for standard deviation is straightforward; however, as was mentioned, the GMAT test will most likely assess your overall understanding of standard deviation rather than ask you to calculate it. Here are some standard deviation properties that will help answer GMAT questions on standard deviation.

- If we increase or decrease all values in a set by the same percentage, the standard deviation will change by the same percentage.
- If we increase or decrease all values in a set by a constant value, the standard deviation WILL NOT change.
- Numbers further away from the mean alter the standard deviation the most, whilst numbers closer to the mean alter the standard deviation the least.

For some reason many people fear statistics and standard deviation GMAT questions, but these questions are very straightforward once you understand the basic principles.

Understanding is made much easier with practice, so be sure to try many examples until you are completely comfortable with analyzing statistical data and standard deviation.

Good luck!

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