# A certain characteristic in a large population has a distribution that is symmetric about the mean m. If 68 percent of the distribution lies within one standard deviation d of the mean, what percent of the distribution is less than m + d? GMAT Explanation, Video Solution, and Additional Practice!

A certain characteristic in a large population has a distribution that is symmetric about the mean m. If 68 percent of the distribution lies within one standard deviation d of the mean, what percent of the distribution is less than m + d?

(A) 16%
(B) 32%
(C) 48%
(D) 84%
(E) 92%

The good news is: you do not need to memorize the standard deviation formula. It is a waste of time and will not help you on the GMAT. Unluckily when I first studied for the exam I was told by my GMAT tutor to memorize that beast. Yeah. Not good. You do though need to have a sense for what standard deviation means and how to work with it in an equation with mean.

Standard deviation is a measure of spread. How spread out is the set. If the set is concentrated around the mean then it has a small spread or small standard deviation. If it is a very diffuse set with numbers scattered way out then the set will have a greater standard deviation. That's about it in terms of what you need to know conceptually.

Beyond that you need to understand how to work an equation with standard deviation and mean. Let's say x is within one standard deviation of the mean that indicates that,

x is between: Mean - standard deviation and Mean + Standard deviation.

If x is two standard deviations above the mean that indicates: x = mean + 2*standard deviation.

If x is one standard deviation below the mean then x = mean - standard deviation. You get the point.

Back to the question: A certain characteristic in a large population has a distribution that is symmetric about the mean m. If 68 percent of the distribution lies within one standard deviation d of the mean, what percent of the distribution is less than m + d?

Distribution is symmetric about the mean??? The distribution is split evenly. 50% above the mean. 50% below the mean.

And then 68 percent of the distribution is within one standard deviation. Aha! So that's what we were talking about above. If within one SD then it's 1 SD above and 1 SD below meaning the 68 percent is divided in half, 34% above and 34% below.

We want to know: what percent of the distribution is less than m + d? That translates to, what percent is +1 standard deviation or below. So we know that between the mean and +1 standard deviation is 34% of the distribution. And we know that below the mean is 50% of the distribution (symmetric) so 34% + 50%  = 84%. ## Additional GMAT Statistics Standard Deviation Practice Questions

Here's a GMAT statistics standard deviation question from the GMAT Prep Tests. It's a bit more straightforward but good practice for understanding the mechanics of mean and standard deviation: A vending machine is designed to dispense 8 ounces of coffee into a cup.

For a certain exam, a score of 58 was 2 standard deviations below mean and a score of 98 was 3 standard deviations above mean. What was the mean score for the exam?

(A) 74
(B) 76
(C) 78
(D) 80
(E) 82

7.51 8.22 7.86 8.36
8.09 7.83 8.30 8.01
7.73 8.25 7.96 8.53

A vending machine is designed to dispense 8 ounces of coffee into a cup. After a test that recorded the number of ounces of coffee in each of 1000 cups dispensed by the vending machine, the 12 listed amounts, in ounces, were selected from the data above. If the 1000 recorded amounts have a mean of 8.1 ounces and a standard deviation of 0.3 ounces, how many of the 12 listed amounts are within 1.5 standard deviation of the mean?

(A) Four
(B) Six
(C) Nine
(D) Ten
(E) Eleven

This question is from the the GMAT Prep Test 1 and 2 so if you haven't worked on those yet go ahead and skip this one for now. It's important to remember that standard deviation is plus or minus. So 1 standard deviation out would be 8.4 on the upper end and 7.8 on the lower end. 1.5 standard deviations out is 8.55 on the upper end and 7.65 on the lower end. Now you just have knock out amounts out of that range. Working backwards is faster since it becomes clear that most of the measurements are falling within 1.5 standard deviations of the mean. Let's also remember that we're not basing all of this on the 8 ounces the vending machine was designed to dispense but on the 8.1 mean that was calculated based on the 1000 cups actually dispensed. Which of the following sets has the same standard deviation as the set that contains k, k + 2, and 2k?

A. k-2, -2, 0

B. 2k, 2k + 4, 4k

C. 1, 2, k

D. k, k-1, k-2

E. k/2, k, 2

# GMAT Question of the Day - Data Sufficiency - Statistics/Standard Deviation

A factory received 19 boxes of widgets. What was the standard deviation of the numbers of widgets in the 19 boxes?

(1) For the 19 boxes of widgets, the median of the numbers of widgets was equal to the mean of the number of widgets.

(2) For the 19 boxes of widgets, the value of the range was less than or equal to the minimum number of widgets.

## GMAT Question of the Day Solution

On the GMAT you don't see standard deviation questions very often. However, for most GMAT tutoring students the topic of standard deviation is a bit of a mystery. The most common question: Do I need to know the formula for standard deviation for the GMAT? Nope. I don't know it and have never used it. I've also never seen a GMAT question that requires knowing the formula for standard deviation. Is it possible that there is a GMAT question out there that requires knowing the standard deviation formula? Maybe but I doubt it. That said you do need to understand the concept of standard deviation and what information you need to calculate it.

So what is standard deviation? It boils down to how spread out a set of numbers is. The set 1, 2, 3 has a smaller standard deviation than the set 1, 1000, 5000. How is standard deviation tested on the GMAT? Most often you'll see standard deviation in the data sufficiency. Most often the question will ask whether you can calculate the standard deviation for a certain set of numbers. There are some special GMAT standard deviation rules that you should memorize. These are situations in which you do not need to know the specific numbers in the set but you will be able to calculate the standard deviation.

If all of the numbers in the set are the same then the standard deviation is zero. Here are some clues that indicate that all of the numbers in a set are the same:

1. The range of the set is zero

2. The max of the set equals the average

3. The min of the set equals the average.

If the set is evenly spaced and you know the spacing and the number of numbers then you can calculate the standard deviation. Why? Well, any evenly spaced set with the same spacing and the same number of numbers has the same standard deviation. The standard deviation of 1, 2, 3 is the same as the standard deviation of 45, 46, 47.

In this GMAT question of the day we aren't quite so lucky as to have a cut and dry case but the information above will help a bit.

(1) The median equal to the mean indicates that we have an evenly spaced set (which could also be a set containing only one value, a spacing of 0). However we do not know the number of numbers in the set so we can't calculate the standard deviation.

(2) The range being less than the minimum doesn't provide us with enough information to solve for the standard deviation. You can give yourself some examples that would obviously create different standard deviations. You could have a set of equal numbers. All 7's for instance. The range is 0, the min is 7, and the standard deviation is 0. You could also have a set with a bunch of 15's and a bunch of 16's. In this case the min is 15, the range is 1, and the standard deviation is not 0. Two different values for standard deviation. Insufficient.

(1) + (2) Putting the statements together we have an evenly spaced set with a range that is less than the minimum. You can either have a set of all the same number or a consecutive set that starts at 19 or above (there are other options but you only need two different ones to prove insufficiency). Insufficient. Let's write out a few examples:

7, 7, 7...

101, 102, 103...

A set made of all 7's will certainly have a different standard deviation from a set made up of 100, 101, 102...119

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