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Standard Deviation for the GMAT

The dispersion of numbers or values in a list is known as the standard deviation. The standard deviation is more significant when the data is more scattered and away from the mean.

The Greek letter sigma (σ) usually represents standard deviation and is calculated using the formula.

standard-deviation

Where X represents each value in the list, x̄ represents the mean of the values in the list, and n is the number values in the list.

Example 1:

To understand how standard deviation may be tested on the GMAT, here is an example from OG 2017, Question No. 367

4, 6, 8, 10, 12, 14, 16, 18, 20, 22

List M (not shown) consists of 8 different integers, each of which is in the list shown. What is the standard deviation of the numbers in list M?

(1) The average (arithmetic mean) of the numbers in list M is equal to the average of the numbers in the list shown.

(2) List M does not contain 22.

Solution:

We are given ten different numbers, 8 of which are in List M. The question asked is: what is the standard deviation of the numbers in list M?

We know that we can find the standard deviation of a set of numbers if we know the names. Thus the question turns out to be: which eight numbers are in the list or which two numbers are not in the list?

Let’s consider Statement (1): The average (arithmetic mean) of the numbers in list M is equal to the percentage of the numbers in the record shown. It follows that the percentage of the two numbers excluded should be the same as the average of the numbers in the list displayed.

The average of the numbers in the list shown is 13, and, therefore, the sum of the two numbers excluded should be 26. You find that the numbers could be 4 and 22, or they could be 12 and 14, or even they could be any three other pairs of that sort (6 and 20, 8 and 18, or 10 and 12).

On observation, one sees that the numbers 2 and 22 are far away from the mean; whereas, 12 and 14 are close to the way. This shows that as we move on to exclude different pairs of numbers, the standard deviations obtained will be different.

In order to solve the question correctly, it is essential to look at the second Statement.

Let’s consider Statement (2): List M does not contain 22. This doesn’t tell us which eight numbers are there in List M. The information provided in Statement (2) alone is also not sufficient for answering the question.

On combining information provided in both the statements, we know for sure that the two numbers that are not in List M are 4 and 22. Hence, to answer the question, using both statements is essential.

The correct answer is C.

Note: Did you notice that in the list shown, the ten numbers are evenly spaced? If you did, it must have been easy for you to find the average and figure out the pairs of numbers that add up to 26.

Observations: Note that to solve the question, we did not have to calculate the standard deviation. It was enough to know that (i) the standard deviation depends on every number/value in a list and (ii) the standard deviation depends on the extent of dispersion of the numbers/values in the list.

 Corollary:

 (i) Standard deviation can’t be negative; the minimum possible standard deviation of the numbers/values in any list is 0 — when all the numbers/values are the same.

(ii) If every number/value in a list is increased or decreased by a fixed amount, the standard deviation remains unchanged.

Conclusion: The more closely packed are the values around the mean, the smaller the standard deviation. The higher the standard deviation, the farther away are the values from the mean.

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