First, here are some challenging practice questions:

**Q1**.

(A)

(B)

(C)

(D)

(E)

**Q2**.

I.

II.

III.

(A) I only

(B) I & II

(C) I & III

(D) II & III

(E) I, II, & III

**Q3**. Consider the following sets:

Rank those three sets from least standard deviation to greatest standard deviation.

(A)

(B)

(C)

(D)

(E)

Do these three questions make your head spin? You have found a good article to help you! Explanations to these will appear at the end.

When we are summarizing a list of numbers, typically we want to know the **center** and the **spread**

The two most typical measures of center are mean and median. Center gives us an idea of where the middle of the distribution of numbers falls.

Measures of spread give us an idea of the spacing of the numbers, how much they are “**spread**” out from each other. A relatively crude measure of spread is the range, which really only tells us about the extreme high and the extreme low, not all the data points in the middle. A more sophisticated measure of spread is the standard deviation.

Every list of numbers has a mean. Therefore, every number on the list has a deviation from the mean: that is how far that number is from the mean.

Technically, numbers below the mean have a negative deviation from the mean, and numbers above the mean have a positive deviation from the mean. In the list

Here is the technical procedure for calculating the standard deviation. We already have List #1, original data set, and List #2, deviations from the mean for each value in List #1. Now, List #3 will be the List #2 squared — the squared deviations from the mean. This is the list we average: that average is something called the “**variance**.” Then, to undo the effects of squaring, we take a square root, and that final answer is the standard deviation. The OG explains this procedure in the Math Review. If you understand and remember this, great, but chances are good that you don't need to know it in all its gory detail if you know the rough and ready facts below.

- 1) The standard deviation gives us an estimate of the size of a typical deviation from the mean. It's a way of “
**averaging**” the deviations from the mean, though it is not strictly the mean of that list. - 2) If every element in the data set is equal, they all equal the mean, each deviation from the mean is
, and the standard deviation is zero. This is the*zero*for any set to have. (That's an excellent QUANT shortcut to know!)*lowest possible standard deviation* - 3) If you add the same number to every number on the list, or if you subtract the same number from every number on a list, or if you subtract each number on the list from the same number, all of the new lists produced would have exactly the same standard deviation as the original. Addition and subtraction slides values up and down the number line, but does not change any of the spacing between the numbers.
- 4) If you multiply the numbers on a list by any values (other than
\pm{1} ), or if you raise the numbers on a list to a power, thatchanges the standard deviation. Multiplying changes the spacing on the list. In particular, if you multiply each number by*always*k , then you multiply the standard deviation by|k| . - 5) If all the numbers on the list are
, that distance is the standard deviation. For example, in the set*the same distance from the mean*\{17, 17, 17, 23, 23, 23\} , themean = 20 , and each number is exactly3 units from the mean, so the standard deviation is3 . - 6) If you do anything that “
**bunches the numbers together**”, that decrease the standard deviation. If you do anything that “**pulls the numbers further apart**”, that increase the standard deviation. - 7) If you include
numbers in the set — that is tricky, because adding in most numbers will change the mean of the entire set, which will change the deviation from the mean for each number on the list, which changes the standard deviation. If you include an additional number or a few additional numbers that are far away from the other numbers, this inclusion will wildly increase the standard deviation.*new* - 8) If you include two new numbers that are
, then that will not change the mean. If the distance of these two numbers from the mean is greater than the standard deviation, adding them will increase the standard deviation (there's a larger “average” distance from the mean). If the distance of these two numbers from the mean is less than the standard deviation, adding them will decrease the standard deviation (there's a smaller “*symmetrical around the mean***average**” distance from the mean). - 9) This is an extreme instance of the last case discussed in the previous point. If you include two new numbers
(and therefore, with a deviation from the mean of zero), of course that decreases the standard deviation, but we can say more than that.*equal to the mean*you could include in a set, the new numbers that will*Of all possible new numbers*the overall standard deviation of the set are new entries equal to the mean. That is the single most efficient way to decrease the standard deviation of a set by including new entries to the list.*most decrease*

I realize that's a great deal of information. The more you understand how standard deviation works, the more you will understand the interconnection of these “**rough and ready**” facts, which will make the entire list easier to remember.

At this point, you may want to go back to the three practice questions at the beginning of this post, and see if you have any insights.

**Q1**. This is a very tricky problem. Starting list has

A. **far away**”, which will wildly decrease the mean, increasing the deviations from the mean of almost every number on the list, and therefore increasing the standard deviation. **WRONG**

B. **far away**”, which will wildly increase the mean, increasing the deviations from the mean of almost every number on the list, and therefore increasing the standard deviation. BTW, (A) & (B) are essentially the same change — add the mean and add one number eight units from the mean. **WRONG**

C. **WRONG**

D. **RIGHT**

E. **WRONG**

**Answer = D**

**Q2**. Original set:

Notice that

Notice that

This one is very tricky, and probably is at the outer limit of what the QUANT could ever expect you to see. The spacing between the numbers in **notice**”, but once you see that, of course adding and subtraction the same number doesn't change the standard deviation.

The correct combination is I and III.

**Answer = C**

**Q3**. OK, well first of all,

Now we have to compare the standard deviations of

**Answer = D**