# Standard Deviation: 2 questions

For fundamental concepts of Standard Deviation, please refer to this page

Many quake in their boots when they hear that there will be Statistics covered on the GRE. They run to their college stats textbooks, dust off the cover, roll up their sleeves, and start computing the standard deviations of a list of twenty, three-digit numbers. Stop, if this in anyway describes you.

The Statistics on the GRE is much simpler, and does not test your aptitude at crunching numbers as much as it does your ability to think about Statistics. That is you will rely more on intuition than computation on statistics questions on the GRE. You shouldn’t be so worried about how many statistics questions there are on the GRE, anyway.

To illustrate take a look at the following question.

Q1. The standard deviation on a test was 12 points, and the mean was 70. If the scores fell along a normal distribution and student X scored 95 points, then student X scored higher than approximately what percent of students?

A. 2\%

B. 13\%

C. 48\%

D. 96\%

E. 98\%

Answering this question correctly requires understanding standard distribution (that refers to the distribution of scores along the familiar bell-curve). To understand how standard deviation relates to the bell-curve take a look below:

Within 1 Standard Deviation Above the Mean = 34\%

Within 1 Standard Deviation Below the Mean = 34\%

Between 1 and 2 Standard Deviations Above the Mean = 13.5\%

Between 1 and 2 Standard Deviations Below the Mean = 13.5\%

Between 2 and 3 Standard Deviations Above the Mean = 2\%

Between 2 and 3 Standard Deviations Below the Mean = 2\%

In the problem above, 34\% of students scored between 70 and 82. Likewise, 34\% of students scored between 58 and 70. This symmetry is very important, and you will notice that the bell curve is symmetrical (or even) on both sides. So, given a large enough sample size, the number of students who scored three standard deviations below the average of 70 (34) is the same as the number who scored three standard deviations above the average (106).

Returning to the actual question, we want to find how many standard deviations above the average a score 95 of points is: 95 - 70 = 25, which is a tiny bit more than two standard deviations. The question is asking for an approximation, so we can round down 25 to 24.

Looking at table above, we can see that two standard deviations above the norm is better than 34\% + 13.5\%. The trick here is to not forget to account for the left side of the bell-curve, which is 50\% (after all, half the score are on the left side and the other half on the right side—don’t forget the symmetry of the bell-curve).

That gives us a total of 50\% + 47.5 = 97.5, which approximates to (E) 98\%.

Let’s try another problem.

Q2. The reaction time of 1000 Rhesus monkeys was measured. The average time it took the monkeys to respond to a quickly moving object in their visual fields was .135 seconds, with a standard deviation of .021 seconds (assume a normal distribution). If one of the geriatric monkeys had a reaction time of .205 seconds, then that monkey’s reaction time is how many standard deviations from the mean?

A. 0 – 1 standard deviations

B. 1 – 2 standard deviations

C. 2 – 3 standard deviations

D. 3 – 4 standard deviations

E. 4 – 5 standard deviations

This is exactly the sort of daunting problem that the GRE likes to throw at you. Believe it or not, there is very little math involved. Again, you want to rely on intuition more than math.

.205 - .135 = .07. If the standard deviation is .021, we can determine the number of standard deviations the monkey’s reaction time is from the mean: \dfrac{.07}{.021}, which equals approximately 3.4. Therefore (D) – the geriatric monkey’s reaction time is 3 - 4 standard deviations from the mean.

## Takeaway

To do well on statistics questions on the GRE, you have to rely more on intuition than on number crunching. Having a strong sense of standard distribution and how standard deviation relates to standard distribution will help you immeasurably.