GMAT Statistics Fundamentals - Mean, Mode, Range, Median and Standard Deviation

Even if you fear statistics by its reputation, it is one of the easiest sections in the GMAT because a standard set of questions is asked and anyone who understands the fundamentals that I shall describe will be able to ace the questions. The three most basic topics in stats are mean, mode, and median. Usually, the GMAT will go one step further into range and standard deviation.

Mean: Mean is the average. Let’s say there are two numbers: 6 and 8. The mean would be:
(6+8)/2 =14/2 =7. If you analyze the number 7, it makes sense that it is average of 6 and 8. Using the same approach, the mean of n numbers a1,a2,a3…….an would be (a1+a2+a3…..+an)/n. If you remember this formula, you should be able to do well with mean questions. We shall discuss some of the standard questions in subsequent blogs, but for right now, remember the key formula and start doing some mean and average questions from Grockit games.

Mode: Let’s say that you are given a set of numbers, such as {4,3,7,9,9,11,10}. In order to find the mode, you have to arrange the numbers in ascending order and find the number which occurs the most. For the set in the example, the ascending order is {3,4,7,9,9,10,11} and 9 occurs the most (two times) and thus is the mode in this example. A set of numbers can also have two or more modes. For example, the set {1,2,5,5,9,9,10,11} has both 5 and 9 as the mode. The most important thing is to arrange the numbers in ascending numbers while tackling such problems.

Median: Taking the set {4,3,7,9,9,11,10} and arranging them in the ascending order {3,4,7,9,9,10,11}; the median would be the middle number. Since there are seven numbers, the fourth number would be the middle number and hence 9 is the median. Here it was a bit easier since there was an odd number of numbers in the sequence. If there is an even number of numbers in the sequence, then there are 2 middle numbers and median would be the average of those two middle numbers. Take, for example, {3,4,7,9,9,10,11,12}. In this sequence, there is one more number (12) than our previous set and thus a total of 8 numbers in the sequence. The median would be the average of the fourth number (9) and fifth number (9); the average of 9 and 9 is of course 9. Consider another sequence:{3,4,7,9,10,11,12,13}. Here there are 8 numbers and the median is 9.5 (the average of the middle numbers, 9 and 10).

Range: When a sequence is provided in the ascending order, the range of a sequence is the difference between the first and last term of the sequence. For {3,4,7,9,10,11,12,13}, the range is 13-3=10. I think this is sufficient to understand all your GMAT range questions. Once again, just remember to put all your numbers in ascending order.

Standard Deviation: Even though the formula for standard deviation may seem complicated, all there is to standard deviation is how much all of the numbers deviate from the mean. I shall  explain this in detail in another blog but for right now, remember that standard deviation is the distance of each number from the mean and is best denoted by the following formula:

Standard deviation = sqrt((x-xa)sqr )/(n-1) where n = total numbers in sequence and xa is the mean of the sequence. If you look closely at the formula, the numerator is square of x-xa, which is actually the distance of each number from the mean.