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Descriptive Statistics


GMAT VarianceVariance plays a major role in interpreting data in statistics. The most common application of variance is in polls. For opinion polls, the data gathering agencies cannot invest in collecting data from the entire population. They set criteria for sampling the population based on ethnicity, income group, regions, education level, salary and religion, so that the population is completely represented by the samples.

With samples, the data can vary from the mean by a large extent. Variance is used to find the variation of the data from the mean. Interestingly, the variance exaggerates the spread, and thus standard deviation was introduced.

Standard Deviation = Square Root (Variance)

Variance Definition

Variance is represented by (Sigma)^2
. In simple mathematical terms, it is the average of the square difference between each data from the mean.

Steps to Calculate Variance

1) Find the mean of the data set
2) For each data, find the difference between the number and the mean
3) Square the resulting number and
...


GMAT Weighted AverageAverage Question is an important topic in GMAT problem solving and data sufficiency. Let us start with the Basics.
 
Average (Arithmetic Mean)


Average of n numbers a1, a2, a3, a4, a5....an

(An) = (a1+a2+.....an) /n

Example: Find the average of 34, 56, 75 and 83

Answer

a1 = 34
a2= 56
a3=75
a4 = 83

Total Number of Elements (n) = 4

Average (An) = (a1+a2+a3+a4)/n
= (34 + 56 + 75 + 83)/4 = 62

Shortcut to Remember:  An x n = a1+a2+.......an

Let us straight away apply this shortcut

Q) The average of four numbers is 20. If one of the numbers is removed, the average of the remaining numbers is 15. What number was removed?

(A) 10
(B) 15
(C) 30
(D) 35
(E) 45

Answer:

Four Numbers = a1, a2, a3, a4

n=4

An = 20

An x n = a1 + a2 + a3 + a4

20...


Set A consists of integers -9, 8, 3, 10, and J; Set B consists of integers -2, 5, 0, 7, -6, and T. If R is the median of Set A and W is the mode of set B, and R^W is a factor of 34, what is the value of T if J is negative?

(A) -2
(B) 0
(C) 1
(D) 2
(E) 5

Solution

This problem demonstrates a helpful note about statistics problems – quite often the key to solving a stats problem is something other than stats: number properties, divisibility, algebra, etc. The statistics nature of these problems is often just a way to make a simpler problem look more difficult.

Here, the phrase “factor of 34? should stand out to you, as there are only four factors of 34, so you can narrow down the possibilities pretty quickly to 1, 2, 17, and 34. And because the number in question must be an exponential term that becomes a factor of 34, it’s even more limited: 2, 17, and 34 can only be created by one integer exponent – “itself” to the first power.

The base of that exponent is going to be the median of Set A, and because we know that the median of Set A will be 3 (a negative term for variable J means that 3 will be the middle term), the question becomes that much clearer. 3^W can only be a factor of 34 if it’s set equal to 1, and the only way to do that is for W to be 0. REMEMBER: anything to the power of...


GMAT StatisticsEven 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...


What is the mode of the set if the average of the set {15, 30, 60, k, 20, 40} is equal to 37.5?

A. 20
B. 25
C. 35
D. 40
E. 60

Answer

The first step is to determine the value of k. We determine the value by taking the sum of the set elements, dividing by 6 and setting them equal to 37.5.

(165 + k)/6 = 37.5.

165 + k = 225.

k = 60.

Since 60 is the most frequently appearing number in the set, 60 must be mode

Correct Answer - Choice E




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