Statistics

If you love sports and watch them on a regular basis, then you know all about averages. A frequently used average is a pitcher’s earned run average (E.R.A.), which is calculated by adding up all of the runs that a pitcher had in the innings he pitched, and then dividing that by the number of innings. For example, if a pitcher had 47 runs in 19 innings pitched, then his E.R.A is 2.47 (47 ÷ 19 = 2.47).

Averages are easy to calculate, as they are simply the sum of values divided by the number of values used.

Here is an example:

Craig is 180 pounds, Jim is 201 pounds, and Francis is 257 pounds. What is the average weight of the three men?

180 + 201 + 257 = 638

638 / 3 = 212.67

The average weight is 212.67 pounds.

Finding the average is simple enough, but can you find the sum of the values with nothing but the average and the number of values? Well, this is also easy, since we have two of the variables we need to find the solution.

Sum of Values = Average Value × Number of Values

212.67 × 3 = 638

As long as you have the two variables – average value and number of values – to complete the equation, it is fairly easy to figure out.

When looking at a series of numbers, with only the average known, how do we solve that problem?

Here is an example:

The average of 3, 5, 6, and x is 6. What is x? Since we know that the average is 6, we figured it out backwards. There are four numbers here: 3, 5, 6, and x. The sum of these numbers divided by four is the average.

( 3 + 5 + 6 + x ) / 4 = 6

3 + 5 + 6 + x = 6 × 4 = 24

x = 24 - 3 - 5 - 6

x = 10

We can then verify this by adding the values:

3 + 5 + 6 + 10 = 24

24 / 4 = 6

Now that we have looked at these examples, what about the average rate? This is often seen in examples that are worded as: Average A per B. Here is an example to help you understand average rate:

Frank packaged 17 boxes in 3 hours and then 37 boxes in 4 hours. What is his average box per hour rate?

Well, average box per hour rate = total boxes / total hours

17 + 37 / 3 + 4

= 54 / 7

= 7.714 boxes per hour is his average.

Ø  More Practice from the GMAT® Review 13th Edition: Questions 12, 16, 30, 53, 208

Standard Deviation

Once again, if you pay attention to sports, then you already have a handle on statistics, and probably a bit of a handle on probability.

With statistics and probability, we will deal with various terms like mean, mode, median, range, and standard deviation.

Standard Deviation is the measure of a set of numbers (how much they deviate from the mean or average). The greater the spread, the higher the standard deviation.

Thankfully, you never have to calculate standard deviation on the GMAT, so we will move on. However, it does help to know what standard deviation means.

Ø  More Practice from the GMAT**®** Review 13****th Edition: Question 112

Probability

Probability, as any gambler knows, is determined for a finite number of outcomes. Obviously, the higher the probability, the greater the possibility that a desirable or undesirable outcome will occur.

We find probability by dividing the number of desired outcomes by the number of total possible outcomes:

P = D / T

(P = probability, D = desired outcome, T = total possible outcomes)

An example: John is reaching into a prize bin with 321 names on it, including seven names of people whom he knows. Therefore, what is the chance that John pulls out the name of someone he knows?

D = 7 and T = 321

P = 7 / 321

P = .022

Therefore, the probability of John selecting someone he knows is .022, or 2.2%

Calculating this form of probability is not so difficult. What is harder is calculating the probability of a certain outcome after multiple repetitions of the same or different experiment. Typically, you will find that these questions come in two different forms. One is where each event must occur in a set way, and another is where each event has different outcomes.

To determine the probability in multiple-event situations, there are two steps we should take:

•       Find out the probability of each event;

•       Multiply the probabilities together.

Here’s an example to help understand this further:

There are 15 Canadians and 15 Russians in the NHL Entry Draft. What is the probability that the first two picks of the draft will both be Canadian?

Looking at the fractions of these two picks, we have 15/30 and 15/30, which can both be reduced to 1/2 .

Once one draft pick is made, there is a 14/29 possibility that the next pick will be a Canadian player. Therefore, we multiply the two fractions:

This then converts to .24 in decimal form.

As a result, the chance of two Canadian players being selected first and second in the draft is 24%.

The previous example calculated the probability that each individual event will occur a certain way. What about situations in which different outcomes may occur?

To calculate this, we must determine the total number of possible outcomes by figuring out the number of possible outcomes for each individual event and multiplying those together. For example:

Audy sees four doors. Behind some is money, behind others is nothing. What are the chances that Audy finds money behind at least three of the four doors?

Since each door has two possible outcomes, four tries have (2 × 2 × 2 × 2) = 16 possible outcomes. We list the possible outcomes where three of the four doors have money behind them.

(****$ = Money ; E = Nothing)

1.     E $ $ $

2.     $ E $ $

3.     $ $ E $

4.     $ $ $ E

5.     $ $ $ $

Now we know that there are five outcomes where money is behind three of the four doors; therefore, the number of possible desired outcomes is 5 and the number of possible total outcomes is 16. Using what we learned a few pages back, we have the following:

5 / 16

= .3125

= 31.25%

The chance of Audy finding money is 31.25%.

Ø  More Practice from the GMAT**®** Review 13****th Edition: Questions 68, 107, 193, 215

Ratios

Ratios are the comparison of two quantities by division.

So, on to the next section!

Actually, there is a bit more to know about ratios than just that.

Typically, ratios can be written in two ways: as a fraction ( y / z ) and with a colon (y:z). You can also say “the ratio of y to z” if you want.

Usually, ratios are expressed as ( y / z ).

Whenever you are dealing with ratios, it is always best to reduce them to their lowest terms.

Craig is 26, Layla is 22.

The ratio of Craig’s age to Layla’s age is 26 / 22 (26 to 22).

Therefore the lowest terms are 13 / 11.

Ratios, which are worded in the way we have seen, “y to z,” should be turned into this format:

The ratio of 17 to 22 is 17 / 22.

When working with ratios, we often hear the word “proportion,” which is simply an equation in which two ratios are equal to one another.

Ratios are two pieces, parts, and wholes, where the whole is the entire set and the part is the portion we are taking out.

Describing a ratio is then worded as “what fraction of the whole is this part?”

Or, to more easily understand this, let’s look at this example:

Out of 37 players on the British & Irish Lions team, 12 are from Ireland.

So, the whole is 37, and the part is 12.

Therefore, the question can now be worded as:

What fraction of the British & Irish Lions team is from Ireland?

12 / 37

= 32.4 percent

Knowing what we do about ratios, let’s move on to part ratios and whole ratios. A ratio can either compare a part to a part or a part to a whole.

The ratio of trucks to cars is 2:7. As a result, what fraction of the total cars and trucks is trucks?

By adding 2 and 7, we know that for every 9 vehicles, 2 are trucks, which means that the fraction or ratio of trucks to total cars and trucks is:

Ratios with more than two terms are usually ratios of various parts, and usually these parts equal the whole, which allows us to find the part : whole ratio.

The ratio of trucks to cars to SUVs is: 2:7:3

What ratios can be determined here?

Ratio of trucks to whole = 2 / (2 + 7 + 3)

Ratio of trucks to whole = 2 / 12

1 / 6

Ratio of cars to whole = 7 / 12

Ratio of SUVs to whole = 3 / 12 = 1 / 4

Ratio of SUVs to cars and trucks = 3:9

Ratio of trucks to cars: 2:7

Ratio of cars to trucks and SUVs = 7:5

As we have seen above, ratios are always reduced to their simplest form, which can cause some confusion. Just because the ratio of trucks to total vehicles is 2:12, this does not mean that only two trucks were sold. The actual ratio may be much higher, like 20:120.

There is one more term you need to know: rates. A rate is a ratio that compares two different types of quantities, often seen in the example of miles per hour:

Henry drove 126 miles in four hours. His average rate is:

= 31.5 miles per hour

The exact formula is distance = rate × time.

That is all there is to rates. They are also more or less the same as averages, which use the formula sum = number of items × average.

Ø  More Practice from the GMAT**®** Review 13****th Edition: Questions 98, 105, 113, 125, 179