SAT Math 1 & 2 Subject Tests
Chapter 10
Statistics and Sets
Math Subject Test questions about statistics and sets deal with the arrangements and combinations of large groups, probability, overlapping groups, and statistical measures like mean, median, and mode. On each Math Subject Test, only about one question in 20 will involve statistics and sets, so spend time on this chapter only after you’ve mastered the more essential material in earlier chapters.
DEFINITIONS
Here are some terms dealing with sets and statistics that appear on the Math Subject Tests. Make sure you’re familiar with them. If the meaning of any of these vocabulary words keeps slipping your mind, add that word to your flash cards.
Mean |
An average—also called an arithmetic mean. |
Median |
The middle value in a list of numbers when the numbers are arranged in order. When there is an even number of values in the list, the median is the average of the two middle values. |
Mode |
The value that occurs most often in a list. |
Range |
The result when you subtract the smallest value from the largest value in a list. |
Standard Deviation |
A measure of the variation of the values in a list. |
Combination |
A grouping of distinct objects in which order is not important. |
Permutation |
An arrangement of distinct objects in a definite order. |
WORKING WITH STATISTICS
The science of statistics is all about working with large groups of numbers and trying to see patterns and trends in those numbers. To look at those numbers in different ways, statisticians use a variety of mathematical tools. And, just to keep you guessing, ETS tests your knowledge of several of these tools. The three most commonly tested statistical measures are the mean, the median, and the mode.
Mean
The mean (or “arithmetic mean”) of a set is simply its average value—the sum of all its elements divided by the number of elements. To calculate averages on the Math Subject Tests, use the average wheel we discussed in Chapter 3.
Median
The median is the middle value of a set. To find a set median, you must first put all of its elements in order. If the set has an odd number of elements, then there will be one value in the exact middle, which is the median value. If the set has an even number of elements, then there will be two middle values; the median value is the average of these two middle values.
Mode
The mode of a set is simply the value that occurs most often in that list.
Many statistics questions require you to work with all three of these measures. The calculations involved are usually not very difficult. However, the real challenge of these questions is simply understanding these terms and knowing how to use them. Similarly, there are two more statistical terms that you may be required to know for certain questions—range and standard deviation.
Range
The range of a set is the positive difference between the set’s highest and lowest values. You can also think of the range as the distance on the number line from the lowest to the highest value in the set. Remember that distances are always positive.
Stem-and-Leaf Plots and Boxplots
ETS may ask you about a stem-and-leaf plot or a boxplot once in a while. The good news is that the questions are usually pretty simple if you understand the basic concepts.
Suppose that a class earned these quiz scores: 65, 70, 70, 78, 80, 81, 84, 86, 89, 89, 93, 93, 93, 98, 100.
A stem-and-leaf plot would show the data like this:
The tens digits are listed vertically, and then each ones digit is listed horizontally. For example, the row that reads “7| 0 0 8” means “70, 70, 78”. This forms a sort of bar graph, but we have actual numbers instead of bars.
A boxplot shows the data broken into quartiles. Using our fifteen quiz scores, this would be the boxplot:
Each part of the boxplot represents 25% of the data. Here, 78 is the first quartile, or Q_{1}, 86 is the median (sometimes called the second quartile, or Q_{2}), and 93 is the third quartile, or Q_{3}. The only other thing you need to know is that the interquartile range is the range of the middle 50%: Q_{3}− Q_{1}, or the width of the box. In this example, that’s 93 − 78 = 35.
Standard Deviation
The standard deviation of a set is a measure of the set’s variation from its mean. A set composed of 10 identical values (having a range of 0) could have the same mean as a set with widely scattered values. The first list would have a much smaller standard deviation than the second.
How Far From the SD?
Some questions may ask
you to figure out how
many deviations above or
below the mean a number
is. In order to figure that
out, find the difference
between the mean and
the number you’re dealing
with. Then divide it by the
standard deviation.
Standard deviation comes up very infrequently on the Math Level 2. Computing a standard deviation is a long, annoying process that you will not be asked to endure. (Aren’t you glad?) Just remember that the standard deviation is a measure of how far the typical value in a set is from the set’s average. The bigger the standard deviation, the more widely dispersed the values are. The smaller the standard deviation, the more closely grouped the values in a set are around the mean. On some questions, you need to know how many standard deviations above or below the mean a certain value is. On other questions, drawing a rough sketch of the data will be enough.
DRILL
Try the following practice questions using these statistical definitions. The answers to these drills can be found in Chapter 12.
25. List M contains ten elements whose sum is zero. Which of the following statements must be true?
I. The mean of the elements in M is zero.
II. The median of the elements in M is zero.
III. The mode of the elements in M is zero.
(A) None
(B) I only
(C) I and II only
(D) II and III only
(E) I, II, and III
42. The subjects in a research study are divided into Group A and Group B. Both groups are given the same test. The mean score in Group A is greater than that in Group B, but the standard deviation of scores in Group A is less than that in Group B. Which of the following must be true?
(A) The range of scores in Group A is equal to the range of scores in Group B.
(B) The median score in Group A is greater than the median score in Group B.
(C) The scores are more closely grouped about the mean in Group A than in Group B.
(D) The highest score in Group A is greater than the highest score in Group B.
(E) The number of subjects in Group A is less than the number of subjects in Group B.
PROBABILITY
Probability is a mathematical expression of the likelihood of an event. The basis of probability is simple. The likelihood of any event is discussed in terms of all of the possible outcomes. To express the probability of a given event, x, you would count the number of possible outcomes, count the number of outcomes that give you what you want, and arrange them in a fraction, like this:
Probability of x =
Not!
You can find the probability
that something WILL
NOT happen by subtracting
the probability that it
WILL happen from 1. For
example, if the weatherperson
tells you that there
is a 0.3 probability of rain
today, then there must be
a 0.7 probability that it
won’t rain, because
1 − 0.3 = 0.7.
Every probability is a fraction. The largest a probability can be is 1. A probability of 1 indicates total certainty. The smallest a probability can be is 0, meaning that it’s something that cannot happen. Most probabilities you’ll be asked to find on the Math Subject Tests are fractions between 0 and 1. Figuring out the probability of any single event is usually simple. When you flip a coin, there are only two possible outcomes, heads and tails. The probability of getting heads is therefore 1 out of 2, or . When you roll a die, there are six possible outcomes, 1 through 6; the odds of getting a 6 is therefore . The odds of getting an even result when rolling a die are since there are three even results in six possible outcomes. Here’s a typical example of a simple probability question.
11. A bag contains 7 blue marbles and 14 marbles that are not blue. If one marble is drawn at random from the bag, what is the probability that the marble is blue?
(A)
(B)
(C)
(D)
(E)
Here’s How to Crack It
Here, there are 21 marbles in the bag, 7 of which are blue. The probability that a marble chosen at random would be blue is therefore , or . The correct answer is (B).
Probability of Multiple Events
Some advanced probability questions require you to calculate the probability of more than one event. Here’s a typical example:
23. If a fair coin is flipped three times, what is the probability that the result will be tails exactly twice?
(A)
(B)
(C)
(D)
(E)
Here’s How to Crack It
When the number of possibilities involved is small enough, the easiest and safest way to do a probability question like this is to write out all of the possibilities and count the ones that give you what you want. Here are all the possible outcomes of flipping a coin three times.
heads, heads, heads |
tails, tails, tails |
heads, heads, tails |
tails, tails, heads |
eads, tails, heads |
tails, heads, tails |
heads, tails, tails |
tails, heads, heads |
As you can see by counting, only three of the eight possible outcomes produce tails exactly twice. The chance of getting exactly two tails is therefore . The correct answer is (C).
Sometimes, however, you’ll be asked to calculate probabilities for multiple events when there are too many outcomes to write out easily. Consider, for example, this variation on an earlier question.
41. A bag contains 7 blue marbles and 14 marbles that are not blue. What is the probability that the first three marbles drawn at random from this bag will be blue?
(A)
(B)
(C)
(D)
(E)
Here’s How to Crack It
Three random drawings from a bag of 21 objects produce a huge number of possible outcomes. It’s not practical to write them all out. To calculate the likelihood of three events combined, you need to take advantage of a basic rule of probability.
The probability of multiple events occurring together is the product of the probabilities of the events occurring individually.
In order to calculate the probability of a series of events, calculate the odds of each event happening separately and multiply them together. This is especially important in processes like drawings, because each event affects the odds of following events. This is how you’d calculate the probability of those three marble drawings.
The first drawing is just like the simple question you did earlier; there are 7 blue marbles out of 21 total—a probability of .
For the second drawing, the numbers are different. There are now 6 blue marbles out of a total of 20, making the probability of drawing another blue marble , or .
For the third drawing, there are now 5 blue marbles remaining out of a total of 19. The odds of getting a blue marble this time are .
To calculate the odds of getting blue marbles on the first three random drawings, just multiply these numbers together.
The odds of getting three blue marbles is therefore , and the answer is (D). This can also be expressed as a decimal, as 0.026. ETS often asks for answers in decimal form on the Math Subject Tests, just to make sure you haven’t forgotten how to push the little buttons on your calculator. Just bear with them.
Drill
Try the following practice questions about probability. The answers to these drills can be found in Chapter 12.
13. If the probability that it will rain is , then what is the probability that it will NOT rain?
(A)
(B)
(C)
(D)
(E) It cannot be determined from the information given.
16. In an experiment, it is found that the probability that a released bee will land on a painted target is . It is also found that when a bee lands on the target, the probability that the bee will attempt to sting the target is . In this experiment, what is the probability that a released bee will land on the target and attempt to sting it?
(A)
(B)
(C)
(D)
(E)
20. The chart above shows the cookie production at MunchCo for three days. What is the probability that a cookie made on one of these three days will be burned?
(A)
(B)
(C)
(D)
(E) It cannot be determined from the information given.
24. If two six-sided dice are rolled, each having faces numbered 1 to 6, what is the probability that the product of the two numbers rolled will be odd?
(A)
(B)
(C)
(D)
(E)
44. In a basketball-shooting contest, if the probability that Heather will make a basket on any given attempt is , then what is the probability that she will make at least one basket in three attempts?
(A)
(B)
(C)
(D) 1
(E)
PERMUTATIONS, COMBINATIONS, AND FACTORIALS
Questions about permutations, combinations, and factorials are fairly rare on the Math Subject Tests and more common on the Math Level 2 than on the Math Level 1. As is the case with many of the odds and ends of precalculus, questions about permutations and combinations are rarely mathematically difficult; they just test your understanding of the concepts and ability to work with them. Both permutations and combinations are simply ways of counting groups of numbers.
Simple Permutations
A permutation is an arrangement of objects of a definite order. The simplest sort of permutation question might ask you how many different arrangements are possible for 6 different chairs in a row, or how many different 4-letter arrangements of the letters in the word FUEL are possible. Both of these simple questions can be answered with the same technique.
Just draw a row of boxes corresponding to the positions you have to fill. In the case of the chairs, there are six positions, one for each chair. You would make a sketch like the following:
Then, in each box, write the number of objects available to put into that box. Keep in mind that objects put into previous boxes are no longer available. For the chair-arranging example, there would be 6 chairs available for the first box; only 5 left for the second box; 4 for the third, and so on until only one chair remained to be put into the last position. Finally, just multiply the numbers in the boxes together, and the product will be the number of possible arrangements, or permutations.
There are 720 possible permutations of a group of 6 chairs. This number can also be written as “6!”—that’s not a display of enthusiasm—the exclamation point means factorial. The number is read “six factorial,” and it means 6 • 5 • 4 • 3 • 2 • 1, which equals 720. A factorial is simply the product of a series of integers counting down to 1 from the specified number. For example, the number 70! means 70 • 69 • 68 … 3 • 2 • 1 .
That’s a Fact!
On a TI-83, you can
calculate a factorial by
hitting the MATH key and
then scrolling over to PRB.
Most scientific calculators
have a factorial feature,
but not all of them do.
The number of possible arrangements of any group with n members is simply n!. In this way, the number of possible arrangements of the letters in FUEL is 4!, because there are 4 letters in the group. That means 4 • 3 • 2 • 1 arrangements, or 24. If you sketched 4 boxes for the 4 letter positions and filled in the appropriate numbers, that’s exactly what you’d get.
Advanced Permutations
Permutations get a little trickier when you work with smaller arrangements. For example, what if you were asked how many 2-letter arrangements could be made from the letters in FUEL? It’s just a modification of the original counting procedure. Sketch 2 boxes for the 2 positions. Then fill in the number of letters available for each position. As before, there are 4 letters available for the first space, and 3 for the second. The only difference is that you’re done after two spaces.
= 12
As you did before, multiply the numbers in the boxes together to get the total number of arrangements. You should find that there are 12 possible 2-letter arrangements from the letters in FUEL.
That’s all there is to permutations. The box-counting procedure is the safest way to approach them. Just sketch the number of positions available, and fill in the number of objects available for each position, from first to last—then multiply those numbers together.
On to Combinations
Combinations differ from permutations in just one way. In combinations, order doesn’t matter. A permutation question might ask you to form different numbers from a set of digits. Order would certainly matter in that case, because 135 is very different from 513. Similarly, a question about seating arrangements would be a permutation question, because the word “arrangements” tells you that order is important. So questions that ask about “schedules” or “orderings” require you to calculate the number of permutations.
Combination questions, on the other hand, deal with groupings in which order isn’t important. Combination questions often deal with the selection of committees. Josh, Lisa, Andy isn’t any different from Andy, Lisa, Josh, as far as committees go. In the same way, a question about the number of different 3-topping pizzas you could make from a 10-topping list would be a combination question, because the order in which the toppings are put on is irrelevant. Questions that refer to “teams” or “pairs” are therefore asking about the number of possible combinations.
Which One to Use?
Combination and permutation
questions can be
very similar in appearance.
Always ask yourself carefully
whether sequence
is important in a certain
question before you
proceed.
Calculating Combinations
Calculating combinations is surprisingly easy. All you have to do is throw out duplicate answers that count as separate permutations, but not as separate combinations. For example, let’s make a full-fledged combination question out of that pizza example.
pepperoni |
sausage |
meatballs |
anchovies |
green peppers |
onion |
mushrooms |
garlic |
tomato |
broccoli |
36. If a pizza must have 3 toppings chosen from the list above, and no topping may be used more than once on a given pizza, how many different kinds of pizza can be made?
(A) 720
(B) 360
(C) 120
(D) 90
(E) 30
Here’s How to Crack It
To calculate the number of possible combinations, start by figuring out the number of possible permutations.
= 720
That tells you that there are 720 possible 3-topping permutations that can be made from a list of 10 toppings. You’re not done yet, though. Because this is a list of permutations, it contains many arrangements that duplicate the same group of elements in different orders. For example, those 720 permutations would include these:
pepperoni, mushrooms, onion |
mushrooms, onion, pepperoni |
pepperoni, onion, mushrooms |
onion, pepperoni, mushrooms |
mushrooms, pepperoni, onion |
onion, mushrooms, pepperoni |
All six of these listings are different permutations of the same group. In fact, for every 3-topping combination, there will be 6 different permutations. You’ve got to divide 720 by 6 to get the true number of combinations, which is 120. The correct answer is (C).
So, how do you know what number to divide permutations by to get combinations? It’s simple. For the 3-position question above, we divided by 6, which is 3!. That’s all there is to it. To calculate a number of possible combinations, calculate the possible permutations first, and divide that number by the number of positions, factorial. Take a look at one more:
29. How many different 4-person teams can be made from a roster of 9 players?
(A) 3,024
(B) 1,512
(C) 378
(D) 254
(E) 126
Here’s How to Crack It
This is definitely a combination question. Start by sketching 4 boxes for the 4 team positions.
Then fill in the number of possible contestants for each position, and multiply them together. This gives you the number of possible permutations.
= 3,024
Finally, divide this number by 4! for the 4 positions you’re working with. This gets rid of different permutations of identical groups. You divide 3,024 by 24 and get the number of possible combinations, 126. The correct answer is (E).
Factorials
On the Math Level 2, ETS occasionally asks you to calculate a factorial itself. If you try to do a factorial question in your head, you’re likely to fall into one of ETS’s traps. Use your calculator and be careful.
18.
(A)
(B)
(C)
(D) 5
(E)
For this question, just use your calculator. 5! = 120 and 6! = 720, so you have = . The answer is (C). It’s supposed to be easy, but don’t try to simplify this in your head. Joe Bloggs might choose any of the answer choices here.
Factoring the Factorial
Sometimes numbers will be too bulky for your calculator, or you’ll realize there’s a faster way. You can factor factorials. Let’s take another look at question 18. Notice that the denominator is 6! − 5!. 6! is the same as 6 • 5!, which means you can factor 5! out of the denominator and you’re left with 5!(6 − 1). Now you can cancel and you end up with .
DRILL
Try the following practice questions about permutations, combinations, and factorials. The answers to these drills can be found in Chapter 12.
27. How many different 4-student committees can be chosen from a panel of 12 students?
(A) 236
(B) 495
(C) 1,980
(D) 11,880
(E) 20,736
31.
(A) x!(x + 2)!
(B) (x^{2} − 1)!
(C) x(x − 2)!
(D) x!(x − 1)
(E) x(x + 1)!
32. In how many different orders may 6 books be placed on a shelf?
(A) 36
(B) 216
(C) 480
(D) 720
(E) 46,656
45. How many 7-person committees consisting of 4 females and 3 males may be assembled from a pool of 17 females and 12 males?
(A) 523,600
(B) 1,560,780
(C) 1.26 × 10^{7}
(D) 7.54 × 10^{7}
(E) 7.87 × 10^{9}
GROUP QUESTIONS
Group questions are a very specific type of counting problem. They don’t come up frequently on the Math Subject Tests, but when they do come up, they’re easy pickings if you’re prepared for them. If you’re not, they can be a bit confusing. Here’s a sample group question.
34. At Bedlam Music School, 64 students are enrolled in the gospel choir, and 37 students are enrolled in the handbell choir. Fifteen students are enrolled in neither group. If there are 100 students at Bedlam, how many students are enrolled in both the gospel choir and the handbell choir?
(A) 12
(B) 16
(C) 18
(D) 21
(E) 27
Here’s How to Crack It
As you can see, part of the difficulty of such problems lies in reading them—they’re confusing. The other trick lies in the actual counting. If there are students in both the gospel choir and the handbell choir, then when you count the members of both groups, you’re counting some kids twice—the kids who are in both groups. To find out how many students are in both groups, just use the group problem formula.
Group Problem Formula
Total = Group 1 + Group 2 + Neither − Both
For question 34, this formula gives you 64 + 37 + 15 − Both = 100. Solve this, and you get Both = 16. The correct answer is (B).
The group problem formula will work for any group question with two groups. Just Plug In the information you know, and solve for the piece that’s missing.
DRILL
Use the group formula on the following practice questions. The answers to these drills can be found in Chapter 12.
25. At Buford Prep School, 253 students are enrolled in French, and 112 students are enrolled in Latin. 23 students are enrolled in both Latin and French. If there are 530 students at Buford Prep School, how many students are enrolled in neither French nor Latin?
(A) 188
(B) 342
(C) 388
(D) 484
(E) 507
28. On the Leapwell gymnastics team, 14 gymnasts compete on the balance beam, 12 compete on the uneven bars, and 9 compete on both the balance beam and the uneven bars. If 37 gymnasts compete on neither the balance beam nor the uneven bars, how many gymnasts are on the Leapwell team?
(A) 45
(B) 51
(C) 54
(D) 63
(E) 72
42. In a European tour group, of the tourists speak Spanish, of the tourists speak French, and of the tourists speak neither language. What fraction of the tourists in the tour group speak both Span ish and French?
(A)
(B)
(C)
(D)
(E)
Summary
· Statistics is about working with large groups of numbers and looking for patterns and trends in those numbers
· The mean is the average value of a set.
· The median is the middle value of a set when the values of the set are in chronological order.
· The mode is the value that occurs the most in a set.
· Probability is the number of ways to get what you want divided by the total number of possible outcomes.
· The probability of multiple events occurring can be calculated either by writing them all out or by multiplying the individual probabilities together.
· Permutations and combinations are more common on the Level 2 test, but do appear on the Level 1 test.
· A permutation is the number of ways you can arrange objects in a definite order.
· A combination is the number of ways you can group objects. Order doesn’t matter. With the same set of objects, the combination will be smaller than the permutation.
· Group questions require one formula: Total = Group 1 + Group 2 + Neither − Both.
· Here are some Level 2−only concepts:
· Standard deviation is a measure of a set’s variation from its mean. It comes up very infrequently on the Level 2, but it does pop up.
· The range of a set in statistics is the difference between the set’s highest and lowest values.
· A factorial is found when multiplying the numbers between 1 and the number you’re looking for. 4! = 4 × 3 × 2 × 1. Use your calculator on these questions.