SAT II Math I : Median

Study concepts, example questions & explanations for SAT II Math I

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Example Questions

Example Question #1 : Median

What is the median of 13, 12, 22, 7, 7, 8, 13, 27, 30?

Possible Answers:

\displaystyle 8

\displaystyle 12

\displaystyle 13

\displaystyle 22

Correct answer:

\displaystyle 13

Explanation:

The median is middle number of a set, when the numbers are arranged in order from least to greatest.

In our case we re arrange our numbers in the following fashion:

7, 7, 8, 12, 13, 13, 22, 27, 30

The middle number in our case is the 5th entry.

Therefore, the median is 13

Example Question #2 : Median

What is the median of 3, 7, 6, 11, 14, 9?

Possible Answers:

\displaystyle 11

\displaystyle 9

\displaystyle 8

\displaystyle 7

Correct answer:

\displaystyle 8

Explanation:

The median is the middle number in a set of numbers when arranged in order from least to greatest.

In our case we get

3, 6, 7, 9, 11, 14

This set has two middle numbers 7 and 9. From here we need to find the average between these two numbers to find our median:

\displaystyle \frac{7+9}{2}=\frac{16}{2}=8

Example Question #1 : Median

What's the median?

\displaystyle 13, 2, 16, 8, 34, 26

Possible Answers:

\displaystyle 11.5

\displaystyle 14.5

\displaystyle 21

\displaystyle 18

\displaystyle 12

Correct answer:

\displaystyle 14.5

Explanation:

Median is to find the middle number arranging from smallest to greatest. So we have \displaystyle 2, 8, 13, 16, 26, 34. The two middle numbers in the set is \displaystyle 13, 16. We then find the average which is \displaystyle \frac{29}{2} or \displaystyle 14.5.

Example Question #691 : Sat Subject Test In Math I

If the lowest value in a set was decreased and the highest value in the same set was increased, how would that affect the median?

Possible Answers:

It would decrease

It would increase

It would stay the same

Not enough information to give an adequate answer

Correct answer:

It would stay the same

Explanation:

The median is the middle number of a set.  If the same amount of values are used and the actual middle number is not changed, the median of the set will not change.  Changing the actual values of the two extremes did not change them position wise in the set when placed in value order.  

Example Question #1 : How To Find The Median For A Set Of Data

Let \displaystyle b be a positive integer.

Find the median of the set.

\displaystyle b,3b,5b,7b,9b,11b,13b,15b

Possible Answers:

\displaystyle 8b

\displaystyle 9b

\displaystyle 10b

\displaystyle 11b

Correct answer:

\displaystyle 8b

Explanation:

The median is the middle value of the set in increasing order.

In this set of 8 (or any even number) entries, the median is the mean of the two middle entries of the set in increasing order

\displaystyle \frac{7b+9b}{2}

or

\displaystyle 8b

Example Question #1 : Median

Consider the following test scores from a typical high school class with \displaystyle 15 students:

\displaystyle 81,83,91,99,100,89,92,96,89,94,83,90,98,89,97

The mean of this data set is_________, and the mode of this data set is _______.

Possible Answers:

\displaystyle 81,\ 100

\displaystyle 91.4,\ 89

\displaystyle 89,\ 91

\displaystyle 91,\ 91.4

\displaystyle 91.4,\ 100

Correct answer:

\displaystyle 91.4,\ 89

Explanation:

The mean is just the average of all the test scores, which is found by adding up the scores and dividing by the number of scores (\displaystyle 15).  This gives \displaystyle 91.4 as the mean.  The mode is the score which occurs most frequently.  In this case, the mode is \displaystyle 89.  The median, the middle score of the sequence, is \displaystyle 91.

Example Question #2 : Median

What is the median of the first 20 even numbers?

Possible Answers:

\displaystyle 10

Cannot be calculated

\displaystyle 20

\displaystyle 21

\displaystyle 11

Correct answer:

\displaystyle 21

Explanation:

Let's think of this list of numbers:

2, 4, 6, ...

Where does it end? The first 5 even numbers goes to 10. That means that the last number in the first 20 will be the number 40. So the question is, "Where is the middle?" Well, this is an even number of values, so there is no actual middle. What we have to do, then is find the 10th and the 11th numbers and take their average. The 10th number is easy, based on what we just said. If the 5th is 10, then the 10th is 20. The 11th will just be two more than that, namely 22. To calculate the median, we just have to find the average of those two numbers:

\displaystyle \frac{20+22}{2}=21

If you prefer to write out the full list:

2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40

Example Question #2 : Median

What is the median of the first ten prime numbers?

Possible Answers:

\displaystyle {19}

\displaystyle 12

\displaystyle 22

\displaystyle 13

\displaystyle 16

Correct answer:

\displaystyle 12

Explanation:

To answer this question, you need to know the first ten prime numbers! Remember, prime numbers are all of the integers that are divisible only by themselves and by 1.  They do not include 1.  So, our list is:

2, 3, 5, 7, 11, 13, 17, 19, 23, 29

The median is the "middle value." There is no proper "middle" since we have an even number of values. We need to take the 5th and the 6th elements (the middle two values) and average them. The 5th term is 11 and the 6th is 13; therefore, the median is:

\displaystyle \frac{11+13}{2}=\frac{24}{2}=12

Example Question #3 : Median

There is a table of flowers prepared for sale. Twelve flowers are \displaystyle 10 inches tall, five are \displaystyle 12 inches tall, and four are \displaystyle 15 inches tall. What is the median height of these flowers?

Possible Answers:

\displaystyle 15\ \text{in}

\displaystyle 12\ \text{in}

\displaystyle 10\ \text{in}

\displaystyle 11\ \text{in}

\displaystyle 13.5\ \text{in}

Correct answer:

\displaystyle 10\ \text{in}

Explanation:

The easiest way to do this is first to find the total number of flowers:

\displaystyle 12 +5+4=21

Now, the median element is the "middle" term. To find the middle, you can divide 21 by 2:

\displaystyle \frac{21}{2}=10.5

Since you have an odd number of elements, you unsurprisingly get a fraction. This means that there 10 items to the left of the median and 10 to its right. The 11th term is your median. Now, your group of flowers looks like this:

1-12: 10 inches

13-17: 12 inches

18-21: 15 inches

The 11th item is going to be in that first group, meaning that the median is 10 inches.

If you prefer to write out all of the terms, it will be:

10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 12, 12, 12, 12, 12, 15, 15, 15, 15

Example Question #1 : Median

There are 1000 magical beanstalks planted in a row. Each stalk is 10 feet taller than the one before it. The smallest stalk is 10 feet tall. What is the median height of the stalks?

Possible Answers:

\displaystyle 5000\ \text{ft}

\displaystyle 750\ \text{ft}

No median can be calculated

\displaystyle 5005\ \text{ft}

\displaystyle 505\ \text{ft}

Correct answer:

\displaystyle 5005\ \text{ft}

Explanation:

The first thing to do is figure out which stalk is in the "middle." Since there are an even number of stalks, there is no exact middle; there are 500 on one side and 500 on the other. This means that the 500th and the 501st are the median. These will have to be averaged.

Now, we need to determine the height of these two stalks. Consider the pattern given:

1st stalk: 10 feet

2nd stalk: 20 feet

3rd stalk: 30 feet

4th stalk: 40 feet

You should see the pattern that emerges for this problem. Each stalk is 10 times that stalk's place in the row. This means that the 500th stalk will be:

\displaystyle 500*10 =5000

The 501st stalk will be:

\displaystyle 501*10=5010

The average of these two numbers is:

\displaystyle \frac{5000+5010}{2} = \frac{10010}{2}=5005

5005 feet is the median.

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