SAT Mean, Median, and Mode

To successfully solve problems about mean, median, and mode on the SAT, you need to understand:

Examples

The average (arithmetic mean) of the lengths of 15 city blocks is $d$ miles. Which of the following is the total length (in miles) of all the city blocks in terms of $d?$

(A) $\ \ d-15$

(B) $\ \ 15+d$

(C) $\ \ \frac{d}{15}$

(D) $\ \ \frac{15+d}{2}$

(E) $\ \ 15d$

Correct Answer: E

Solution:

Tip: The average of $n$ numbers is the sum of the numbers divided by $n.$
Let the total length be $L.$ Then, $\frac{L}{15} = d$ and multiplying both sides by 15 we obtain $L=15d$ miles.

Incorrect Choices:

(A)
This answer is meant to confuse you. Note that it is meaningless since we are adding quantities with different units. 15 is the number of city blocks, and $d$ miles is the average length of a city block.

(B)
This sum is meaningless because to the number of city blocks is added a distance.

(C)
If you divide, instead of multiply both sides by 15 in $\frac{L}{15} = d,$ you will get this wrong answer.

(D)
This answer is meant to confuse you. It is the average of 15 and $d.$ Note that this is meaningless since 15 is the number of city blocks, and d is the average length of a city block.

In a set of 9 different numbers, which of the following CANNOT affect the value of the median?

(A) $\ \$ Multiplying all numbers by 2.
(B) $\ \$ Increasing each number by 2.
(C) $\ \$ Introducing a 10th number.
(D) $\ \$ Eliminating one of the numbers.
(E) $\ \$ Decreasing the smallest number.

Correct Answer: E

Solution 1:

Tip: If $n$ numbers are arranged in increasing order, the median is the middle value if $n$ is odd, and it is the average of the two middle values if $n$ is even. We analyze each of the answer choices.

(A) If all numbers are multiplied by 2, so is the median. We eliminate this choice.
(B) If all numbers are increased by 2, so is the median. We eliminate this choice.
(C) If the 9 numbers are arranged in increasing order, the median is the 5th number. Introducing a 10th number means the new median will be the average of the 5th and 6th numbers. Unless the 5th number equals the 6th number, the median will be affected. We eliminate this choice.
(D) If the 9 numbers are arranged in increasing order, the median is the 5th number. Eliminating one of the numbers means the new median will be the average of the 4th and 5th numbers. And since those are different (given), the new median will not be the same as the old one. We eliminate this choice.
(E) The median is the middle value. Half of the numbers are greater than the median and half of the numbers are smaller than the median. If we decrease the smallest number, the same half of the numbers as before will be smaller than the median and the same half of the numbers as before will be greater than the median. This is the correct answer.

Solution 2:

Tip: Replace variables with numbers.
Let the numbers be 1, 2, 3, 4, 5, 6, 7, 8, and 9. The median is the middle value, 5.

(A) Multiplying all numbers by 2, we obtain a new list: 2, 4, 6, 8, 10, 12, 14, 16, and 18, and a new median: 10. But $10\neq 5.$ This choice is wrong.
(B) Increasing all numbers by 2, we obtain a new list: 3, 4, 5, 6, 7, 8, 9, 10, and 11, and a new median: 7. But $7\neq5.$ This choice is wrong.
(C) We show that this action can affect the median by finding a counter-example: Let the 10th number be 10. Then the new median is $\frac{5+6}{2}=5.5.$ But, $5.5 \neq5.$ This choice is wrong.
(D) Let's eliminate the last number, 9. The new median would be $\frac{4+5}{2} = 4.5.$ But $4.5 \neq 5.$ We eliminate this choice.
(E) Since we've eliminated all other choices, this one must be correct. If we change 1 to any number smaller than 1, the median would still be 5.

Incorrect Choices:

(A), (B), (C), and (D)
See the either solution for why these choices are wrong.

Review

If you thought these examples difficult and you need to review the material, these links will help:

SAT Tips for Mean, Median, and Mode

The average of $n$ numbers is the sum of the numbers divided by $n.$

If the average of a set of numbers is $A$ and a new number $x=A$ is introduced to the set, the new average will also equal $A.$

If $n$ numbers are arranged in increasing order, the median is the middle value if $n$ is odd, and it is the average of the two middle values if $n$ is even.

In a set of numbers, the mode is the number that appears most frequently.

To find the weighted mean of a some numbers, find the product of each number and its weight, then divide the sum of these products by the sum of the weights.

SAT General Tips

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