Algebra · Chapter 18

Mean, Median, Mode

Three ways to describe the middle of a list: the mean is a balance point, the median sits in the middle, the mode is the crowd favorite. Then the geometric, harmonic and quadratic means, and how they line up.

What's this chapter about?

18.0a Mean, Median and Mode
18.0b Other Means
18.1 Practice Problems

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18.0a0%

Mean, Median and Mode

In this section you will be able to find a mean, median and mode, turn a mean into a sum, and split a median problem into cases.

Add the number 1 to a set and its mean drops by 13. Add 2001 instead and the mean rises by 27. How many numbers are in the set, and what is their mean?

The mean is a balance point: put a weight at every number and the see-saw balances at the mean. The median is the middle of the sorted list. The mode is the value that shows up most often.

Look: a mean is really a sum in disguise. Sum $=$ mean $\times$ count. Most contest problems about means become easy once you write down the sums.

The median is different. It only cares about order, so a median problem usually splits into cases: where does the unknown fall in the sorted list?

Exploreplay first, formulas later

Try this: Drag x from −10 up to 20. The median stays at 6, then follows x, then stops at 8. Why those three stages?
Try this: Each step of x moves the mean by 1/5. Where do the median and the mean meet? Watch for the green badge.
Try this: Push x all the way to −10, then to 20. Which statistic barely moves, and which one gets dragged along?

Name it

Definition 18.0.1 · Mean/Average

\text{Mean} = \text{average of all terms} = \frac{\text{sum of all terms}}{\text{number of terms}}

Turned around: $\text{sum} = \text{mean} \times \text{number of terms}$ . Most mean problems are really sum problems.

Definition 18.0.2 · Median

After arranging the numbers in increasing or decreasing order:

If the number of terms is odd, the median is the middle number.
If the number of terms is even, the median is the average of the middle two numbers.

Definition 18.0.3 · Mode

\text{Mode} = \text{most common term(s)}

Watch itpredict, then reveal

Example 18.1

AIME

A finite set $\mathcal{S}$ of distinct real numbers has the following properties: the mean of $\mathcal{S} \cup \{1\}$ is $13$ less than the mean of $\mathcal{S}$ , and the mean of $\mathcal{S} \cup \{2001\}$ is $27$ more than the mean of $\mathcal{S}$ . Find the mean of $\mathcal{S}$ .

1
Warm up on the see-saw: a block of $4$ numbers with mean $10$ , and a new number $0$ joins them. What is the new mean?

Predict first:
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Example 18.2

AMC 10

What is the sum of all real numbers $x$ for which the median of the numbers $4, 6, 8, 17,$ and $x$ is equal to the mean of those five numbers?

1
What is the mean of the five numbers, in terms of $x$ ?

Predict first:
Your prediction

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Example 18.3

AMC 10

When the mean, median, and mode of the list

10, 2, 5, 2, 4, 2, x

are arranged in increasing order, they form a non-constant arithmetic progression. What is the sum of all possible real values of $x$ ?

1
What is the mode, whatever $x$ is?

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Problem 18.1.6(additional)

AMC 10

The numbers $3, 5, 7, a,$ and $b$ have an average (arithmetic mean) of $15$ . What is the average of $a$ and $b$ ?

Problem 18.1.3

AMC 10/12

Ms. Blackwell gives an exam to two classes. The mean of the scores of the students in the morning class is $84$ , and the afternoon class's mean score is $70$ . The ratio of the number of students in the morning class to the number of students in the afternoon class is $\frac{3}{4}$ . What is the mean of the scores of all the students?

Problem 18.1.7(additional)

AMC 10

The mean, median, and mode of the $7$ data values $60, 100, x, 40, 50, 200, 90$ are all equal to $x$ . What is the value of $x$ ?

Contest cornerreal contest problems

Problem 18.1.1

AMC 10

The quiz scores of a class with $k > 12$ students have a mean of $8$ . The mean of a collection of $12$ of these quiz scores is $14$ . What is the mean of the remaining quiz scores in terms of $k$ ?

Problem 18.1.4

AMC 10/12

In the following list of numbers, the integer $n$ appears $n$ times in the list for $1 \le n \le 200$ .

1, 2, 2, 3, 3, 3, 4, 4, 4, 4, \ldots, 200, 200, \ldots, 200

What is the median of the numbers in this list?

Problem 18.1.5

AMC 12

Last academic year Yolanda and Zelda took different courses that did not necessarily administer the same number of quizzes during each of the two semesters. Yolanda's average on all the quizzes she took during the first semester was $3$ points higher than Zelda's average on all the quizzes she took during the first semester. Yolanda's average on all the quizzes she took during the second semester was $18$ points higher than her average for the first semester and was again $3$ points higher than Zelda's average on all the quizzes Zelda took during her second semester. Which one of the following statements cannot possibly be true?

Problem 18.1.8(additional)

AMC 12

Every high school in the city of Euclid sent a team of $3$ students to a math contest. Each participant in the contest received a different score. Andrea's score was the median among all students, and hers was the highest score on her team. Andrea's teammates Beth and Carla placed $37$ th and $64$ th, respectively. How many schools are in the city?

Problem 18.1.9(additional)

AMC 10

What is the median of the following list of $4040$ numbers?

1, 2, 3, \ldots, 2020, 1^2, 2^2, 3^2, \ldots, 2020^2

Problem 18.1.11(additional)

AMC 10

Melanie computes the mean $\mu$ , the median $M$ , and the modes of the $365$ values that are the dates in the months of $2019$ . Thus her data consist of $12$ $1$ s, $12$ $2$ s, $\ldots$ , $12$ $28$ s, $11$ $29$ s, $11$ $30$ s, and $7$ $31$ s. Let $d$ be the median of the modes. Which of the following statements is true?

Checkpoint for section 18.0a

Get 2 of 3 right to master this section. One try each, no hints.

18.0b0%

Other Means

In this section you will be able to compute geometric, harmonic and quadratic means, and see why QM ≥ AM ≥ GM ≥ HM.

The arithmetic mean of two numbers is a two-digit integer. Reverse its digits and you get their geometric mean. How far apart are the two numbers?

The ordinary mean adds. The geometric mean multiplies and then takes a root. The harmonic mean averages the reciprocals. The quadratic mean averages the squares.

Look at the semicircle: the radius is the arithmetic mean $\frac{a+b}{2}$ , and the half-chord at the split is the geometric mean $\sqrt{ab}$ . A half-chord can never be longer than the radius, so $\text{AM} \ge \text{GM}$ . They are equal only when the split is at the center: $a = b$ .

Exploreplay first, formulas later

Try this: Drag the split point P slowly to the middle of the diameter. What happens to the four bars?
Try this: Which mean is always the smallest? Which is always the largest? Try a = 1, b = 20.
Try this: Set a = 4 and b = 9. Check the bars against the formulas: AM = 6.5, GM = √36 = 6.

Name it

Definition 18.0.5 · Geometric Mean

The geometric mean of the positive numbers $a_1, a_2, a_3, \ldots, a_n$ is

\sqrt[n]{a_1 \times a_2 \times a_3 \times \cdots \times a_n}

Definition 18.0.6 · Harmonic Mean

The harmonic mean of the positive numbers $a_1, a_2, a_3, \ldots, a_n$ is

\frac{1}{\dfrac{\frac{1}{a_1} + \frac{1}{a_2} + \cdots + \frac{1}{a_n}}{n}} = \frac{n}{\frac{1}{a_1} + \frac{1}{a_2} + \cdots + \frac{1}{a_n}}

In words: the reciprocal of the average of the reciprocals.

Definition 18.0.7 · Quadratic Mean

The quadratic mean (root mean square) of $a_1, a_2, a_3, \ldots, a_n$ is

\sqrt{\frac{a_1^2 + a_2^2 + a_3^2 + \cdots + a_n^2}{n}}

For positive numbers, $\text{QM} \ge \text{AM} \ge \text{GM} \ge \text{HM}$ , with equality exactly when all the numbers are equal (Chapter 40 proves it).

Contest cornerreal contest problems

Problem 18.1.13(additional)

AMC 10

The arithmetic mean of two distinct positive integers $x$ and $y$ is a two-digit integer. The geometric mean of $x$ and $y$ is obtained by reversing the digits of the arithmetic mean. What is $|x - y|$ ?

Checkpoint for section 18.0b

Get 2 of 3 right to master this section. One try each, no hints.

18.1

Practice Problems

In this section you will be able to mix sums, medians and modes on harder contest problems.

Four tougher problems: two are sum puzzles, two ask how far a mode can be pushed. When a problem says “largest” or “least”, build the extreme list.

Want every problem from this chapter on one page? Open the problem set.

Contest cornerreal contest problems

Problem 18.1.2

AMC 10

This problem also appears in the book as 15.1.3.

Suppose that $S$ is a finite set of positive integers. If the greatest integer in $S$ is removed from $S$ , then the average value (arithmetic mean) of the integers remaining is $32$ . If the least integer in $S$ is also removed, then the average value of the integers remaining is $35$ . If the greatest integer is then returned to the set, the average value of the integers rises to $40$ . The greatest integer in the original set $S$ is $72$ greater than the least integer in $S$ . What is the average value of all the integers in the set $S$ ?

Problem 18.1.10(additional)

AMC 10

A list of $2018$ positive integers has a unique mode, which occurs exactly $10$ times. What is the least number of distinct values that can occur in the list?

Problem 18.1.12(additional)

AIME

Let $S$ be a list of positive integers—not necessarily distinct—in which the number $68$ appears. The average (arithmetic mean) of the numbers in $S$ is $56$ . However, if $68$ is removed, the average of the remaining numbers drops to $55$ . What is the largest number that can appear in $S$ ?

Problem 18.1.14(additional)

AIMEChallenge

A sample of $121$ integers is given, each between $1$ and $1000$ inclusive, with repetitions allowed. The sample has a unique mode (most frequent value). Let $D$ be the difference between the mode and the arithmetic mean of the sample. What is the largest possible value of $\lfloor D \rfloor$ ? (For real $x$ , $\lfloor x \rfloor$ is the greatest integer less than or equal to $x$ .)