Algebra · Chapter 15

Mean, Median, Mode

Mean, median, mode and range: four ways to describe a list of numbers, and how to use them as clues.

What's this chapter about?

15.1 Mean, Median, Mode Fundamentals
15.2 Mean Median Mode Conditions Examples
15.3 Practice Problems

You'll need

12.1 Algebraic manipulations

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15.10%

Mean, Median, Mode Fundamentals

In this section you will be able to find the mean, median, mode and range of a list, and see how each one reacts when a number changes.

Your class takes a quiz. Is a “typical” score the average? The one in the middle? The one most kids got? All three are useful, and they are often different!

The mean is the balance point of the dots: the triangle under the line. The median is the middle dot once you sort them. The mode is the tallest stack. The range is how far apart the two ends are.

Look:a mean problem is really a sum problem. If you know the mean and how many numbers there are, you know the total.

Exploreplay first, formulas later

Try this: Drag the dot at 9 to 10. Which of the four numbers changed? Which stayed put?
Try this: Add a value so the mean goes up but the median stays the same. Where did you put it?
Try this: Add one more value. Now there is an even count. Which two sorted numbers does the median use?
Try this: Make two stacks the tallest. How many modes are there now?

Name it

Definition 15.1.1 · Mean/Average

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

Definition 15.1.3 · Mode

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

Definition 15.1.5 · Median

After arranging the numbers in increasing or decreasing order:

If the number of terms is odd,

\text{Median} = \text{middle number}

If the number of terms is even,

\text{Median} = \text{average of middle two numbers}

Definition 15.1.6 · Range

\text{Range} = \text{Largest number} - \text{Smallest Number}

Watch itpredict, then reveal

Example 15.1

Find the sum of the mean, median, mode, and range of 1, 9, 7, 1, 3, 5, 2.

1
What is the first step we should do to make analyzing the numbers easier?

Predict first:

Example 15.2

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 $S$ ?

1
Say $S$ has $n$ numbers. Using the mean, what is the sum of all of them?

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Example 15.3

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

1
There are 4040 numbers, an even count. The median is the average of which two positions in sorted order? Give the first one.

Predict first:
Your prediction

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Problem 15.3.5

AMC 8

What is the sum of the mean, median, and mode of the numbers 2, 3, 0, 3, 1, 4, 0, 3?

Problem 15.3.14(additional)

AMC 8 Modified

Here is a list of the numbers of fish that Tyler caught in nine outings last summer:

2, 0, 1, 3, 0, 3, 3, 1, 2.

What is the ordering of the mean, median, and mode of the number of fish caught?

Problem 15.3.15(additional)

EMCC

The mean of the numbers 2, 0, 1, 5, and $x$ is an integer. Find the smallest possible positive integer value for $x$ .

Problem 15.3.10

AMC 8

The harmonic mean of a set of non-zero numbers is the reciprocal of the average of the reciprocals of the numbers. What is the harmonic mean of 1, 2, and 4?

Contest cornerreal contest problems

Problem 15.3.1

AMC 8

Four students take an exam. Three of their scores are 70, 80, and 90. If the average of their four scores is 70, then what is the remaining score?

The fulcrum sits at the average, 70. Which way does it tip so far?

Problem 15.3.2

AMC 8

The following bar graph represents the length (in letters) of the names of 19 people. What is the median length of these names?

Problem 15.3.3

AMC 8

Mr. Garcia asked the members of his health class how many days last week they exercised for at least 30 minutes. The results are summarized in the following bar graph, where the heights of the bars represent the number of students.

What was the mean number of days of exercise last week, rounded to the nearest hundredth, reported by the students in Mr. Garcia’s class?

Problem 15.3.4

AMC 8

Hammie is in the 6th grade and weighs 106 pounds. Her quadruplet sisters are tiny babies and weigh 5, 5, 6, and 8 pounds. Which is greater, the average (mean) weight of these five children or the median weight, and by how many pounds?

Problem 15.3.7

AMC 8

The diagram shows the number of students at soccer practice each weekday during last week. After computing the mean and median values, Coach discovers that there were actually 21 participants on Wednesday. Which of the following statements describes the change in the mean and median after the correction is made?

Problem 15.3.16(additional)

AMC 8

Mr. Ramos gave a test to his class of 20 students. The dot plot below shows the distribution of test scores.

Later Mr. Ramos discovered that there was a scoring error on one of the questions. He regraded the tests, awarding some of the students 5 extra points, which increased the median test score to 85. What is the minimum number of students who received extra points?

(Note that the median test score equals the average of the 2 scores in the middle if the 20 test scores are arranged in increasing order.)

Dot plot of the 20 test scores

Checkpoint for section 15.1

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

15.20%

Mean Median Mode Conditions Examples

In this section you will be able to build a list that meets several conditions at once, and push one number as high as it can go.

Six different numbers. Their mean is 7, their median is 6, and their range is 9. Can you find them? Each condition cuts down the choices.

Turn every condition into something you can count with. The mean gives you the total. The median pins the middle numbers. The range ties the largest to the smallest.

To make one number as big as possible, make all the others as small as the rules allow.

Exploreplay first, formulas later

Try this: Example 15.4 wants mean 7, median 6 and range 9. Which of the three does this list already meet?
Try this: Move the 4 and the 8 to 5 and 7. Did the median change? Did the mean?
Try this: Now fix the mean: what must all six dots add up to? Drag the big dots until the mean reads 7.

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Example 15.4

Omega Learn

A list of 6 distinct positive integers has a mean of 7, a median of 6, and a range of 9. Find the sum of the 2nd and 5th smallest numbers.

1
What are the possibilities for the middle 2 numbers, or the 3rd and 4th smallest numbers? First, what must they add up to?

Predict first:
Your prediction

Example 15.5

AMC 10

The mean, median, unique mode, and range of a collection of eight integers are all equal to 8. The largest integer that can be an element of this collection is what?

1
The mean is 8. What do the eight integers add up to?

Predict first:
Your prediction

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Problem 15.3.6

AMC 8

The mean, median, and unique mode of the positive integers 3, 4, 5, 6, 6, 7, and $x$ are all equal. What is the value of $x$ ?

Problem 15.3.17(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 15.3.8

AMC 8

The mean of a set of five different positive integers is 15. The median is 18. What is the maximum possible value of the largest of these five integers?

Problem 15.3.9

AMC 8

One day the Beverage Barn sold 252 cans of soda to 100 customers, and every customer bought at least one can of soda. What is the maximum possible median number of cans of soda bought per customer on that day?

Problem 15.3.11

AMC 8

How many subsets of two elements can be removed from the set $\{1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11\}$ so that the mean (average) of the remaining numbers is 6?

Problem 15.3.12

AMC 12

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 $x$ ?

Checkpoint for section 15.2

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

15.3

Practice Problems

In this section you will be able to mix every idea from this chapter on harder contest problems.

These ones take a few steps. Name the unknowns, turn each mean into a sum, and write one equation per sentence. Every problem has three hints if you get stuck.

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

Contest cornerreal contest problems

Problem 15.3.13

AMC 10

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 15.3.18(additional)

AMC 10

When 15 is appended to a list of integers, the mean is increased by 2. When 1 is appended to the enlarged list, the mean of the enlarged list is decreased by 1. How many integers were in the original list?

Problem 15.3.19(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 1s, 12 2s, . . . , 12 28s, 11 29s, 11 30s, and 7 31s. Let $d$ be the median of the modes. Which of the following statements is true?

Problem 15.3.20(additional)

MATHCOUNTS

The mean of a list of nine numbers is 17, and the modes are $a$ , $b$ and $c$ . If $a + 4$ , $1 + b$ and $c - 8$ are distinct numbers in the list, and none of them are modes of the list, then what is the value of $3(a + b + c)$ ?

Problem 15.3.21(additional)

Omega Learn

6 positive integers have a median of 5.5, two distinct modes, a range of 5, and a mean of $\frac{35}{6}$ . Find the sum of all possible values for the largest integer.