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Statistics & Data

Mean, median, mode, range, data tables, charts, weighted averages

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Statistics & DataAMC 10 Step by Step

Name: Date: October 28, 2026

20 problems, easiest first

What you need to know

Mean $=\dfrac{\text{sum}}{\text{count}}$ , so sum $=$ mean $\times$ count. Almost every problem about a mean is really about the total. Example: five numbers with mean $12$ add up to $60$ .
Median: put the numbers in order first. With an odd count, the median is the middle number. With an even count, it is the average of the two middle numbers. Example: $8,3,5,9,1$ sorts to $1,3,5,8,9$ , with median $5$ . And $2,7,4,10$ sorts to $2,4,7,10$ , with median $\frac{4+7}2=5.5$ .
Mode: the value that appears most often. "Unique mode" means it appears strictly more often than every other value; a tie is not allowed. Example: $2,3,3,3,5,5$ has unique mode $3$ , but $2,3,3,5,5,7$ has no unique mode, since $3$ and $5$ tie.
Range $=$ largest $-$ smallest. Example: $2,3,3,5,5,7$ has range $7-2=5$ .
Combining two groups (a weighted average). If a group of $a$ people has mean $m_1$ and a group of $b$ people has mean $m_2$ , the mean of everyone is $\dfrac{am_1+bm_2}{a+b}$ . Example: $20$ students averaging $70$ and $30$ averaging $80$ give $\frac{1400+2400}{50}=76$ .

How AMC 10 tests it

Problems 1–8: find one statistic from another. The hard part is not mixing up the mean and the median.
"After the next test my average will be $k$ ": change the averages into totals and subtract. Example: to go from an average of $85$ on $4$ tests to $87$ on $5$ , you need $5\cdot87-4\cdot85=95$ .
Middle-of-the-paper problems where the mean, median, mode, and range are all given and you find the largest possible number in the list. Solve them with a bound plus a list that reaches it.

Standard approaches

Rewrite every mean as a sum, and every median as a fact about a certain position in the sorted list.
When numbers are added or removed, keep track of totals, not averages.
For largest/smallest problems, first place the values that are forced (the median spot, the repeated mode). Then push the free values as far as they can go.
Bound first, then build: show the larger answer is impossible, then write down a list that reaches the next one.

Worked example

A list of five positive integers has mean $12$ , median $13$ , and unique mode $15$ . What is the largest possible range of the list?

(A) $8$ (B) $9$ (C) $10$ (D) $11$ (E) $12$

Solution. Write the list in order as $a\le b\le c\le d\le e$ .

Use the mean and median. The mean is $12$ , so the sum is $5\cdot12=60$ . The median is the middle number, so $c=13$ .

Use the mode. The mode $15$ must appear at least twice. Only $d$ and $e$ can be $15$ , since $a$ , $b$ , $c$ are at most $13$ . So $d=e=15$ , and $a+b=60-13-15-15=17$ .

Make the range big. The range is $15-a$ , so we want $a$ as small as possible, which means $b$ as large as possible. Since $15$ appears exactly twice and is the unique mode, no other value may appear twice: $a\ne b$ and $b\ne13$ . As $b\le c=13$ , this leaves $b\le12$ , so $a=17-b\ge5$ .

Check that it works. The list $5,12,13,15,15$ has sum $60$ , median $13$ , and unique mode $15$ . Its range is $15-5=10$ . The answer is $\boxed{\textbf{(C)}\ 10}$ .

Pitfalls

Giving the mean when the sum is asked, or the median when the mean is asked.
Forgetting to sort before finding the median. Example: the middle of $9,2,7,4,3$ as written is $7$ , but the median is $4$ .
Ignoring "unique" in "unique mode." Example: $4,13,13,15,15$ has sum $60$ and median $13$ , but $13$ ties with $15$ , so its range of $11$ is not allowed.
Averaging group averages without weighting by group size. Example: above, $\frac{70+80}{2}=75$ is wrong; the true mean is $76$ .

Traps that recur

Treating the median as 5.2 or one of the given numbers; with nine values the fifth must be an integer, so one of x, y, z has to sit in the middle. (2024 AMC 10B #15)
Using five 8s to fill the bottom, which forces the other values up and yields 24; fewer 8s lets 1s in and gives a bigger maximum. (2014 AMC 10B #18)
Ignoring that the mode must be unique, or forgetting the mean fixes the total at 64. (2002 AMC 10A #21)
Assuming the mean is always the largest of the three statistics; when the median is x you must test all three orderings, and when the median is 4 you must discard solutions that leave the case's range. (2000 AMC 10 #23)

1.2016 AMC 10B #5
The mean age of Amanda's $4$ cousins is $8$ , and their median age is $5$ . What is the sum of the ages of Amanda's youngest and oldest cousins?
1. A) $13$
2. B) $16$
3. C) $19$
4. D) $22$
5. E) $25$
2.2014 AMC 10A #5
On an algebra quiz, $10\%$ of the students scored $70$ points, $35\%$ scored $80$ points, $30\%$ scored $90$ points, and the rest scored $100$ points. What is the difference between the mean and median score of the students' scores on this quiz?
1. A) $1$
2. B) $2$
3. C) $3$
4. D) $4$
5. E) $5$
3.2010 AMC 10A #1
Mary’s top book shelf holds five books with the following widths, in centimeters: $6$ , $\dfrac{1}{2}$ , $1$ , $2.5$ , and $10$ . What is the average book width, in centimeters?
1. A) $1$
2. B) $2$
3. C) $3$
4. D) $4$
5. E) $5$
4.2005 AMC 10A #6
The average (mean) of $20$ numbers is $30$ , and the average of $30$ other numbers is $20$ . What is the average of all $50$ numbers?
1. A) $23$
2. B) $24$
3. C) $25$
4. D) $26$
5. E) $27$
5.2001 AMC 10 #1
The median of the list $n, n + 3, n + 4, n + 5, n + 6, n + 8, n + 10, n + 12, n + 15$ is $10$ . What is the mean?
1. A) $4$
2. B) $6$
3. C) $7$
4. D) $10$
5. E) $11$
6.2024 AMC 10A #12
Zelda played the Adventures of Math game on August 1 and scored $1,700$ points. She continued to play daily over the next $5$ days. The bar chart below shows the daily change in her score compared to the day before. (For example, Zelda's score on August 2 was $1,700 + 80 = 1,780$ points.) What was Zelda's average score in points over the $6$ days?
1. A) $1700$
2. B) $1702$
3. C) $1703$
4. D) $1713$
5. E) $1715$
7.2022 AMC 10A #8
A data set consists of $6$ (not distinct) positive integers: $1$ , $7$ , $5$ , $2$ , $5$ , and $X$ . The average (arithmetic mean) of the $6$ numbers equals a value in the data set. What is the sum of all possible values of $X$ ?
1. A) $10$
2. B) $26$
3. C) $32$
4. D) $36$
5. E) $40$
8.2021 AMC Fall 10A #10
A school has $100$ students and $5$ teachers. In the first period, each student is taking one class, and each teacher is teaching one class. The enrollments in the classes are $50, 20, 20, 5,$ and $5$ . Let $t$ be the average value obtained if a teacher is picked at random and the number of students in their class is noted. Let $s$ be the average value obtained if a student was picked at random and the number of students in their class, including the student, is noted. What is $t-s$ ?
1. A) ${-}18.5$
2. B) ${-}13.5$
3. C) $0$
4. D) $13.5$
5. E) $18.5$
9.2021 AMC 10B #6
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?
1. A) $74$
2. B) $75$
3. C) $76$
4. D) $77$
5. E) $78$
10.2016 AMC 10A #7
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 $x$ ?
1. A) $50$
2. B) $60$
3. C) $75$
4. D) $90$
5. E) $100$
11.2013 AMC 10B #6
The average age of 33 fifth-graders is 11. The average age of 55 of their parents is 33. What is the average age of all of these parents and fifth-graders?
1. A) $22$
2. B) $23.25$
3. C) $24.75$
4. D) $26.25$
5. E) $28$
12.2001 AMC 10 #16
The mean of three numbers is $10$ more than the least of the numbers and $15$ less than the greatest. The median of the three numbers is $5$ . What is their sum?
1. A) $5$
2. B) $20$
3. C) $25$
4. D) $30$
5. E) $36$
13.2022 AMC 10B #10
Camila writes down five positive integers. The unique mode of these integers is $2$ greater than their median, and the median is $2$ greater than their arithmetic mean. What is the least possible value for the mode?
1. A) $5$
2. B) $7$
3. C) $9$
4. D) $11$
5. E) $13$
14.2021 AMC 10A #16
In the following list of numbers, the integer $n$ appears $n$ times in the list for $1 \leq n \leq 200$ . $1, 2, 2, 3, 3, 3, 4, 4, 4, 4, \dots, 200, 200, \dots , 200$ What is the median of the numbers in this list?
1. A) $100.5$
2. B) $134$
3. C) $142$
4. D) $150.5$
5. E) $167$
15.2021 AMC 10B #19
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 $S$ ?
1. A) $36.2$
2. B) $36.4$
3. C) $36.6$
4. D) $36.8$
5. E) $37$
16.2020 AMC 10A #11
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. A) $1974.5$
2. B) $1975.5$
3. C) $1976.5$
4. D) $1977.5$
5. E) $1978.5$
17.2019 AMC 10A #12
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 consists of $12$ $1\text{s}$ , $12$ $2\text{s}$ , . . . , $12$ $28\text{s}$ , $11$ $29\text{s}$ , $11$ $30\text{s}$ , and $7$ $31\text{s}$ . Let $d$ be the median of the modes. Which of the following statements is true?
1. A) $\mu < d < M$
2. B) $M < d < \mu$
3. C) $d = M =\mu$
4. D) $d < M < \mu$
5. E) $d < \mu < M$
18.2019 AMC 10B #13
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. A) $-5$
2. B) $0$
3. C) $5$
4. D) $\frac{15}{4}$
5. E) $\frac{35}{4}$
19.2018 AMC 10B #14
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?
1. A) $202$
2. B) $223$
3. C) $224$
4. D) $225$
5. E) $234$
20.2012 AMC 10A #13
An iterative average of the numbers $1$ , $2$ , $3$ , $4$ , and $5$ is computed the following way. Arrange the five numbers in some order. Find the mean of the first two numbers, then find the mean of that with the third number, then the mean of that with the fourth number, and finally the mean of that with the fifth number. What is the difference between the largest and smallest possible values that can be obtained using this procedure?
1. A) $\frac{31}{16}$
2. B) $2$
3. C) $\frac{17}{8}$
4. D) $3$
5. E) $\frac{65}{16}$

Problems © Mathematical Association of America (MAA), American Mathematics Competitions. Reproduced for non-commercial educational use. The topic note, tags and key insights are original to this site.