Mean, Median, Mode: all problems

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

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?

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?

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

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 $x$?

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?

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?

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?

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?

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?

When the mean, median, and mode of the list

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

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 $S$?

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

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

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

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.)

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 $x$?

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?

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?

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 $3(a + b + c)$?

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.