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Statistics

Mean, Median, Mode

Lesson

Three of the most common ways to describe “the middle” of a data set:

  • Mean (the average): add up every value and divide by how many there are.
    xˉ=sum of all valuesnumber of values\bar{x} = \frac{\text{sum of all values}}{\text{number of values}}
  • Median: the middle value once the data is sorted. With an even count, it’s the average of the two middle values.
  • Mode: the value that appears most often.

For problems where the answer isn’t a clean number, round to two decimal places.

Worked example 1 — mean

Data: 12, 15, 18, 11, 14.

xˉ=12+15+18+11+145=705=14\bar{x} = \frac{12 + 15 + 18 + 11 + 14}{5} = \frac{70}{5} = 14

Worked example 2 — median, even count

Data: 6, 2, 9, 4, 1, 8. Sort first: 1, 2, 4, 6, 8, 91,\,2,\,4,\,6,\,8,\,9. Six values, so the median is the average of the 3rd and 4th:

median=4+62=5\text{median} = \frac{4 + 6}{2} = 5

How to type your answer

Type a single number. Use a decimal point for fractions if needed, and round to two decimal places when the answer isn’t exact. Examples: 14, 22.5, 17.67, -2.

Practice

Work through these. Stuck? Click Get a hint.

Warm-Up

Quick problems to get going.

Problem 1

Mean of 2, 4, 6, 8, 10\text{Mean of } 2,\ 4,\ 6,\ 8,\ 10

Problem 2

Mean of 1, 3, 5, 7, 9\text{Mean of } 1,\ 3,\ 5,\ 7,\ 9

Problem 3

Median of 3, 8, 1, 4, 7\text{Median of } 3,\ 8,\ 1,\ 4,\ 7

Problem 4

Mode of 5, 2, 7, 5, 9\text{Mode of } 5,\ 2,\ 7,\ 5,\ 9

Practice

Standard problems matching the lesson.

Problem 5

Mean of 12, 15, 18, 11, 14\text{Mean of } 12,\ 15,\ 18,\ 11,\ 14

Problem 6

Mean of −2, 4, −1, 3, 6\text{Mean of } -2,\ 4,\ -1,\ 3,\ 6

Problem 7

Median of 6, 2, 9, 4, 1, 8\text{Median of } 6,\ 2,\ 9,\ 4,\ 1,\ 8

Problem 8

Median of 15, 20, 25, 30\text{Median of } 15,\ 20,\ 25,\ 30

Problem 9

Mode of 4, 7, 4, 2, 9, 4, 5\text{Mode of } 4,\ 7,\ 4,\ 2,\ 9,\ 4,\ 5

Problem 10

Mean of 10, 20, 30, 40, 50\text{Mean of } 10,\ 20,\ 30,\ 40,\ 50

Problem 11

Median of 7, 3, 9, 5, 1, 7\text{Median of } 7,\ 3,\ 9,\ 5,\ 1,\ 7

Problem 12

Mean of 2.5, 3.5, 4.5, 5.5, 4.0\text{Mean of } 2.5,\ 3.5,\ 4.5,\ 5.5,\ 4.0

Problem 13

Mode of 12, 15, 18, 12, 20, 12, 17\text{Mode of } 12,\ 15,\ 18,\ 12,\ 20,\ 12,\ 17

Problem 14

Median of 100, 200, 150, 175, 125\text{Median of } 100,\ 200,\ 150,\ 175,\ 125

Challenge

Harder problems — edge cases, trickier numbers, multiple steps.

Problem 15

Mean of 17, 19, 14, 22, 18, 16 (round to 2 dp)\text{Mean of } 17,\ 19,\ 14,\ 22,\ 18,\ 16\ \text{(round to 2 dp)}

Problem 16

Mean of 3, 7, 11, 4, 8, 6, 9 (round to 2 dp)\text{Mean of } 3,\ 7,\ 11,\ 4,\ 8,\ 6,\ 9\ \text{(round to 2 dp)}

Problem 17

Median of 12, 7, 19, 4, 8, 15, 11, 6\text{Median of } 12,\ 7,\ 19,\ 4,\ 8,\ 15,\ 11,\ 6

Problem 18

Mode of 8, 5, 8, 3, 5, 8, 2, 5, 8\text{Mode of } 8,\ 5,\ 8,\ 3,\ 5,\ 8,\ 2,\ 5,\ 8

Problem 19

Mean of −5, −2, 0, 3, 7, 1 (round to 2 dp)\text{Mean of } -5,\ -2,\ 0,\ 3,\ 7,\ 1\ \text{(round to 2 dp)}

Problem 20

Median of 1.2, 3.5, 2.8, 4.1, 2.0\text{Median of } 1.2,\ 3.5,\ 2.8,\ 4.1,\ 2.0

Problem 21

If the mean of x, 5, 8, 10, 12 is 9, find x.\text{If the mean of } x,\ 5,\ 8,\ 10,\ 12 \text{ is } 9,\ \text{find } x.

Problem 22

Test scores: 85, 78, 92, 88. Score on next test to bring mean to 87?\text{Test scores: } 85,\ 78,\ 92,\ 88.\ \text{Score on next test to bring mean to } 87\text{?}