Grade: 6 Subject: Mathematics Unit: Statistics Introduction SAT Domain: ProblemSolving+DataAnalysis ACT Section: Math

Mean, Median, and Mode

Master the three fundamental measures of central tendency used to describe data sets.

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When analyzing data, we often want to find a single value that represents the "center" or "typical value" of a data set. The three most common measures of central tendency are the mean, median, and mode.

Mean (Average)

The mean is the sum of all values divided by the number of values. It represents the "balance point" of the data.

Mean = (Sum of all values) / (Number of values)

Median (Middle Value)

The median is the middle value when data is arranged in order from least to greatest.

  • Odd number of values: The median is the middle number
  • Even number of values: The median is the average of the two middle numbers

Mode (Most Frequent)

The mode is the value that appears most often in a data set.

  • A data set can have no mode (all values appear once)
  • A data set can have one mode (unimodal)
  • A data set can have multiple modes (bimodal, multimodal)

When to Use Each Measure

Measure Best Used When... Sensitive To...
Mean Data is symmetric without outliers Outliers (extreme values)
Median Data has outliers or is skewed Not affected by outliers
Mode Finding the most common value May not exist or be unique

Step-by-Step Process

To find the Mean:

  1. Add all the values together
  2. Count how many values you have
  3. Divide the sum by the count

To find the Median:

  1. Arrange all values in order (least to greatest)
  2. If odd count: Find the middle value
  3. If even count: Average the two middle values

To find the Mode:

  1. Count how often each value appears
  2. The value(s) appearing most often is the mode

Examples

Example 1: Finding Mean, Median, and Mode

Problem: Find the mean, median, and mode of: 12, 15, 11, 15, 18, 15, 14

Mean:

Sum = 12 + 15 + 11 + 15 + 18 + 15 + 14 = 100

Count = 7

Mean = 100 / 7 = 14.29 (rounded to 2 decimal places)

Median:

Ordered: 11, 12, 14, 15, 15, 15, 18

Middle position (4th of 7): 15

Mode:

15 appears 3 times (most frequent): 15

Example 2: Even Number of Values

Problem: Find the median of: 8, 3, 5, 9, 6, 4

Step 1: Order the data: 3, 4, 5, 6, 8, 9

Step 2: Find middle positions (3rd and 4th of 6): 5 and 6

Step 3: Average: (5 + 6) / 2 = 5.5

Example 3: Effect of Outliers

Problem: Test scores: 85, 88, 90, 87, 89, 25. Compare the mean and median.

Mean: (85 + 88 + 90 + 87 + 89 + 25) / 6 = 464 / 6 = 77.3

Median: Ordered: 25, 85, 87, 88, 89, 90

Middle values: (87 + 88) / 2 = 87.5

Analysis: The outlier (25) pulled the mean down significantly. The median (87.5) better represents the "typical" score.

Example 4: SAT-Style Problem

Problem: The mean of 5 numbers is 20. Four of the numbers are 15, 18, 22, and 25. What is the fifth number?

Step 1: If the mean is 20 with 5 numbers, the sum must be 20 x 5 = 100

Step 2: Current sum: 15 + 18 + 22 + 25 = 80

Step 3: Fifth number: 100 - 80 = 20

Example 5: Multiple Modes

Problem: Find the mode of: 4, 7, 4, 9, 7, 2, 5

Count each value:

2: 1 time | 4: 2 times | 5: 1 time | 7: 2 times | 9: 1 time

Both 4 and 7 appear most often (2 times each).

Answer: This data set is bimodal with modes 4 and 7.

Practice

Solve these problems to build your skills with measures of central tendency.

1. Find the mean of: 24, 18, 32, 26, 20

2. Find the median of: 45, 38, 52, 41, 48, 55, 43

3. Find the mode of: 8, 5, 8, 3, 8, 6, 5, 8

4. The heights (in inches) of 6 students are: 62, 65, 58, 70, 64, 65. Find the mean, median, and mode.

5. A student's test scores are: 78, 85, 92, 88, 82. What score must the student get on the 6th test to have a mean of 85?

6. The median of 7 consecutive integers is 15. What is the largest integer?

7. Which measure of center would be most affected if the value 100 was added to this data set: 12, 14, 15, 16, 18?

8. The mean of 4 numbers is 12. If three of the numbers are 8, 10, and 14, what is the fourth number?

9. Find the median of: 3.5, 4.2, 3.8, 4.5, 3.2, 4.0

10. A data set has no mode. Which of the following could be the data set?
A) 2, 3, 3, 4, 5
B) 1, 2, 3, 4, 5
C) 5, 5, 5, 5, 5
D) 2, 2, 3, 3, 4

Click to reveal answers
  1. Mean = (24 + 18 + 32 + 26 + 20) / 5 = 120 / 5 = 24
  2. Ordered: 38, 41, 43, 45, 48, 52, 55. Median = 45 (4th of 7)
  3. 8 appears 4 times (most). Mode = 8
  4. Mean = 384/6 = 64; Median = (64+65)/2 = 64.5; Mode = 65
  5. Need sum = 85 x 6 = 510. Current sum = 425. Sixth score = 85
  6. If median is 15, sequence is 12, 13, 14, 15, 16, 17, 18. Largest = 18
  7. The mean would be most affected (outliers affect mean more than median or mode)
  8. Sum must be 48. 48 - (8+10+14) = 48 - 32 = 16
  9. Ordered: 3.2, 3.5, 3.8, 4.0, 4.2, 4.5. Median = (3.8+4.0)/2 = 3.9
  10. B - each value appears exactly once, so there is no mode

Check Your Understanding

Question 1: What is the difference between mean and median?

Show answer

The mean is calculated by adding all values and dividing by the count - it's the arithmetic average. The median is the middle value when data is ordered. The mean is affected by extreme values (outliers), while the median is resistant to outliers.

Question 2: When would median be a better measure of center than mean?

Show answer

Median is better when data contains outliers or is skewed. For example, with incomes (where a few very high earners can skew the average) or home prices, the median gives a more accurate picture of the "typical" value.

Question 3: Can a data set have no mode?

Show answer

Yes! If every value in a data set appears exactly the same number of times (especially just once each), the data set has no mode. Example: 1, 2, 3, 4, 5 has no mode.

Question 4: How do you find the median of an even number of values?

Show answer

First, order the values from least to greatest. Then find the two middle values and calculate their average. For example, with 6 values, average the 3rd and 4th values.

Next Steps

  • Practice calculating mean, median, and mode with different data sets
  • Learn to recognize when each measure is most appropriate
  • Move on to learn about Reading Data Displays