Mean, Median, Mode, and Range – ASVAB Math Review

1.Mean (Average)

The sum of all values divided by the number of values.

x̄ = Σx / n

Σx: Sum of data. n: Number of items.

Example

What is the mean of the data set: 4, 8, 6, 2?

Show solution

Step 1: Add all the numbers together: $$4 + 8 + 6 + 2 = 20$$

Step 2: Count the total number of items: $$n = 4$$

Step 3: Divide the sum by the count: $$\frac{20}{4} = 5$$

Practice problems

Problem 1
Find the average of the following scores on a math quiz: 70, 80, 90, 100.

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Answer: 85

Step 1: Calculate the sum of the scores: $$70 + 80 + 90 + 100 = 340$$
Step 2: Count the number of items: 4
Step 3: Divide the total sum by the count: $$\frac{340}{4} = 85$$
Problem 2
A driver tracks their daily mileage over 5 days: 12, 18, 15, 20, and 10 miles. What is the mean daily mileage?

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Answer: 15 miles

Step 1: Add the daily mileages together: $$12 + 18 + 15 + 20 + 10 = 75$$
Step 2: Count the number of days recorded: 5
Step 3: Divide the total mileage by the number of days: $$\frac{75}{5} = 15$$ miles
Problem 3
The temperatures recorded during a 3-day weekend were 72°F, 78°F, and 84°F. What was the average temperature?

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Answer: 78°F

Step 1: Calculate the total sum of the temperatures: $$72 + 78 + 84 = 234$$
Step 2: Count the number of days: 3
Step 3: Divide the sum by the count: $$\frac{234}{3} = 78^\circ\text{F}$$

Remember

Always find the exact sum of all values before dividing by the total count of items.

2.Median

The middle value when data is ordered from least to greatest.

Middle Position

If n is odd: Middle value. If n is even: Average of two middle values.

Example

Find the median of the data set: 11, 3, 7, 5, 9.

Show solution

Step 1: Arrange the data from least to greatest: 3, 5, 7, 9, 11

Step 2: Count the items (5 items, which is odd) and locate the middle number

Step 3: Identify the median value in the center: 7

Practice problems

Problem 4
What is the median value of the following list of numbers: 22, 14, 18, 25, 12, 19, 20?

Show solution

Answer: 19

Step 1: Sort the list in ascending order: 12, 14, 18, 19, 20, 22, 25
Step 2: Count the items (7 items, which is odd)
Step 3: Locate the fourth number which is in the exact middle: 19
Problem 5
Find the median of this data set: 6, 2, 8, 4.

Show solution

Answer: 5

Step 1: Arrange the data from least to greatest: 2, 4, 6, 8
Step 2: Count the items (4 items, which is even). Locate the two middle numbers: 4 and 6
Step 3: Find the average of the two middle numbers: $$\frac{(4 + 6)}{2} = 5$$
Problem 6
A store counts the number of customers hourly: 40, 48, 42, 41, 49, 45. What is the median number of customers?

Show solution

Answer: 43.5

Step 1: Rearrange the numbers in order: 40, 41, 42, 45, 48, 49
Step 2: Identify the two middle elements from the 6 values: 42 and 45
Step 3: Calculate their average: $$\frac{(42 + 45)}{2} = \frac{87}{2} = 43.5$$

Remember

You must arrange the numbers in ascending order before locating the middle position.

3.Mode

The value(s) that appear most frequently in a data set.

Most Frequent

A set can have one mode, more than one (multimodal), or no mode.

Example

Find the mode of the data set: 3, 7, 3, 5, 9, 7, 3.

Show solution

Step 1: Count how many times each value appears: 3 appears three times; 7 appears twice; 5 and 9 appear once

Step 2: Compare frequencies to find the highest count

Step 3: Identify the value with the most occurrences: 3

Practice problems

Problem 7
What is the mode of the following set of package weights: 5 lbs, 8 lbs, 5 lbs, 12 lbs, 8 lbs, 5 lbs?

Show solution

Answer: 5 lbs

Step 1: Count the occurrences: 5 appears three times, 8 appears twice, 12 appears once
Step 2: Identify the highest frequency count
Step 3: The mode is 5 lbs because it appears the most
Problem 8
Determine the mode of this sequence: 14, 15, 16, 17, 18

Show solution

Answer: No mode

Step 1: Count the occurrences of each number in the set
Step 2: Note that every single number appears exactly once
Step 3: Conclude that there is no mode since no number repeats more than others
Problem 9
Find the mode for the survey responses: Yes, No, Yes, Maybe, No, Yes.

Show solution

Answer: Yes

Step 1: Tally the entries: Yes appears 3 times, No appears 2 times, Maybe appears 1 time
Step 2: Compare the total frequencies of each response
Step 3: The entry with the highest frequency is Yes

Remember

Count the occurrences of each number carefully; a set can have no mode if all values appear equally.

4.Range

The difference between the highest and lowest values.

Max−Min

Shows the spread or distribution of the data.

Example

Find the range of the data set: 14, 6, 22, 18, 9.

Show solution

Step 1: Locate the maximum value: $$\text{Max} = 22$$

Step 2: Locate the minimum value: $$\text{Min} = 6$$

Step 3: Subtract the minimum from the maximum: $$22 - 6 = 16$$

Practice problems

Problem 10
What is the range of a student's test scores: 65, 88, 94, 72, 81?

Show solution

Answer: 29

Step 1: Identify the maximum score: 94
Step 2: Identify the minimum score: 65
Step 3: Subtract the minimum score from the maximum score: $$94 - 65 = 29$$
Problem 11
A local business recorded bank deposits of $200, $850, $150, and $400. Calculate the range of these deposits.

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Answer: $700

Step 1: Find the maximum deposit value: $850
Step 2: Find the minimum deposit value: $150
Step 3: Compute the difference: 850 - 150 = $700
Problem 12
Find the range of the following group of negative numbers: -3, -8, -2, -10.

Show solution

Answer: 8

Step 1: Determine the maximum value (closest to zero): $$\text{Max} = -2$$
Step 2: Determine the minimum value (farthest from zero): $$\text{Min} = -10$$
Step 3: Subtract the minimum from the maximum: $$-2 - (-10) = -2 + 10 = 8$$

Remember

Always check the full data set to isolate the absolute maximum and absolute minimum numbers.