Average

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Quick recall

• Find the total first, then divide by the number of observations.
• Weight group averages by group size.

On this page

Formula Cards

1. Mean

$$\bar{x}=\frac{\sum x}{n}$$

2. Add one value

$$\text{New average}=\frac{n\,\text{Old average}+x}{n+1}$$

3. Remove one value

$$\text{New average}=\frac{n\,\text{Old average}-x}{n-1}$$

Requires n > 1.

4. Weighted average

$$\bar{x}=\frac{n_1a_1+n_2a_2}{n_1+n_2}$$

n₁ and n₂ are group sizes; a₁ and a₂ are their averages.

5. Consecutive numbers

$$\text{Average}=\frac{\text{First}+\text{Last}}{2}$$

Worked Examples

Example 1

Question: Five numbers have average 10. Add 20. Find the new average.

Step 1: Recover the original sum.

$$5\times10=50$$

Step 2: Add the value and divide by the new count.

$$\bar{x}=\frac{50+20}{6}=\frac{35}{3}\approx11.67$$

Answer: Approximately 11.67.

Example 2

Question: Ten numbers have average 15. Remove 25. Find the new average.

Step 1: Recover the original sum.

$$10\times15=150$$

Step 2: Subtract the value and reduce the count.

$$\bar{x}=\frac{150-25}{9}=\frac{125}{9}\approx13.89$$

Answer: Approximately 13.89.

Example 3

Question: Class A: 20 students, average 60. Class B: 30 students, average 70. Find the combined average.

Step 1: Find the combined total.

$$20\times60+30\times70=3300$$

Step 2: Divide by all 50 students.

$$\bar{x}=\frac{3300}{50}=66$$

Answer: 66.

Example 4

Question: Find the average of the first 10 natural numbers.

Step 1: Identify the first and last values.

$$1,\quad10$$

Step 2: Use the midpoint shortcut for consecutive numbers.

$$\bar{x}=\frac{1+10}{2}=5.5$$

Answer: 5.5.

Drill

Try each question before opening its answer.

Question 1

Average of 6 numbers is 20. Find their sum.

• Show answer

  120

Question 2

Average of 8 numbers is 12. A 9th number, 30, is added. New average?

• Show answer

  14

Question 3

Average age of 30 students is 12 years. If teacher's age (40) is included, new average?

• Show answer

  12.9

Question 4

Average of 5 consecutive numbers is 23. Find the largest.

• Show answer

  25

Question 5

Two sections of 25 and 35 students score averages 80 and 60. Combined average?

• Show answer

  68.3

Question 6

Average of first 20 odd numbers?

• Show answer

  20

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