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Calculating
Averages

Master the arithmetic mean and learn how to find the average of any data set: a core skill for the ISEE Quantitative Reasoning section.

ISEE Middle Level

What Is an Average?

An average (also called the arithmetic mean) is a single number that represents the "center" of a group of numbers. It tells you what each value would be if every value in the set were exactly the same.

The Average Formula

The formula for the arithmetic mean is:

This can also be written as: \(\bar{x} = \frac{x_1 + x_2 + \cdots + x_n}{n}\), where \(n\) is how many values you have and each \(x_i\) is a value in your data set.

Visualizing the Mean

Imagine the average as the "balance point" of a number line.

0 1 2 3 4 5 6 7 8 9 2 4 6 8
\(\text{Mean} = 5\)

The Core Rule: Three Parts

Every average problem uses three quantities. Know any two and you can find the third.

Sum Average Count (n)
\(\div\)
\(\text{Sum} \div \text{Count} = \text{Average}\)
\(\text{Average} \times \text{Count} = \text{Sum}\)

Rearranging for Missing Values

The ISEE often gives you the average and asks you to find the sum or a missing value. The trick: rearrange the formula.

1
Find the Sum: Multiply the average by the count: \(\text{Sum} = \text{Average} \times n\)
2
Find a missing value: Calculate the sum from the average, then subtract the known values.
3
Find the count: Divide the sum by the average: \(n = \frac{\text{Sum}}{\text{Average}}\)

Average vs. Median vs. Mode

These three terms all describe the "center" of data, but they work differently.

Average (Mean)
Add all values, divide by how many. Affected by extreme values (outliers).
Median
The middle value when data is ordered. Not affected by outliers.
Mode
The value that appears most often. A data set can have no mode, or multiple modes.

How to Calculate an Average

1
List all values in the data set. Make sure you haven't skipped any.
2
Add them up to find the sum. Double-check your arithmetic.
3
Count the values to find \(n\). Include every value, even repeats.
4
Divide: \(\text{Average} = \frac{\text{Sum}}{n}\). Simplify or express as a decimal.

Example 1: Basic Average

Find the average of the data set: {12, 18, 7, 15, 23}

12 18 7 15 23 + + + +
\(\text{Sum} = 75\)
\(\div 5 \text{ values} = 15\)

Example 2: Decimal Average

A student scores 82, 91, 76, 88 on four quizzes. What is her average score?

1
Sum: \(82 + 91 + 76 + 88 = 337\)
2
Count: \(n = 4\) quizzes
3
Divide: \(\frac{337}{4} = 84.25\)

Example 3: Finding a Missing Value

The average of 5 numbers is 40. Four of the numbers are 35, 42, 38, 50. What is the fifth number?

1
Find the total Sum: \(\text{Sum} = \text{Average} \times n = 40 \times 5 = 200\)
2
Sum of known values: \(35 + 42 + 38 + 50 = 165\)
3
Subtract: \(200 - 165 = 35\). The missing number is 35.

Example 4: Combining Groups

A class of 10 boys has an average height of 58 inches. A class of 15 girls has an average height of 56 inches. What is the average height of all 25 students?

1
Boys' total: \(58 \times 10 = 580\) inches
2
Girls' total: \(56 \times 15 = 840\) inches
3
Combined average: \(\frac{580 + 840}{10 + 15} = \frac{1420}{25} = \) 56.8 inches

Common Mistakes to Avoid

✕ Miscounting Values
Forgetting to count a repeated value or accidentally skipping one. Count carefully, especially if values repeat.
✕ Averaging Averages
Averaging two group averages only works if the groups are the same size. Otherwise, find each group's sum first.
✕ Arithmetic Errors
Addition mistakes are the top cause of wrong answers. Add in pairs or group numbers that sum to 10s or 100s.

Summary Cheat Sheet

Problem Type Given Find Formula
Basic average All values Average \(\frac{\text{Sum}}{n}\)
Missing value Average + some values Unknown value \(\text{Avg} \times n - \text{known sum}\)
Combined groups Two group averages + counts Overall average \(\frac{\text{Sum}_1 + \text{Sum}_2}{n_1 + n_2}\)
Target average Current avg, want new avg Needed score \(\text{New Avg} \times (n+1) - \text{current sum}\)

Ready to Practice?

Here are 5 questions to test your understanding. Remember:

Sum First
When given an average, multiply by the count to get the total sum before doing anything else.
Sanity Check
Your answer must fall between the smallest and largest values in the set.

Question 1 of 5

What is the average of 14, 22, 8, 16, 30?

Question 2 of 5

The average of 6 numbers is 15. What is the sum of the 6 numbers?

Question 3 of 5

The average of four numbers is 25. Three of the numbers are 20, 30, 22. What is the fourth number?

Question 4 of 5

A student has quiz scores of 85, 90, 78, 92. What score does she need on the 5th quiz to have an average of 88?

Question 5 of 5

Group A has 8 students with an average score of 70. Group B has 12 students with an average score of 80. What is the combined average of all 20 students?