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\(\bar{x}\)

Calculating the
Mean

Master the arithmetic mean: how to find the average of any data set, interpret what it tells you, and avoid common traps on the SSAT.

SSAT Upper Level: Quantitative

What Is the Mean?

The mean (also called the arithmetic average) is the single number that represents the "center" of a data set. It answers the question: if every value were the same, what would that value be?

The Formula

In words: add up all the values, then divide by how many values there are. That's it.

Visualizing the Balance Point

2 4 6 8 10 12 3 5 7 9 11 Mean = 7 Data: {3, 5, 7, 9, 11} Sum = 35, Count = 5, Mean = 35 ÷ 5 = 7

Why Does the Mean Matter?

Summarize
A single number that represents an entire data set.
Compare
Compare groups: "Class A averaged 85, Class B averaged 78."
Predict
Estimate what a "typical" new value might be.

The Core Rule

Sum Mean Count (n)
\( \div n \)
\( \div \text{Mean} \)
 × each other
\( \text{Sum} = \text{Mean} \times n \)

Rearranging for Sum and Count

1
Find the mean: \( \bar{x} = \dfrac{\text{Sum}}{n} \)
2
Find the sum: \( \text{Sum} = \bar{x} \times n \). Useful when a problem gives the mean and count, and asks for the total.
3
Find the count: \( n = \dfrac{\text{Sum}}{\bar{x}} \). Useful when a problem gives the mean and sum, and asks how many values.

Mean vs Median vs Mode

Mean Median Mode
Definition Sum ÷ count Middle value when sorted Most frequent value
Affected by outliers? Yes No No
Always unique? Yes Yes No (can have multiple)
Uses every value? Yes No No
Example: {2, 3, 3, 8, 100} 23.2 3 3

Step-by-Step Approach

1
List all values. Write them out so nothing is missed.
2
Add them up. Find the sum of all values.
3
Count them. Determine \( n \), the number of values.
4
Divide. Compute \( \bar{x} = \dfrac{\text{Sum}}{n} \).
5
Check. Does your answer make sense? It should fall between the smallest and largest values.

Example 1: Basic Mean

Problem: Find the mean of {12, 8, 15, 10, 5}.

12 8 15 10 5
\( \text{Mean} = 10 \)

Example 2: Missing Value

Problem: The mean of five numbers is 14. Four of the numbers are 10, 12, 18, 16. What is the fifth number?

1
Find the total sum: \( \text{Sum} = \bar{x} \times n = 14 \times 5 = \) 70
2
Add known values: \( 10 + 12 + 18 + 16 = \) 56
3
Subtract: \( 70 - 56 = \) 14

Example 3: Combined Groups

Problem: Group A has 6 students with a mean score of 80. Group B has 4 students with a mean score of 90. What is the overall mean?

1
Group A total: \( 80 \times 6 = \) 480
2
Group B total: \( 90 \times 4 = \) 360
3
Combined mean: \( \dfrac{480 + 360}{6 + 4} = \dfrac{840}{10} = \) 84

Example 4: Adding a New Value

Problem: The mean of 8 numbers is 25. A ninth number, 43, is added. What is the new mean?

1
Original sum: \( 25 \times 8 = \) 200
2
New sum: \( 200 + 43 = \) 243
3
New mean: \( \dfrac{243}{9} = \) 27

Common Mistakes

✕ Wrong Count
Miscounting the number of values. If a problem says "five test scores," double-check you have exactly 5 numbers.
✕ Averaging Averages
Taking the mean of two means without accounting for group sizes. You must weight by count.
✕ Forgetting to Add All
Skipping a value in the sum. Write each value down and check off as you add.

Summary Table

Problem Type Strategy Key Formula
Find the mean Add all values, divide by count \( \bar{x} = \frac{\text{Sum}}{n} \)
Find a missing value Use mean × count to get sum, subtract known values \( x = \bar{x} \cdot n - \text{known sum} \)
Combined groups Get each group's total, then divide by total count \( \bar{x} = \frac{\text{Sum}_A + \text{Sum}_B}{n_A + n_B} \)
Add/remove a value Adjust the sum and count, then recalculate \( \bar{x}_{\text{new}} = \frac{\text{Sum} \pm v}{n \pm 1} \)

Question 1 of 5

What is the mean of {4, 7, 10, 3, 6}?

Question 2 of 5

The mean of six numbers is 15. What is the sum of the six numbers?

Question 3 of 5

The mean of four numbers is 20. Three of the numbers are 15, 22, and 25. What is the fourth number?

Question 4 of 5

Group X has 5 students with a mean score of 70. Group Y has 3 students with a mean score of 90. What is the combined mean of all 8 students?

Question 5 of 5

The mean of 10 numbers is 8. An eleventh number, 30, is added to the set. What is the new mean?