Practice Maths

L50 — The Mean

Key Ideas

Key Terms

mean
The sum of all values divided by the number of values. Also called the average. Mean = sum ÷ count.
average
Another name for the mean. Used to represent the "typical" or "balanced" value in a dataset.
sum
The total obtained by adding all values in the dataset together.
outlier
A data value that is much higher or lower than all other values. Even one outlier can significantly distort the mean.

Calculating and Using the Mean

The mean (also called the average) is found by:

Mean = Sum of all values ÷ Number of values

Use the mean when:

  • The data has no extreme outliers that would pull the average up or down
  • You want one number to represent the "typical" value in a balanced dataset

Be careful when data has outliers — one very large or very small value can make the mean unrepresentative of most values.

Hot Tip: The mean doesn't have to be a whole number — that's completely fine! If the sum doesn't divide evenly, leave your answer as a decimal or fraction.

Worked Example

Example 1 — Calculate the mean of: {4, 7, 2, 9, 3}

Sum = 4 + 7 + 2 + 9 + 3 = 25

Count = 5 values

Mean = 25 ÷ 5 = 5

Example 2 — Find the missing value:

The mean of 5 numbers is 8. Four of the numbers are 6, 9, 4, 11. What is the fifth?

Total sum needed = mean × count = 8 × 5 = 40

Sum of known values = 6 + 9 + 4 + 11 = 30

Missing value = 40 − 30 = 10

What Does the Mean Actually Tell You?

The mean (or average) is a single number that represents the "middle" of a dataset — it's where the data would balance if you put it on a number line scale. If five friends scored 3, 7, 2, 9, and 4 on a quiz, the mean of 5 tells you that if everyone had scored the same, they would all have scored 5. It doesn't mean anyone actually scored 5 — it's a representation of the group's overall performance.

The mean is the most commonly used measure of centre, and it's what most people mean when they say "average."

How to Calculate the Mean

The formula is simple: Mean = Sum of all values divided by the number of values.

Step 1: Add up all the values. Step 2: Count how many values there are. Step 3: Divide the sum by the count.

For example, with scores 8, 5, 12, 3, 7: Sum = 8 + 5 + 12 + 3 + 7 = 35. Count = 5. Mean = 35 ÷ 5 = 7.

The mean doesn't have to be a whole number. If the answer is a decimal, that's completely fine — leave it as a decimal.

Finding a Missing Value Using the Mean

Sometimes you're told the mean and asked to find a missing value. Work backwards: if the mean is known and so is the count, you can find the total sum (mean × count). Then subtract the values you do know to find the missing one.

For example: The mean of 4 numbers is 9. Three of the numbers are 6, 10, and 8. Total needed = 9 × 4 = 36. Known sum = 6 + 10 + 8 = 24. Missing value = 36 − 24 = 12.

The Effect of Outliers on the Mean

An outlier is a value that is much higher or lower than all the others. Even a single outlier can pull the mean up or down significantly, making it a poor representation of a typical value. Imagine seven students score 60, 65, 63, 68, 62, 66, and 64 — the mean is about 64. But if one student is sick and scores 10, the mean drops to around 57. That's not representative of how most students actually performed. In situations like this, the median (middle value) is usually a better choice.

Exam tip — show your working: Always write your sum and your count clearly before dividing. Examiners award marks for working, so even if you make a small arithmetic error, you can still earn most of the marks if your method is clear. Write "Sum = ..., Count = ..., Mean = Sum ÷ Count = ..." every time.

  1. Calculate the Mean

    Find the mean of each dataset. Show your working (sum of all values ÷ number of values).

    1. {3, 7, 5, 9, 1}
    2. {12, 15, 8, 14, 11}
    3. {6, 4, 8, 2, 10, 6}
    4. {23, 17, 31, 29, 20}
    5. {5.2, 3.8, 6.4, 4.6}
    6. {102, 98, 105, 107, 88}
    7. {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}
    8. {14, 16, 18, 20, 22}

  2. Effects on the Mean & Missing Values

    1. The mean of {6, 8, 10, 12} is 9. A fifth value of 14 is added. Without fully recalculating, predict whether the new mean will be higher or lower than 9. Then verify by calculating.
    2. The mean of {6, 8, 10, 12} is 9. The value 8 is removed. Will the new mean be higher or lower? Calculate the new mean.
    3. The mean of 4 numbers is 15. Three of the numbers are 12, 18, 14. Find the fourth number.
    4. The mean of 6 numbers is 20. Five of the numbers are 18, 22, 15, 24, 19. Find the sixth number.
    5. A student scored 72, 68, and 80 on three tests. What score do they need on a fourth test to have a mean of 75?
    6. The mean of five numbers is 30. If each number is increased by 5, what is the new mean?
    7. Find the value of x if the mean of {x, 8, 12, 6} is 9.
    8. The mean of {4, 7, n, 5, 9} is 7. Find n.

  3. When is the Mean Misleading?

    1. The dataset {5, 6, 6, 7, 7, 8, 55} has a mean of about 12.9. Explain why the mean does not represent the typical value in this dataset.
    2. Identify the outlier in: {31, 28, 33, 29, 35, 27, 98}. Calculate the mean with and without the outlier. What is the difference?
    3. A real estate agent advertises that the "average house price" in a suburb is $1.2 million. The actual prices are: $450,000, $480,000, $460,000, $470,000, $4,800,000. Explain why the mean is misleading here. What is the actual mean?
    4. True or False: "The mean is always the best measure to describe a typical value in a dataset." Explain your answer.

  4. Mean in Context

    1. A class of 5 students scored: 64, 71, 58, 79, 68 on a maths test. What is the class mean? If a sixth student who scored 92 joins the class, what is the new mean?
    2. Rainfall (mm) recorded each month for 6 months: 45, 62, 38, 55, 71, 49. What is the average monthly rainfall? How much more rain would need to fall in month 7 to raise the average to 55 mm?
    3. A basketball player's points per game over 5 games: 18, 22, 15, 27, 13. After the 6th game, the mean increases to 20. How many points did the player score in the 6th game?
    4. The mean weight of 8 parcels is 2.5 kg. Three of the parcels weigh 1.8 kg, 3.2 kg, and 2.6 kg. A fourth parcel weighs x kg. If the remaining four parcels each weigh 2.5 kg, find x.

  5. Complete the Calculation

    Each calculation below has been started. Fill in the missing steps and give the final mean.

    1. Data: {9, 15, 12, 6, 18}    Sum = 9 + 15 + 12 + 6 + 18 = ?    Count = ?    Mean = ?
    2. Data: {40, 50, 60}    Sum = ?    Mean = ? ÷ 3 = ?
    3. Data: {7, 7, 7, 7, 7, 7}    Sum = ?    Mean = ?
    4. Data: {100, 200, 300, 400}    Mean = (? + ? + ? + ?) ÷ 4 = ? ÷ 4 = ?

  6. Calculate from a Data Table

    The table below shows five students' quiz scores. Calculate the mean score for the group.

    StudentScore
    Alex14
    Bella18
    Carlos14
    Diana20
    Ethan9
    1. What is the total of all five scores?
    2. Calculate the mean score.
    3. Which students scored above the mean? Which scored below?

  7. True or False?

    State whether each statement is True or False and give a reason.

    1. The mean of {10, 10, 10, 10} is 10.
    2. Adding a value equal to the current mean will not change the mean.
    3. The mean must always be one of the values in the dataset.
    4. If all values in a dataset are doubled, the mean is also doubled.
    5. The mean of {1, 2, 3, 4, 100} is greater than 20.
    6. If you remove the largest value from a dataset, the mean will always decrease.

  8. Spot the Error

    A student calculated the mean of {6, 9, 3, 12, 10} as follows:

    Student's working:

    Sum = 6 + 9 + 3 + 12 + 10 = 40

    Mean = 40 ÷ 4 = 10

    1. Identify the error the student made.
    2. Write the correct working and give the correct mean.
    3. Why is it important to count the number of values carefully before dividing?

  9. Which is Correct?

    Two students each calculated the mean of the dataset {3, 8, 5, 12, 7, 9}.

    Student A

    Sum = 3 + 8 + 5 + 12 + 7 + 9 = 44

    Count = 6

    Mean = 44 ÷ 6 ≈ 7.33

    Student B

    Sum = 3 + 8 + 5 + 12 + 7 + 9 = 44

    Count = 5

    Mean = 44 ÷ 5 = 8.8

    1. Which student is correct? How do you know?
    2. What mistake did the other student make?
    3. Calculate the correct mean and round your answer to two decimal places.

  10. Extended Investigation

    A sports coach records the number of goals scored by a team in their last 8 games:

    Game12345678
    Goals31425144
    1. Calculate the mean number of goals per game.
    2. In game 9, the team scores 0 goals. Recalculate the new mean.
    3. How many goals would the team need to score in game 10 (after game 9's 0) to bring the mean back up to 3?
    4. The coach says: "We average 3 goals per game." Is this an accurate claim based on the first 8 games? Explain.
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