Arithmetic Mean Formula: Definition, Average Formula, Methods and Examples

The Arithmetic Mean Formula is x̄ = Σx / n, where Σx is the sum of all values and n is the total number of values.
For grouped data, the arithmetic mean is calculated using x̄ = Σfᵢxᵢ / Σfᵢ, where fᵢ is frequency and xᵢ is the value or class midpoint.

The Arithmetic Mean Formula is used when a question asks students to find the average of marks, runs, heights, weights, scores, observations or grouped data. In simple data, students add all values and divide by the number of values. In frequency distribution, they multiply each value by its frequency, add the products, and divide by the total frequency.

In Class 9, Class 10 and Class 11 Maths or Statistics, arithmetic mean appears in data handling, statistics and measures of central tendency. CBSE, ICSE, state board and competitive foundation questions often ask students to calculate mean using direct method, assumed mean method or step deviation method.

Key Takeaways

• Arithmetic Mean Formula: The basic formula is x̄ = Σx / n.
• Meaning: Arithmetic mean gives the average value of a dataset.
• Simple Arithmetic Mean Formula: It is used for ungrouped data where all values have equal importance.
• Grouped Data: The arithmetic mean formula for grouped data is x̄ = Σfᵢxᵢ / Σfᵢ.
• Frequency: In grouped data, Σfᵢ gives the total number of observations.

Arithmetic Mean Formula Structure 2026

What is Arithmetic Mean Formula?

The Arithmetic Mean Formula calculates the average of a set of values. It is found by dividing the sum of all observations by the total number of observations.

Formula:

x̄ = Σx / n

Where:

• x̄ = arithmetic mean
• Σx = sum of all observations
• n = total number of observations
• Σ = sigma symbol, meaning sum of

Example:

For values {4, 8, 12, 16}:

Σx = 4 + 8 + 12 + 16

Σx = 40

n = 4

So:

x̄ = 40 / 4

x̄ = 10

The arithmetic mean is 10.

Arithmetic Mean Formula

The main Arithmetic Mean Formula is used for raw or ungrouped data. It gives one central value that represents the whole dataset.

Formula:

Arithmetic Mean = Sum of all values / Number of values

Symbol form:

x̄ = Σx / n

Expanded form:

x̄ = (x₁ + x₂ + x₃ + ... + xₙ) / n

Where:

• x₁, x₂, x₃, ... xₙ = individual observations
• n = number of observations

Example:

If marks are 12, 15, 18, 20, 25, then:

x̄ = (12 + 15 + 18 + 20 + 25) / 5

x̄ = 90 / 5

x̄ = 18

The arithmetic mean of the marks is 18.

Mean Formula

The mean formula and arithmetic mean formula are the same in school-level statistics. Both are used to calculate the average value of a dataset.

Formula:

Mean = Total of observations / Number of observations

or

Mean = Σx / n

This formula is useful when every observation has equal importance.

For example, if a student scores 70, 80, 90 in three tests:

Mean = (70 + 80 + 90) / 3

Mean = 240 / 3

Mean = 80

The mean score is 80.

Average Formula

The average formula gives the central value of a group of numbers. Arithmetic mean is the most commonly used average in school Maths.

Formula:

Average = Sum of quantities / Number of quantities

For example, the average of 6, 10, 14, 18 is:

Average = (6 + 10 + 14 + 18) / 4

Average = 48 / 4

Average = 12

The average value is 12.

Average is used in questions based on marks, speed, runs, income, rainfall, temperature and expenses.

Arithmetic Average Formula

The arithmetic average formula is another name for the arithmetic mean formula. It is used when values are added together and divided by the total count.

Formula:

Arithmetic Average = Sum of observations / Number of observations

or

x̄ = Σx / n

Example:

For values 3, 6, 9 and 12:

x̄ = (3 + 6 + 9 + 12) / 4

x̄ = 30 / 4

x̄ = 7.5

The arithmetic average is 7.5.

Arithmetic Mean Formula for Ungrouped Data

Ungrouped data contains individual observations listed separately. The direct Arithmetic Mean Formula is used for this type of data.

Formula:

x̄ = Σx / n

Steps:

1. Add all observations.
2. Count the number of observations.
3. Divide the sum by the count.

Example:

Find the arithmetic mean of 5, 7, 9, 11, 13.

Sum:

Σx = 5 + 7 + 9 + 11 + 13

Σx = 45

Number of observations:

n = 5

Mean:

x̄ = 45 / 5

x̄ = 9

The arithmetic mean is 9.

Simple Arithmetic Mean Formula

The simple arithmetic mean formula is used when all observations have equal weight. It is the most basic form of arithmetic mean.

Formula:

x̄ = Σx / n

Here:

• Each value is counted once.
• No frequency or weight is used.
• The formula works for raw data.

Example:

Data:

10, 20, 30

Mean:

x̄ = (10 + 20 + 30) / 3

x̄ = 60 / 3

x̄ = 20

The simple arithmetic mean is 20.

Arithmetic Mean Formula for Discrete Frequency Distribution

Discrete frequency distribution gives values along with their frequencies. Each value is multiplied by its frequency before finding the mean.

Formula:

x̄ = Σfx / Σf

Where:

• x̄ = arithmetic mean
• x = value of observation
• f = frequency
• Σfx = sum of products of value and frequency
• Σf = total frequency

Example:

Now:

Σfx = 6 + 20 + 12

Σfx = 38

Σf = 3 + 5 + 2

Σf = 10

So:

x̄ = 38 / 10

x̄ = 3.8

The arithmetic mean is 3.8.

Arithmetic Mean Formula for Grouped Data

Grouped data is arranged in class intervals such as 0-10, 10-20 and 20-30. The midpoint of each class interval is used as xᵢ.

Formula:

x̄ = Σfᵢxᵢ / Σfᵢ

Where:

• x̄ = arithmetic mean
• fᵢ = frequency of the class
• xᵢ = midpoint of the class interval
• Σfᵢxᵢ = sum of frequency × midpoint
• Σfᵢ = total frequency

Midpoint formula:

xᵢ = (Lower limit + Upper limit) / 2

Example:

For class interval 10-20:

xᵢ = (10 + 20) / 2

xᵢ = 15

The midpoint is used because it represents all values in that class interval.

Weighted Arithmetic Mean Formula

Weighted arithmetic mean is used when values have different importance or weights. It is common in marks, index numbers and weighted score calculations.

Formula:

x̄w = Σwx / Σw

Where:

• x̄w = weighted arithmetic mean
• w = weight
• x = value
• Σwx = sum of weighted values
• Σw = sum of weights

Example:

If scores are 80, 70, 90 and weights are 2, 3, 5, then:

Σwx = (2 × 80) + (3 × 70) + (5 × 90)

Σwx = 160 + 210 + 450

Σwx = 820

Σw = 2 + 3 + 5

Σw = 10

So:

x̄w = 820 / 10

x̄w = 82

The weighted mean is 82.

Assumed Mean Method Formula

The assumed mean method makes calculation easier when the values or class midpoints are large. Students choose one convenient value as assumed mean A.

Formula:

x̄ = A + Σfd / Σf

Where:

• A = assumed mean
• d = x − A
• f = frequency
• Σfd = sum of frequency × deviation
• Σf = total frequency

Example:

If values are close to 50, students can choose:

A = 50

Then:

d = x − 50

This method reduces long multiplication and speeds up exam calculations.

Step Deviation Method Formula

The step deviation method is used when deviations have a common factor. It makes calculation shorter for grouped data with equal class width.

Formula:

x̄ = A + h(Σfu / Σf)

Where:

• A = assumed mean
• h = class width
• u = (x − A) / h
• f = frequency
• Σfu = sum of frequency × step deviation
• Σf = total frequency

This method is useful for large class intervals such as 0-10, 10-20, 20-30, where the class width is constant.

Arithmetic Mean Formula Statistics

In statistics, arithmetic mean is one of the most common measures of central tendency. It gives a single representative value for a dataset.

The arithmetic mean formula statistics questions usually involve:

• Raw data
• Frequency tables
• Grouped class intervals
• Weighted averages
• Assumed mean method
• Step deviation method

The formula changes with the format of the data, but the main idea remains the same: total value divided by total count.

Arithmetic Mean Formula Properties

Arithmetic mean has some important properties used in Statistics and Maths. These properties help in proof-based and shortcut questions.

Important properties:

• The algebraic sum of deviations from the mean is zero.
• The mean is affected by every observation.
• The mean lies between the smallest and largest values in a balanced dataset.
• The mean can be a decimal even when all observations are whole numbers.
• The mean is useful for comparing two or more datasets.

Main property:

Σ(x − x̄) = 0

Example:

For values 2, 4, 6, mean is:

x̄ = 4

Deviations:

2 − 4 = −2

4 − 4 = 0

6 − 4 = 2

Sum of deviations:

−2 + 0 + 2 = 0

Difference Between Arithmetic Mean and Median

Arithmetic mean and median are both measures of central tendency. Mean uses all values, while median uses the middle position after arranging values.

Example:

Data:

2, 4, 6, 8, 100

Mean:

x̄ = 120 / 5 = 24

Median:

Middle value = 6

This shows how an extreme value can pull the mean upward.

How to Find Arithmetic Mean

To find arithmetic mean, first identify whether the data is ungrouped, discrete frequency or grouped frequency data. Then choose the correct formula.

Case 1: Ungrouped data

Use:

x̄ = Σx / n

Example:

Data:

4, 8, 12, 16

Sum:

Σx = 40

Count:

n = 4

Mean:

x̄ = 40 / 4

x̄ = 10

Case 2: Discrete frequency data

Use:

x̄ = Σfx / Σf

Example:

Now:

Σfx = 20 + 60 + 150

Σfx = 230

Σf = 2 + 3 + 5

Σf = 10

Mean:

x̄ = 230 / 10

x̄ = 23

Case 3: Grouped data

Use:

x̄ = Σfᵢxᵢ / Σfᵢ

Example:

Now:

Σfᵢxᵢ = 20 + 90 + 250

Σfᵢxᵢ = 360

Σfᵢ = 4 + 6 + 10

Σfᵢ = 20

Mean:

x̄ = 360 / 20

x̄ = 18

Solved Examples on Arithmetic Mean Formula

Arithmetic Mean Formula questions usually ask students to calculate the average from raw data, frequency data or grouped data. Check the data format before applying the formula.

Example 1: Find the arithmetic mean of {4, 8, 12, 16}

Given:

Data = {4, 8, 12, 16}

Formula:

x̄ = Σx / n

Sum of values:

Σx = 4 + 8 + 12 + 16

Σx = 40

Number of values:

n = 4

Substitute:

x̄ = 40 / 4

x̄ = 10

Answer:

The arithmetic mean is 10.

Example 2: Find the arithmetic mean of 5, 10, 15, 20, 25

Given:

Data = 5, 10, 15, 20, 25

Formula:

x̄ = Σx / n

Sum:

Σx = 5 + 10 + 15 + 20 + 25

Σx = 75

Count:

n = 5

Mean:

x̄ = 75 / 5

x̄ = 15

Answer:

The arithmetic mean is 15.

Example 3: Find the mean for the following frequency distribution

Make a calculation table:

Now:

Σfx = 6 + 24 + 27

Σfx = 57

Σf = 2 + 4 + 3

Σf = 9

Formula:

x̄ = Σfx / Σf

Substitute:

x̄ = 57 / 9

x̄ = 6.33

Answer:

The arithmetic mean is 6.33.

Example 4: Find the mean of grouped data

Find class midpoints:

Now:

Σfᵢxᵢ = 25 + 120 + 175

Σfᵢxᵢ = 320

Σfᵢ = 5 + 8 + 7

Σfᵢ = 20

Formula:

x̄ = Σfᵢxᵢ / Σfᵢ

Substitute:

x̄ = 320 / 20

x̄ = 16

Answer:

The arithmetic mean is 16.

Common Mistakes in Arithmetic Mean Formula

Many arithmetic mean errors happen when students add values correctly and divide by the wrong count. The divisor must match the number of observations or total frequency.

Important checks:

• Use n for ungrouped data.
• Use Σf for frequency data.
• Use midpoint xᵢ for grouped class intervals.
• Multiply f × x before adding in frequency tables.
• Use fᵢ × xᵢ before adding in grouped data tables.
• Use the same class width for step deviation method.
• Check whether the final mean is reasonable for the dataset.

For grouped data, dividing Σfᵢxᵢ by the number of class intervals gives an incorrect mean. Divide by Σfᵢ.

Applications of Arithmetic Mean Formula

The Arithmetic Mean Formula is used in Maths, Statistics, Economics, Science and daily calculations. It gives a simple representative value for a group of observations.

Main applications:

• It calculates average marks in exams.
• It finds average runs in cricket.
• It helps compare class performance.
• It is used in business sales analysis.
• It supports rainfall and temperature data analysis.
• It helps calculate average income or expenditure.
• It is used in Statistics chapters on central tendency.