Mean Formula

The mean formula is used to calculate the arithmetic mean, a fundamental measure of central tendency in descriptive statistics. It determines the average value of a dataset by dividing the total sum of observations by the number of observations.

This page covers all major statistical mean formulas, including:

• Arithmetic mean formula

• Sample mean formula

• Population mean formula

• Mean formula for grouped data

• Assumed mean formula

• Weighted mean formula

Arithmetic Mean Formula (Basic Formula)

The basic arithmetic mean formula is:

Mean = Σx / n

Where:

• Σ (Sigma) = Summation symbol

• x = Individual observation

• Σx = Sum of all observations

• n = Number of observations

This formula is used for ungrouped raw numerical data.

Sample Mean Formula

The sample mean represents the average of a subset of a population.

x̄ = Σx / n

Where:

• x̄ (x-bar) = Sample mean

• Σx = Sum of sample observations

• n = Sample size

The sample mean is commonly used in statistical estimation and inferential statistics.

Population Mean Formula

The population mean represents the true average of an entire population.

μ = Σx / N

Where:

• μ (Mu) = Population mean

• Σx = Sum of all population values

• N = Total number of elements in the population

The population mean is considered a population parameter.

Sample Mean vs Population Mean (Formula Comparison)

This distinction is fundamental in statistical analysis.

Mean Formula for Grouped Data (Frequency Distribution)

When data is organized into a frequency table, the mean formula becomes:

Mean = Σfx / Σf

Where:

• x = mean

• 𝑓 = Frequency of each class

• 𝑥 = Class midpoint

• ∑fx = Sum of ( frequency × midpoint )

• ∑f = Total frequency

Steps:

1. Determine the midpoint of each class interval

2. Multiply each midpoint by its frequency

3. Sum all fx values

4. Divide by total frequency

This formula is used in statistical tables and frequency distributions.

Assumed Mean Formula (Shortcut)

The assumed mean method simplifies grouped data calculations.

Mean = A + (Σfd / Σf)

Where:

• A = Assumed mean

• d = Deviation from assumed mean (x − A)

• f = Frequency

• Σfd = Sum of frequency × deviation

• Σf = Total frequency

This method reduces computational complexity for large datasets.

Weighted Mean Formula

When values carry different levels of importance (weights), the weighted mean formula is used:

Weighted Mean = Σwx / Σw

Where:

• w = Weight assigned to each value

• x = Observation

• Σwx = Sum of weighted values

• Σw = Total weight

The weighted mean is widely used in finance, economics, and academic grading systems.

Statistical Symbols Used in Mean Formulas

Statistical notation is essential for applying mean formulas correctly.

The mean formula is foundational for calculating variance, standard deviation, and other statistical measures.

Frequently Asked Questions About Mean Formula

What is the formula of mean?

The arithmetic mean formula is:

Mean = Σx / n

It divides the sum of observations by the total number of observations.

What is the difference between sample mean and population mean formula?

Sample mean uses x̄ = Σx / n, while population mean uses μ = Σx / N. 

What does Σ mean in statistics?

Σ (Sigma) represents summation. 

What is the shortcut formula of mean?

Mean = A + (Σfd / Σf)

It is used in grouped data to simplify calculations using an assumed mean.