How Was the Skewness Calculated?

Skewness describes the asymmetry of a dataset around its mean. Our calculator shows the calculation step by step so you can see how the final skewness value is obtained.

The calculation uses the dataset you enter. The values, intermediate results, and final answer are updated automatically.

Step 1: Find the Mean

First, calculate the arithmetic mean by adding all values and dividing the total by the number of values.

xˉ=∑xin\bar{x}=\frac{\sum x_i}{n}

Where:

• xˉ\bar{x} = mean

• xix_i = each value in the dataset

• nn = number of values

Your calculation:

xˉ=sum of your valuesnumber of values\bar{x}=\frac{\text{sum of your values}}{\text{number of values}}

Mean = [calculated mean]

Step 2: Calculate Deviations From the Mean

Next, subtract the mean from each value in the dataset.

xi−xˉx_i-\bar{x}

The calculator displays each value and its deviation so you can see how far each observation is from the mean.

Step 3: Calculate Cubed Deviations

Each deviation is raised to the third power.

(xi−xˉ)3(x_i-\bar{x})^3

Cubing the deviations is important because positive and negative deviations behave differently when calculating the third central moment. Larger deviations also have a stronger effect on the result.

Step 4: Calculate the Standard Deviation

The standard deviation measures how spread out the values are around the mean.

For population data, the standard deviation is:

σ=∑(xi−xˉ)2n\sigma=\sqrt{\frac{\sum(x_i-\bar{x})^2}{n}}

For sample data, the standard deviation is:

s=∑(xi−xˉ)2n−1s=\sqrt{\frac{\sum(x_i-\bar{x})^2}{n-1}}

The calculator uses the standard deviation required by the selected skewness method and displays the resulting value.

Standard deviation = [calculated standard deviation]

Step 5: Calculate Skewness

Finally, the calculator combines the deviations and standard deviation to determine the skewness.

The exact formula depends on the skewness type selected in the calculator.

Population Skewness

For population data, the standardized third central moment is:

γ1=1n∑(xi−xˉ)3σ3\gamma_1= \frac{\frac{1}{n}\sum(x_i-\bar{x})^3}{\sigma^3}

Where:

• γ1\gamma_1 = population skewness

• nn = number of observations

• xˉ\bar{x} = population mean

• σ\sigma = population standard deviation

Sample Skewness

For sample data, the bias-corrected sample skewness is:

G1=n(n−1)(n−2)∑(xi−xˉs)3G_1= \frac{n}{(n-1)(n-2)} \sum \left(\frac{x_i-\bar{x}}{s}\right)^3

Where:

• G1G_1 = bias-corrected sample skewness

• nn = sample size

• xˉ\bar{x} = sample mean

• ss = sample standard deviation

Your skewness calculation:

G1=[substitute the user’s calculated values here]G_1= \text{[substitute the user’s calculated values here]}

Skewness = [final skewness value]

What Does the Result Mean?

The sign and size of the skewness value help describe the shape of the distribution.

• Skewness near 0: The dataset is approximately symmetric.

• Positive skewness: The distribution has a longer or heavier right tail.

• Negative skewness: The distribution has a longer or heavier left tail.

• Larger absolute skewness: Indicates greater asymmetry.

Important: Sample skewness and population skewness are not interchangeable. This calculator clearly identifies which formula is being used and does not silently mix sample and population calculations.

Detailed Skewness Calculation

The calculation table shows how each value in your dataset contributes to the final skewness result. For every observation, the calculator displays its deviation from the mean, squared deviation, and cubed deviation.

This makes it easier to follow the calculation and understand how values farther from the mean affect the skewness.

What the Columns Mean

• Value: The original observation entered into the dataset.

• Deviation: The difference between the value and the dataset mean.

• Squared Deviation: The deviation multiplied by itself.

• Cubed Deviation: The deviation raised to the third power. This value is especially important when determining the direction and magnitude of skewness.

All values in this table should update automatically whenever you change the dataset.

Skewness Visualization

See the Distribution of Your Data

The skewness value is easier to understand when you can see the shape of the dataset. The visualization displays your data distribution and helps show how the observations are positioned around the center.

The chart dynamically updates when you enter or change your dataset.

The visualization should show:

• Individual data values or their distribution

• The mean

• The median

• The direction of skewness

• An optional normal-distribution reference for comparison

How to Read the Skewness Chart

Right-skewed distribution:
A longer tail extends toward higher values. This usually produces a positive skewness value.

Left-skewed distribution:
A longer tail extends toward lower values. This usually produces a negative skewness value.

Approximately symmetric distribution:
The data is distributed relatively evenly around its center, so the skewness value is generally close to 0.

The visualization gives you a quick way to connect the numerical skewness result with the actual shape of your data.

Current distribution: [Right-skewed / Left-skewed / Approximately symmetric]

Mean: [Calculated mean]
Median: [Calculated median]
Skewness: [Calculated skewness]

What Does Your Skewness Result Mean?

Skewness describes the direction and degree of asymmetry in a dataset. The sign of the result shows which side has the longer tail, while the magnitude indicates how strongly the distribution differs from symmetry.

Skewness ≈ 0

A skewness value close to 0 indicates an approximately symmetric distribution. The data has a relatively balanced shape around its center.

A value near zero does not necessarily mean the dataset is perfectly symmetric. The overall distribution and the context of the data should also be considered.

Positive Skewness

A positive skewness value indicates that the distribution has a longer tail toward higher values, also called the right side of the distribution.

A few relatively large observations can contribute to this right-side tail.

Negative Skewness

A negative skewness value indicates that the distribution has a longer tail toward lower values, also called the left side of the distribution.

A few relatively small observations can contribute to this left-side tail.

Interpreting the Magnitude

The larger the absolute value of skewness, the greater the degree of asymmetry generally indicated by the statistic.

However, there is no single universal cutoff that should be treated as a strict rule for every dataset. The meaning of a particular skewness value depends on the distribution, sample size, subject area, and purpose of the analysis.

For this reason, use the skewness value together with the data distribution and relevant statistical context rather than relying on an arbitrary cutoff alone.

Skewness Formula

The formula used depends on whether the calculation is based on a population or a sample and on the specific definition selected by the calculator.

For bias-corrected sample skewness, the standard formula is:

[
G_1=
\frac{n}{(n-1)(n-2)}
\sum
\left(\frac{x_i-\bar{x}}{s}\right)^3
]

Where:

• (G_1) = bias-corrected sample skewness
• (n) = number of observations
• (x_i) = individual observation
• (\bar{x}) = sample mean
• (s) = sample standard deviation

The calculator applies the selected skewness definition consistently and identifies whether the result represents sample skewness or population skewness.

Why Skewness Results Can Differ

Different textbooks, statistical programs, and calculators may use different definitions of skewness. Some use the population standardized third moment, while others use a bias-corrected sample formula.

As a result, two tools can sometimes produce slightly different skewness values from the same dataset. This does not automatically mean that one calculation is incorrect.

For accurate comparison, make sure you are using the same skewness definition and calculation method in both tools.

Important: Always check whether a reported skewness value represents sample skewness, population skewness, or another statistical definition before comparing results.

What Is Skewness?

Skewness is a statistical measure that describes the asymmetry of a dataset or probability distribution.

In simple terms, skewness helps you understand whether the data is distributed fairly evenly around its center or whether one side has a longer tail.

A distribution can generally be described as symmetric, right-skewed, or left-skewed.

Symmetric Distribution

A symmetric distribution has a similar shape on both sides of its center. The observations are distributed relatively evenly around the middle.

In an approximately symmetric dataset, the mean and median are often close to each other.

Visual: Show a balanced bell-shaped distribution with the mean and median marked near the center.

Right-Skewed Distribution

A right-skewed distribution has a longer tail toward higher values.

Most observations may be concentrated toward the lower or middle part of the range, while a smaller number of larger values extend the distribution to the right.

In many right-skewed datasets, the mean is greater than the median because larger values can pull the mean toward the right.

Visual: Show most data points grouped toward the left with a clearly extended tail toward higher values.

Left-Skewed Distribution

A left-skewed distribution has a longer tail toward lower values.

Most observations may be concentrated toward the higher or middle part of the range, while a smaller number of lower values extend the distribution to the left.

In many left-skewed datasets, the mean is less than the median because smaller values can pull the mean toward the left.

Visual: Show most data points grouped toward the right with a clearly extended tail toward lower values.

Types of Skewness

Positive Skewness

Positive skewness means the distribution has a longer right tail.

The skewness value is greater than zero, and the degree of positive skewness increases as the distribution becomes more asymmetric.

Visual: A simple distribution curve concentrated on the left with a long tail extending to the right. Label the mean and median near the center.

Negative Skewness

Negative skewness means the distribution has a longer left tail.

The skewness value is less than zero, and the distribution becomes more asymmetric as the magnitude of the negative skewness increases.

Visual: A simple distribution curve concentrated on the right with a long tail extending to the left. Label the mean and median near the center.

Zero or Near-Zero Skewness

Zero or near-zero skewness indicates that the distribution is approximately symmetric.

A skewness value close to zero suggests that neither side has a substantially longer tail. The mean and median may also be close, although skewness alone does not guarantee that they are exactly equal.

Visual: A balanced distribution centered around the middle, with the mean and median shown close together.

Quick Comparison