Calculator
Example Data Table
| Observation Group | Value | Frequency | Meaning |
|---|---|---|---|
| A | 12 | 1 | Low value |
| B | 21 | 2 | Common center area |
| C | 28 | 1 | Upper middle value |
| D | 55 | 1 | Right tail value |
Formula Used
The main adjusted sample skewness formula is:
G₁ = √[n(n − 1)] / (n − 2) × g₁
The moment coefficient is:
g₁ = m₃ / m₂^(3/2)
The central moments are:
m₂ = Σ(xᵢ − x̄)² / n
and
m₃ = Σ(xᵢ − x̄)³ / n
Pearson second skewness is:
Sk₂ = 3(mean − median) / sample standard deviation
A positive result means the right tail is longer. A negative result means the left tail is longer.
How to Use This Calculator
- Choose raw values or frequency table mode.
- Enter your dataset in the matching input box.
- Select the skewness method needed for your report.
- Choose decimal places for cleaner output.
- Press the calculate button.
- Review the result, interpretation, moments, and deviation table.
- Download the CSV or PDF file when needed.
Understanding Sample Skewness
What Skewness Shows
Sample skewness shows how a dataset bends around its mean. A balanced dataset has left and right tails that look similar. A positive value means the right tail is longer. A negative value means the left tail is longer. This calculator helps you measure that shape without manual moment work.
Why It Matters
Skewness is useful because the mean alone can hide direction. Two datasets may share the same average. Their tails may still behave very differently. Sales, delivery times, grades, returns, and measurements often show this pattern. A single high value can pull the mean to the right. A single low value can pull it left. The skewness result makes that pull easier to explain.
Input Choices
This tool accepts raw values and frequency rows. Raw values are best for small lists. Frequency rows are useful when repeated values appear often. You can choose an adjusted sample coefficient, a moment coefficient, or Pearson’s median based measure. The adjusted method is usually preferred for sample data. It corrects part of the small sample bias when there are enough observations.
Checking the Result
Always inspect the data before trusting the number. Skewness is sensitive to extreme observations. An outlier may be real, or it may be a typing error. The example table and summary metrics help you review the count, mean, median, variance, and standard deviation. These checks make the final coefficient easier to defend in reports.
Practical Meaning
A result near zero suggests near symmetry. It does not prove normality. It only describes third moment balance. Mild skewness may be acceptable in many practical tasks. Strong skewness can suggest a transformation, a nonparametric method, or a clearer explanation of the data story.
Saving Work
Use the download buttons to save results. The CSV file is useful for spreadsheets. The PDF file is useful for submissions and records. Keep the original values with the output. That makes your calculation transparent.
Advanced Review
For advanced work, compare skewness with a histogram, box plot, and domain knowledge. A finance series, a response time list, and a laboratory batch may need different decisions. The number is a guide. It should support judgment, not replace it. Clear notes make the result easier to repeat later during future reviews too.
FAQs
What is sample skewness?
Sample skewness measures the direction and strength of asymmetry in sample data. It shows whether the longer tail is on the left or right side of the distribution.
What does positive skewness mean?
Positive skewness means the right tail is longer or heavier. The mean is often greater than the median because high values pull the average upward.
What does negative skewness mean?
Negative skewness means the left tail is longer or heavier. Low values pull the mean downward, so the mean may be lower than the median.
Which method should I use?
Use adjusted sample skewness for most sample datasets. Use the moment coefficient for direct third moment analysis. Use Pearson’s method when median based shape is important.
How many values are required?
At least three numeric observations are required for adjusted sample skewness. More values usually give a more reliable shape estimate and reduce small sample instability.
Can skewness prove normality?
No. Skewness only measures asymmetry. A dataset can have low skewness and still fail normality because of peaks, tails, clusters, or outliers.
Why is skewness undefined for equal values?
When all values are equal, the standard deviation is zero. Skewness divides by spread, so the calculation has no valid denominator.
Can I export the result?
Yes. Use the CSV button for spreadsheet use. Use the PDF button for reports, assignments, documentation, or quick sharing.