Skewness Calculator

Pearson's Coefficient of Skewness (First and Second)

Pearson's coefficients are simple, rule-of-thumb measures of skewness. They are particularly useful when you have a clear mode or median in your data.

Pearson's First Coefficient (Mode Skewness): SK₁ = (Mean - Mode) / Standard Deviation

This version is used when the mode is clearly defined. It measures the distance between the mean and the mode, scaled by the standard deviation.

Pearson's Second Coefficient (Median Skewness): SK₂ = 3(Mean - Median) / Standard Deviation

This version is more commonly used because the median always exists and is unique, unlike the mode. It's a good approximation when the distribution is moderately skewed. It highlights how far the mean is from the median, relative to the spread of the data.

• Simple and intuitive measure: Easy to calculate and understand, providing a quick glance at data asymmetry.
• Based on central tendency: Directly uses the mean, median, and mode, which are familiar statistical concepts.
• Robust to outliers (for median version): The median is less affected by extreme values than the mean, making SK₂ more stable for some datasets.
• Range: typically -3 to +3: While not strictly bounded, most practical datasets will have Pearson's coefficients within this range.
• Zero indicates symmetry: A value of zero suggests a perfectly symmetrical distribution where the mean, median, and mode are equal.

Sample Skewness (Moment-based)

Sample skewness is a more precise, moment-based measure that uses the third power of the deviations from the mean. It's often preferred for rigorous statistical analysis.

g₁ = [n / ((n-1)(n-2))] * Σ[(xᵢ - x̄)³ / s³]

where:

• n = number of data points (sample size)
• xᵢ = individual data point
• x̄ = sample mean
• s = sample standard deviation
• Σ = summation

• Moment-based measure: Derived from the third standardized moment, providing a more formal statistical definition.
• Adjusts for sample size: The `n / ((n-1)(n-2))` factor is a bias correction, making it a better estimator for the true population skewness, especially for smaller samples.
• More sensitive to outliers: Because it cubes the deviations, extreme values have a much larger impact on the result.
• Unbiased estimator: This formula provides an unbiased estimate of the population skewness, meaning that on average, it will correctly estimate the true skewness of the larger population from which the sample was drawn.
• Used in small to moderate samples: This is the standard formula for calculating skewness from a sample dataset.

Population Skewness (Third Standardized Moment)

Population skewness is the theoretical measure of skewness for an entire population, often denoted by the Greek letter gamma (γ₁).

γ₁ = E[(X - μ)³] / σ³

where:

• E = Expected value (average over the entire population)
• X = Random variable
• μ = Population mean
• σ = Population standard deviation

• Third standardized moment: This is the true, theoretical measure of skewness for a complete population distribution.
• No bias correction: Since it applies to the entire population, there's no need for sample size adjustments.
• Theoretical measure: Represents the ideal skewness of the underlying data generation process.
• Used in large samples or theoretical contexts: When dealing with very large datasets that approximate a population, or in theoretical statistical discussions.
• Distribution properties: Directly reflects the inherent asymmetry of the probability distribution itself.

Positive Skewness (Right-Skewed Distribution)

A positive skewness value indicates that the tail on the right side of the distribution is longer or fatter than the left side. This means there are more extreme high values (outliers) pulling the mean to the right.

• Right-tailed distribution: The "tail" of the histogram or probability density function extends further to the right.
• Mean > Median > Mode: In a positively skewed distribution, the mode (most frequent value) is typically the smallest, followed by the median, and then the mean (which is pulled towards the longer tail).
• Longer right tail: Visually, the distribution appears to have a stretched-out right side.
• Common in income data: For example, the distribution of household incomes is often positively skewed, with most people earning moderate incomes and a few earning very high incomes.
• Asset returns: Returns on investments can sometimes exhibit positive skewness, indicating a higher chance of small gains and a smaller chance of very large gains.
• Response times: In psychological experiments, response times are often positively skewed, as most responses are quick, but some take much longer.

Negative Skewness (Left-Skewed Distribution)

A negative skewness value indicates that the tail on the left side of the distribution is longer or fatter than the right side. This means there are more extreme low values pulling the mean to the left.

• Left-tailed distribution: The "tail" of the histogram or probability density function extends further to the left.
• Mean < Median < Mode: In a negatively skewed distribution, the mode is typically the largest, followed by the median, and then the mean (which is pulled towards the longer left tail).
• Longer left tail: Visually, the distribution appears to have a stretched-out left side.
• Exam scores: If an exam is very easy, most students might score high, leading to a negatively skewed distribution of scores, with a few students scoring very low.
• Age distributions: The age of death in a developed country might be negatively skewed, with most people living to an old age and fewer dying very young.
• Quality measures: In manufacturing, if a product's quality is consistently high, the distribution of defects might be negatively skewed, with very few items having many defects.

Symmetric Distribution (Zero Skewness)

A skewness value close to zero suggests that the distribution is symmetrical, meaning both sides of the distribution are mirror images of each other. The data is evenly distributed around the center.

• Zero skewness: A value of 0 indicates perfect symmetry. In practice, values very close to zero are considered symmetrical.
• Mean = Median = Mode: For a perfectly symmetrical distribution, the mean, median, and mode all coincide at the center.
• Normal distribution: The classic example of a symmetrical distribution is the normal distribution, often called the "bell curve."
• Equal tails: Both the left and right tails of the distribution are of similar length and shape.
• Bell curve shape: The most common visual representation of a symmetrical distribution, where the data peaks in the middle and tapers off equally on both sides.

Financial Analysis and Risk Management

In finance, skewness is crucial for assessing risk and making investment decisions. Investors often prefer positively skewed returns (many small gains, few large losses) over negatively skewed returns (many small losses, few large gains).

• Return distributions: Analyzing the skewness of stock returns helps investors understand the likelihood of extreme gains or losses.
• Risk assessment: Negatively skewed returns indicate higher "downside risk" – a greater chance of significant losses.
• Portfolio analysis: Skewness helps in constructing diversified portfolios that balance risk and potential returns, considering the shape of return distributions.
• Option pricing: Skewness is a factor in advanced option pricing models, as it affects the probability of an option expiring in or out of the money.
• Market indicators: Changes in market skewness can sometimes signal shifts in investor sentiment or market stability.

Quality Control and Manufacturing

Skewness helps engineers and quality managers monitor production processes and identify potential issues that could lead to defects or inefficiencies.

• Process capability: Assessing the skewness of product measurements helps determine if a manufacturing process is consistently producing items within specifications.
• Defect analysis: Skewness in defect rates can indicate if problems are occurring more frequently at one end of a production parameter.
• Measurement systems: Evaluating the skewness of measurement errors helps ensure the accuracy and reliability of testing equipment.
• Control charts: Skewness can be used as an additional metric on control charts to detect non-random patterns in production.
• Specification limits: Understanding skewness helps in setting appropriate upper and lower specification limits for product quality.

Research Methods and Data Science

For researchers and data scientists, skewness is a key descriptive statistic that informs data preprocessing, model selection, and the interpretation of results.

• Data screening: Skewness is one of the first things to check when exploring a new dataset, as it reveals the underlying distribution shape.
• Assumption testing: Many statistical tests (e.g., t-tests, ANOVA, regression) assume that data is normally distributed (i.e., symmetrical). Skewness helps identify violations of this assumption.
• Distribution fitting: Knowing the skewness helps in choosing the appropriate probability distribution (e.g., normal, log-normal, exponential) to model the data.
• Outlier detection: Highly skewed distributions often contain outliers in their long tails, which need to be addressed.
• Transformation selection: If data is highly skewed, transformations (like logarithmic or square root) can be applied to make it more symmetrical, improving the performance of statistical models.