Free Skewness Calculator

Introduction to Skewness and Kurtosis Analysis

A skewness calculator and kurtosis calculator combined in one tool provides a comprehensive statistical analysis for understanding the shape of your data distribution. This free online statistical analysis tool enables you to quickly assess whether your dataset deviates from normality by measuring its asymmetry (skewness) and its tail heaviness (kurtosis). Beyond automated calculations, the tool also reveals the formulas used, allowing you to verify the computations manually if needed.

Before diving into skewness and kurtosis, it is essential to summarize your dataset using descriptive statistics (mean, median, standard deviation). Those summary measures give a first impression of central tendency and spread, which sets the stage for deeper distribution analysis.

What Do Skewness and Kurtosis Tell Us?

In probability theory and statistics, the normal distribution is symmetric and bell‑shaped, with no tilt toward left or right. Many natural phenomena approximately follow this pattern—such as test scores of a large student group or adult shoe sizes. However, real‑world data often deviates from this ideal.

Coefficient of Skewness (Asymmetry)

Skewness quantifies the degree of asymmetry in the distribution. A perfectly symmetric distribution has a coefficient of skewness equal to zero. If the left tail is longer than the right (indicating more extreme low values), the skewness is negative; if the right tail is longer, the skewness is positive. In other words, the sign of skewness tells you which direction the distribution “leans.”

Coefficient of Kurtosis (Tailedness)

Kurtosis focuses on the tails of the distribution rather than the center. It measures how much of the data is concentrated in the tails compared to a normal distribution. A positive kurtosis value indicates heavy tails and a sharper peak (more outlier‑prone), while a negative kurtosis suggests light tails and a flatter shape. Note that kurtosis ignores the central peak—it purely describes the tail behavior.

Formulas Used for Skewness and Kurtosis

The formulas employed in this skewness and kurtosis calculator match those used by Microsoft Excel, ensuring consistency with common statistical software.

Skewness Formula

$$\text{Skewness} = \frac{N \sum_{i=1}^{N} (x_i - \bar{x})^3}{(N-1)(N-2) \, s^3}$$

where:

• $x_i$ are the individual data points,
• $\bar{x}$ is the sample mean,
• $s$ is the sample standard deviation,
• $N$ is the number of observations.

The denominator requires at least three observations for a valid calculation.

Kurtosis (Excess Kurtosis) Formula

$$\text{Kurtosis} = \frac{N (N+1) \sum_{i=1}^{N} (x_i - \bar{x})^4}{(N-1)(N-2)(N-3) \, s^4} - \frac{3 (N-1)^2}{(N-2)(N-3)}$$

Again, $x_i$, $\bar{x}$, $s$, and $N$ represent the same quantities. This formula gives the excess kurtosis (relative to a normal distribution), so a normal distribution has a kurtosis of 0. The denominator demands at least four observations.

How to Interpret the Skewness Coefficient

After computing the skewness value, you can assess the shape of your distribution using these common guidelines:

• Skewness = 0: Perfect symmetry.
• Negative skewness: Left‑skewed (the left tail is longer, most data cluster on the right).
• Positive skewness: Right‑skewed (the right tail is longer, most data cluster on the left).
• $-0.5$ to $0.5$: Approximately symmetric.
• $-1$ to $-0.5$ or $0.5$ to $1$: Moderately skewed.
• Less than $-1$ or greater than $1$: Highly skewed.

How to Interpret the Kurtosis Coefficient

For excess kurtosis (the value produced by this calculator), use the following rules of thumb:

• Kurtosis = 0: The tail heaviness matches a normal distribution.
• Positive kurtosis: The distribution has heavier tails and a sharper peak than normal (leptokurtic).
• Negative kurtosis: The distribution has lighter tails and a flatter shape than normal (platykurtic).
• Kurtosis > 1: Excessively peaked (very heavy tails).
• Kurtosis < -1: Excessively flat (very light tails).

Practical Example: Height of Sixth‑Grade Boys

To illustrate the use of the data distribution analysis tool, consider the heights (in cm) of 18 boys in a sixth‑grade class:

$$140,\;146,\;146,\;148,\;152,\;153,\;154,\;156,\;156,\;160,\;162,\;162,\;163,\;165,\;166,\;166,\;168,\;172$$

With $N = 18$, the sample mean is $\bar{x} = 157.5$ and the sample standard deviation is $s \approx 8.833$. Plugging these values into the skewness and kurtosis formulas yields:

• Skewness $\approx -0.3212$
• Kurtosis $\approx -0.7195$

Interpretation: The negative skewness indicates a slight left‑skew, suggesting more shorter boys than taller ones in this group. However, because the skewness magnitude is small (between $-0.5$ and $0$), the distribution is nearly symmetric. The negative kurtosis tells us the distribution is flatter than normal—the heights are relatively uniform across the class, with no strong concentration around any particular value. The absolute kurtosis value exceeds $-1$, but it is within the moderate range, confirming a rather flat shape.

By using this statistical analysis calculator, you can perform similar evaluations on any dataset to understand its distributional characteristics at a glance.