Skewness And Kurtosis Formula Calculator

Skewness and Kurtosis Formulas:

\[ \text{Skewness} = \frac{\sum(x_i - \mu)^3 / n}{\sigma^3} \] \[ \text{Kurtosis} = \frac{\sum(x_i - \mu)^4 / n}{\sigma^4} \]

Skewness:

Kurtosis:

Unit Converter ▲

Unit Converter ▼

From:	To:

1. What is Skewness and Kurtosis?

Skewness and Kurtosis are statistical measures that describe the shape of a probability distribution. Skewness measures the asymmetry of the distribution, while Kurtosis measures the "tailedness" or peakiness of the distribution.

2. How Does the Calculator Work?

The calculator uses the following formulas:

\[ \text{Skewness} = \frac{\sum(x_i - \mu)^3 / n}{\sigma^3} \] \[ \text{Kurtosis} = \frac{\sum(x_i - \mu)^4 / n}{\sigma^4} \]

Where:

\( x_i \) — Individual data points
\( \mu \) — Mean of the data
\( \sigma \) — Standard deviation of the data
\( n \) — Number of data points

Explanation: Skewness uses the third moment about the mean, while Kurtosis uses the fourth moment about the mean, both normalized by the standard deviation.

3. Importance of Skewness and Kurtosis

Details: These measures help identify deviations from normal distribution. Skewness indicates whether data is symmetric (skewness ≈ 0), right-skewed (positive), or left-skewed (negative). Kurtosis indicates whether data has heavy tails (leptokurtic, kurtosis > 3) or light tails (platykurtic, kurtosis < 3) compared to normal distribution.

4. Using the Calculator

Tips: Enter data points as comma-separated values. The calculator will compute the mean, standard deviation, skewness, and kurtosis automatically. Ensure you have at least 3 data points for meaningful results.

5. Frequently Asked Questions (FAQ)

Q1: What does positive skewness indicate?
A: Positive skewness indicates the distribution has a longer right tail, with most data concentrated on the left side.

Q2: What is the kurtosis of a normal distribution?
A: A normal distribution has kurtosis of 3. Excess kurtosis (kurtosis - 3) is often reported, where 0 indicates normal tail behavior.

Q3: When are skewness and kurtosis most useful?
A: They are particularly useful in finance for risk assessment, in quality control for process monitoring, and in research for checking normality assumptions.

Q4: What are acceptable ranges for skewness and kurtosis?
A: For normal distribution approximation, skewness between -2 and +2 and kurtosis between -7 and +7 are often considered acceptable.

Q5: Can these measures be used for small sample sizes?
A: While calculable, skewness and kurtosis estimates from small samples (n < 20) may be unreliable due to high sampling variability.