How to Calculate Skewness and Kurtosis

Skewness and Kurtosis Formulas:

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

Skewness:

Kurtosis:

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1. What Are 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:

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

Where:

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

Explanation: Skewness measures the degree of asymmetry in the distribution, while Kurtosis measures whether the data are heavy-tailed or light-tailed relative to a normal distribution.

3. Importance of Skewness and Kurtosis

Details: These measures help identify departures from normality in statistical data. Skewness indicates if data are symmetric (skewness ≈ 0), right-skewed (positive), or left-skewed (negative). Kurtosis indicates if data have more extreme values (leptokurtic, kurtosis > 3) or fewer extreme values (platykurtic, kurtosis < 3) than a normal distribution.

4. Using the Calculator

Tips: Enter numerical data values separated by commas. The calculator will compute the mean, standard deviation, skewness, and kurtosis automatically. Ensure all values are valid numbers.

5. Frequently Asked Questions (FAQ)

Q1: What does positive skewness indicate?
A: Positive skewness indicates the distribution has a longer right tail, meaning most data are concentrated on the left with some extreme high values.

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

Q3: When are skewness and kurtosis useful?
A: They are essential in data analysis, quality control, risk management, and when checking assumptions for statistical tests that require normality.

Q4: What are acceptable ranges for skewness and kurtosis?
A: For normality, skewness should be between -2 and +2, and kurtosis between -7 and +7, though these are general guidelines.

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