Skew And Kurtosis Calculator

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 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:

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

Where:

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

Explanation: Skewness quantifies how symmetrical the distribution is, while Kurtosis quantifies 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 the direction and degree of asymmetry, while Kurtosis indicates the presence of outliers and the peakedness of the distribution.

4. Using the Calculator

Tips: Enter numerical data points separated by commas. The calculator requires at least 3 data points for meaningful results. The output provides both Skewness and Kurtosis values, which are dimensionless measures.

5. Frequently Asked Questions (FAQ)

Q1: What does positive/negative skewness indicate?
A: Positive skewness indicates a right-skewed distribution (tail extends to the right), while negative skewness indicates a left-skewed distribution (tail extends to the left).

Q2: What are typical values for skewness and kurtosis?
A: For a normal distribution, skewness is 0 and kurtosis is 3. Values significantly different from these indicate non-normal distributions.

Q3: What does high/low kurtosis mean?
A: High kurtosis (>3) indicates heavy tails and sharp peak (leptokurtic), while low kurtosis (<3) indicates light tails and flat peak (platykurtic).

Q4: How many data points are needed?
A: At least 3 data points are required for skewness calculation, but more data points provide more reliable estimates of these statistics.

Q5: Are there different types of kurtosis?
A: Yes, this calculator provides excess kurtosis (kurtosis - 3), where 0 represents a normal distribution. Some software may report different kurtosis definitions.