How To Calculate Skewness And Kurtosis Of Grouped Data

Grouped Data Skewness and Kurtosis Formulas:

\[ Skewness = \frac{\sum f (x - \mu)^3 / N}{\sigma^3} \] \[ Kurtosis = \frac{\sum f (x - \mu)^4 / N}{\sigma^4} \]

Skewness:

Kurtosis:

Mean (μ):

Standard Deviation (σ):

Unit Converter ▲

Unit Converter ▼

From:	To:

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 compared to a normal distribution.

2. How Does The Calculator Work?

The calculator uses the following formulas for grouped data:

\[ Skewness = \frac{\sum f (x - \mu)^3 / N}{\sigma^3} \] \[ Kurtosis = \frac{\sum f (x - \mu)^4 / N}{\sigma^4} \]

Where:

\( f \) — Frequency of each class
\( x \) — Midpoint of each class
\( \mu \) — Mean of the distribution
\( N \) — Total frequency (\( \sum f \))
\( \sigma \) — Standard deviation of the distribution

Explanation: These formulas calculate the third and fourth standardized moments about the mean, providing measures of distribution shape that are independent of scale.

3. Importance Of Skewness And Kurtosis

Details: Skewness helps identify if data is symmetric (skewness ≈ 0), right-skewed (positive), or left-skewed (negative). Kurtosis indicates whether data has heavy tails (leptokurtic, kurtosis > 3), light tails (platykurtic, kurtosis < 3), or matches normal distribution (mesokurtic, kurtosis ≈ 3).

4. Using The Calculator

Tips: Enter frequencies and midpoints as comma-separated values. Ensure both lists have the same number of elements. The calculator will compute mean, standard deviation, skewness, and kurtosis automatically.

5. Frequently Asked Questions (FAQ)

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

Q2: What is the difference between population and sample kurtosis?
A: Population kurtosis uses N in denominator, while sample kurtosis uses N-1 for unbiased estimation. This calculator uses population formulas.

Q3: What are typical values for skewness and kurtosis?
A: For normal distribution: skewness ≈ 0, kurtosis ≈ 3. Values beyond ±2 for skewness or beyond 2-8 for kurtosis may indicate non-normal distribution.

Q4: When should I use grouped data calculations?
A: Use grouped data calculations when working with frequency distributions or when individual data points are not available, only class intervals and frequencies.

Q5: How do I interpret kurtosis values?
A: Kurtosis > 3 indicates heavier tails than normal (leptokurtic), < 3 indicates lighter tails (platykurtic), and ≈ 3 indicates normal tail behavior (mesokurtic).