How do I interpret skewness values?

Skewness measures asymmetry. A skewness near 0 indicates a symmetric distribution. Positive skewness means the right tail is longer (distribution skewed right), while negative skewness means the left tail is longer (distribution skewed left).

What does excess kurtosis tell me about my data?

Excess kurtosis measures tail heaviness relative to a normal distribution. Positive excess kurtosis (leptokurtic) indicates heavier tails and more outliers. Negative excess kurtosis (platykurtic) indicates lighter tails. A value of 0 matches a normal distribution.

How are percentiles calculated in this tool?

This calculator uses linear interpolation with the formula h = (n+1) * p, where n is the sample size and p is the percentile (e.g., 0.25 for Q1). When h falls between data points, the percentile is linearly interpolated. For small samples, results are clamped to valid data bounds.

Descriptive Statistics Calculator: Mean, Variance, Standard Deviation

Enter Data

Dataset

Accepts comma, space, tab, or newline-separated values

Statistics Type

Sample (n-1) Population (n)

Input Examples

2, 4, 4, 4, 5, 5, 7, 9
10 20 30 40 50
Values on separate lines
Scientific notation: 1.5e-3

Results

Summary Statistics

Count (n)

Sum

Range

Minimum

Maximum

Central Tendency

Mean
-

Median
-

Mode

Dispersion Sample

Variance
-

Std. Deviation
-

MAD

CV (%)

Shape (Sample Adjusted)

Skewness

Excess Kurtosis

Quartiles

Q1 (25%)

Q2 (50%)

Q3 (75%)

IQR

Model Assumptions

Assumes observations are independent
Sample formulas use Bessel's correction (n-1 denominator)
Skewness and kurtosis use Fisher's adjusted sample formulas
Percentiles use linear interpolation: h = (n+1) * p
CV is most meaningful for positive ratio-scale data

This calculator is for educational purposes only. It uses simplified assumptions and is not investment advice. Results do not constitute a recommendation. Consult a qualified professional for personalized financial analysis.

Understanding Descriptive Statistics

Central Tendency Measures

Mean: x̄ = (1/n) ∑ x_i

Median: Middle value when data is sorted

Mode: Most frequently occurring value

The mean is sensitive to outliers, while the median is robust. For skewed distributions, the median often better represents the typical value.

Dispersion Measures

Sample Variance: s² = ∑(x_i - x̄)² / (n-1)

Population Variance: σ² = ∑(x_i - μ)² / n

Standard Deviation: √Variance

Bessel's correction (dividing by n-1) provides an unbiased estimate when working with samples.

Skewness

Measures asymmetry in the distribution:

Positive skewness: Right tail longer, mean > median
Negative skewness: Left tail longer, mean < median
Zero skewness: Symmetric distribution

Kurtosis

Measures tail heaviness relative to normal distribution:

Leptokurtic (>0): Heavier tails, more outliers
Platykurtic (<0): Lighter tails, fewer outliers
Mesokurtic (=0): Similar to normal distribution

Frequently Asked Questions

Population statistics describe the entire group you are interested in, while sample statistics describe a subset. Sample variance uses n-1 in the denominator (Bessel's correction) to provide an unbiased estimate, while population variance uses n. Use sample statistics when your data represents a subset of a larger population you want to make inferences about.

Dividing by n-1 instead of n corrects for the bias that occurs when estimating a population parameter from a sample. This is called Bessel's correction. When calculating variance from a sample, you use the sample mean as an estimate of the population mean. Since the sample mean is computed from the same data, the deviations around it are systematically smaller than they would be around the true population mean, leading to underestimation without the correction.

Skewness measures asymmetry in your data distribution. A skewness near 0 indicates a symmetric distribution. Positive skewness (right-skewed) means the right tail is longer, with more extreme values above the mean. Negative skewness (left-skewed) means the left tail is longer. In finance, asset returns often exhibit negative skewness, meaning large losses occur more frequently than large gains of the same magnitude.

Excess kurtosis measures the heaviness of tails relative to a normal distribution (which has excess kurtosis of 0). Positive excess kurtosis (leptokurtic) indicates heavier tails, meaning more probability of extreme values and outliers. Negative excess kurtosis (platykurtic) indicates lighter tails with fewer extremes. Financial returns often exhibit positive excess kurtosis, meaning extreme market moves happen more often than a normal distribution would predict.

This calculator uses linear interpolation with the formula h = (n+1) * p, where n is the sample size and p is the percentile as a decimal (0.25 for Q1, 0.75 for Q3). When h falls between data points, the percentile is linearly interpolated between adjacent values. For very small samples, results are clamped to valid data bounds. This method is consistent with many statistical software packages and provides smooth percentile estimates.

The coefficient of variation (CV) is the ratio of standard deviation to mean, expressed as a percentage. It measures relative variability, allowing comparison between datasets with different units or scales. For example, you could compare the relative volatility of stock prices in different currencies. CV is most meaningful for positive ratio-scale data where zero represents complete absence of the quantity. It becomes undefined or misleading when the mean is zero or near zero.

Descriptive Statistics Calculator

Enter Data

Input Examples

Results

Summary Statistics

Central Tendency

Dispersion Sample

Shape (Sample Adjusted)

Quartiles

Model Assumptions

Understanding Descriptive Statistics

Central Tendency Measures

Dispersion Measures

Skewness

Kurtosis

Frequently Asked Questions

What is the difference between sample and population statistics?

Why does sample variance use n-1 instead of n?

How do I interpret skewness values?

What does excess kurtosis tell me about my data?

How are percentiles calculated in this tool?

What is the coefficient of variation used for?

Disclaimer

Related Calculators

Confidence Interval Calculator

Normal Distribution Probability Calculator

Hypothesis Testing Calculator

MarketXLS Excel Add-in

Contact Me