What is the difference between population and sample standard deviation?

The key difference is the denominator in the formula. For a population, you divide by the total number of data points (N). For a sample, you divide by 'n-1' (the sample size minus one). Using 'n-1' for a sample provides an unbiased estimate of the population variance.

How do you compute a z-score?

A z-score measures how many standard deviations a data point is from the mean. The formula is z = (x - μ) / σ, where 'x' is the data point, 'μ' is the population mean, and 'σ' is the population standard deviation.

What is the Pearson correlation coefficient (r)?

The Pearson correlation coefficient (r) is a measure of the linear relationship between two variables. It ranges from -1 to +1. A value of +1 indicates a perfect positive linear relationship, -1 indicates a perfect negative linear relationship, and 0 indicates no linear relationship.

How is the linear regression slope calculated?

The slope (β1) of a simple linear regression line is calculated as the covariance of the two variables (X and Y) divided by the variance of the independent variable (X). Formula: β1 = cov(X,Y) / var(X).

How do you compute percentiles?

This calculator uses the linear interpolation method, a common approach recommended by NIST. It involves sorting the data and calculating a rank. If the rank is not an integer, it interpolates between the two adjacent data points to find the precise value.

Can this tool handle large datasets?

Yes, for a client-side tool. It uses numerically stable algorithms like Welford's algorithm to calculate variance accurately with large inputs without catastrophic cancellation errors. However, for extremely large datasets (millions of points), browser performance may become a limiting factor.

Statistics Calculator — Mean, Median, Mode, Variance & More

Statistics Calculator

A comprehensive tool to compute descriptive statistics, correlation, regression, and more.

Enter Data (comma, space, or newline separated)

Data remains on this device. Uses numerically stable algorithms for accuracy.

Variable X Data

Variable Y Data

Enter the same number of data points for both X and Y.

Data Point (x)

Population Mean (μ)

Population Standard Deviation (σ)

Treat as a Sample (n-1)

Decimal Places

What Are Descriptive Statistics?

Descriptive statistics are summary statistics that quantitatively describe or summarize features from a collection of information. They provide simple summaries about the sample and the measures. Together with simple graphics analysis, they form the basis of virtually every quantitative analysis of data.

Measures of Central Tendency

Measures of central tendency describe the center of a dataset. They represent a typical or central value for a probability distribution.

Mean (Average): The sum of all values divided by the number of values. It's sensitive to outliers. The formula for the sample mean is &bar;x = Σx / n.
Median: The middle value in a dataset that has been sorted in ascending order. If the dataset has an even number of values, the median is the average of the two middle values. It is less affected by outliers than the mean.
Mode: The value that appears most frequently in a dataset. A dataset can have one mode (unimodal), two modes (bimodal), or more (multimodal).

Measures of Spread (Variability)

Measures of spread describe how similar or varied the set of observed values are for a particular variable.

Range: The difference between the highest and lowest values in a dataset.
Variance: The average of the squared differences from the Mean. A high variance indicates that the data points are very spread out from the mean. The sample variance formula is s² = Σ(x - &bar;x)² / (n-1), using n-1 to provide an unbiased estimate of the population variance.
Standard Deviation: The square root of the variance. It is a measure of the amount of variation or dispersion of a set of values. A low standard deviation indicates that the values tend to be close to the mean.
Interquartile Range (IQR): The range between the first quartile (25th percentile) and the third quartile (75th percentile). It is a robust measure of spread, as it is not influenced by outliers.

Correlation & Linear Regression

These tools are used to analyze the relationship between two variables.

Pearson Correlation (r): Measures the strength and direction of a linear relationship between two continuous variables. The value ranges from -1 to +1. A value close to 1 implies a strong positive linear relationship, close to -1 implies a strong negative linear relationship, and close to 0 implies a weak or no linear relationship.
Simple Linear Regression: A statistical method that allows us to summarize and study relationships between two continuous (quantitative) variables. It finds the best-fitting straight line (ŷ = β₀ + β₁x) through the data points. The R-squared (R²) value represents the proportion of the variance for a dependent variable that's explained by an independent variable.

Frequently Asked Questions

Here are answers to some common statistical questions.

What is the difference between sample and population formulas?

When you have data for an entire population, you use the population formulas (dividing variance by N). When you have a sample (a subset of a population), you use sample formulas (dividing variance by n-1). The 'n-1' correction, known as Bessel's correction, provides a more accurate (unbiased) estimate of the true population variance from the sample data.

Why is the median sometimes a better measure of center than the mean?

The mean is sensitive to extreme values, or outliers. A single very large or very small number can significantly skew the mean. The median, being the middle value, is not affected by outliers, making it a more robust measure of central tendency for skewed datasets (like income data).

Disclaimer: This tool provides educational statistical summaries and basic inference. For rigorous analysis or clinical/critical decisions, consult a qualified statistician.