The Pearson correlation coefficient (r) measures the strength and direction of the linear relationship between two continuous 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. The formula is r = Σ((xᵢ - x̄)(yᵢ - ȳ)) / √(Σ(xᵢ - x̄)² · Σ(yᵢ - ȳ)²), where x̄ and ȳ are the sample means. The coefficient of determination (r²) represents the proportion of variance in one variable explained by the other. The t-statistic tests whether the correlation is statistically significant. This calculator is widely used in statistics, data science, psychology, economics, and many other fields to quantify relationships between variables.

The Spearman rank correlation coefficient (ρ, often denoted as rₛ) is a non-parametric measure of the strength and direction of the monotonic relationship between two variables. Unlike Pearson's correlation, Spearman's ρ does not assume a linear relationship; it measures how well the relationship can be described by a monotonic function. The calculation involves converting the original data to ranks, then computing the Pearson correlation on the ranked data. The formula is ρ = 1 - (6·Σd²) / (n·(n²-1)), where d is the difference between the ranks of each pair, and n is the number of pairs. Values range from -1 to +1, with interpretation similar to Pearson's r. This calculator is widely used when data is ordinal, non-normally distributed, or when the relationship is not linear.

The binomial distribution is a discrete probability distribution that models the number of successes in a fixed number of independent trials, each with the same probability of success. It is widely used in statistics, quality control, medicine, and social sciences. The theoretical formula for the probability of exactly k successes in n trials is: P(X = k) = C(n,k) · p^k · (1-p)^(n-k), where C(n,k) is the binomial coefficient (number of ways to choose k successes from n trials), p is the probability of success on a single trial, and (1-p) is the probability of failure. This calculator computes the exact probability for a given k, as well as cumulative probabilities (P(X ≤ k), P(X < k), P(X ≥ k)). It also provides the mean (μ = n·p), variance (σ² = n·p·(1-p)), and standard deviation. The binomial distribution is essential for hypothesis testing, A/B testing, and predicting outcomes in scenarios with binary outcomes.

A frequency distribution table organizes data values by showing how often each value occurs. This tool calculates the frequency, relative frequency, and percentage for each unique value in your dataset. It also provides key statistics: total count, mean, median, mode, min, max, and range. Frequency analysis is fundamental in statistics for understanding data distribution, identifying patterns, and making informed decisions. This calculator is useful for students, researchers, analysts, and anyone working with data. The frequency table helps visualize the spread and central tendency of the data, making it easier to interpret and communicate findings.

Avogadro's number (Nₐ = 6.02214076 × 10²³ mol⁻¹) is the number of particles (atoms, molecules, ions, or electrons) in one mole of a substance. It serves as the fundamental conversion factor between the macroscopic scale (grams) and the microscopic scale (individual particles). The relationship is: particles = moles × Nₐ, moles = mass / molar mass, and mass = moles × molar mass. This constant is named after Amedeo Avogadro, who first proposed that equal volumes of gases contain the same number of molecules. The current exact value was established by CODATA and is fixed in the SI system since 2019. This calculator helps chemists, students, and researchers quickly convert between mass, moles, and particle count for any substance, using its molar mass. It is essential for stoichiometry, solution preparation, and understanding atomic-scale quantities.

The Sortino ratio is a performance metric that measures the risk-adjusted return of an investment, focusing only on downside risk (negative returns). Unlike the Sharpe ratio, which penalizes both upside and downside volatility, the Sortino ratio distinguishes harmful volatility from beneficial volatility. It is calculated as: Sortino Ratio = (R_p - R_f) / σ_d, where R_p is the average periodic return, R_f is the risk-free rate, and σ_d is the downside deviation (standard deviation of negative returns below a target threshold). The target return is typically the minimum acceptable return, often set to 0% or the risk-free rate. A higher Sortino ratio indicates better risk-adjusted performance relative to downside risk. This metric is particularly valuable for evaluating investments with asymmetrical risk profiles, such as hedge funds, private equity, and options strategies, where investors are more concerned about downside losses than upside volatility.

The lognormal distribution is a continuous probability distribution of a random variable whose logarithm is normally distributed. If Y = ln(X) follows a normal distribution with mean μ and standard deviation σ, then X follows a lognormal distribution. This distribution is characterized by positive skewness and is bounded below by zero, making it suitable for modeling quantities that cannot be negative, such as stock prices, income distributions, asset prices, failure times, and environmental concentrations. The probability density function (PDF) is f(x) = (1/(x·σ·√(2π))) · exp(-(ln(x)-μ)²/(2σ²)). The cumulative distribution function (CDF) is derived from the standard normal CDF: F(x) = Φ((ln(x)-μ)/σ). The lognormal distribution has important properties: the mean is exp(μ + σ²/2), the median is exp(μ), and the mode is exp(μ - σ²). This calculator computes CDF, PDF, mean, median, mode, variance, and standard deviation for given parameters.

The median is the middle value in a dataset when the numbers are arranged in ascending or descending order. Unlike the mean (average), the median is not affected by extremely large or small values, making it a robust measure of central tendency. To find the median, first sort all numbers from lowest to highest. If there is an odd count of numbers, the median is the middle number. If there is an even count, the median is the average of the two middle numbers. For example, in the set {1, 3, 5}, the median is 3. In the set {1, 3, 5, 7}, the median is (3+5)/2 = 4. The median is widely used in statistics, economics, and data analysis, especially when dealing with income distributions, house prices, or test scores where outliers could skew the average.

Student's t-distribution is a continuous probability distribution used when estimating the mean of a normally distributed population with a small sample size and unknown standard deviation. It is the foundation of many statistical tests, including the t-test. The theoretical calculation involves the probability density function (PDF) and cumulative distribution function (CDF). The CDF gives the probability that a random variable following the t-distribution is less than or equal to a given t-value. The distribution is symmetric and bell-shaped, similar to the normal distribution but with heavier tails. As degrees of freedom increase, the t-distribution approaches the standard normal distribution. This calculator computes one-tailed and two-tailed p-values from a t-value, and also finds critical t-values for a given significance level (alpha=0.05). It is widely used in hypothesis testing, confidence intervals, and regression analysis.