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21 results for “bar”

Formulae (21)

Average Value of a Function

\bar{f} = \frac{1}{b-a}\int_a^b f(x)\,dx

Mean value of a continuous function over an interval.

Statistics & ProbabilityBasic

Arithmetic Mean

\bar{x} = \frac{1}{n}\sum_{i=1}^{n} x_i

The average of a data set.

Statistics & ProbabilityIntermediate

Variance

\sigma^2 = \frac{1}{n}\sum_{i=1}^{n}(x_i - \bar{x})^2

Average squared deviation from the mean.

Statistics & ProbabilityIntermediate

Standard Deviation

\sigma = \sqrt{\frac{1}{n}\sum_{i=1}^{n}(x_i-\bar{x})^2}

Square root of the variance; typical spread.

Statistics & ProbabilityAdvanced

Pearson Correlation

r = \frac{\sum (x_i-\bar{x})(y_i-\bar{y})}{\sqrt{\sum(x_i-\bar{x})^2\sum(y_i-\bar{y})^2}}

Measures linear correlation between two variables.

Statistics & ProbabilityIntermediate

Coefficient of Variation

CV = \frac{\sigma}{\bar{x}}\times 100\%

Relative variability as a percentage of the mean.

Statistics & ProbabilityAdvanced

Covariance

\text{cov}(X,Y) = \frac{1}{n}\sum (x_i-\bar{x})(y_i-\bar{y})

Measures how two variables vary together.

Statistics & ProbabilityAdvanced

Confidence Interval (Mean)

\bar{x} \pm z\frac{\sigma}{\sqrt{n}}

Range likely to contain the population mean.

Statistics & ProbabilityAdvanced

Z-Test Statistic

z = \frac{\bar{x} - \mu_0}{\sigma/\sqrt{n}}

Tests whether a sample mean differs from a hypothesized value.

Statistics & ProbabilityIntermediate

Weighted Mean

\bar{x}_w = \frac{\sum w_i x_i}{\sum w_i}

Average where values carry different weights.

Statistics & ProbabilityIntermediate

Sample Variance

s^2 = \frac{1}{n-1}\sum(x_i-\bar{x})^2

Unbiased estimate of population variance.

Probability & CombinatoricsAdvanced

Stars and Bars

\binom{n+k-1}{k-1}

Ways to distribute n identical items into k distinct groups.

Data Science & MLIntermediate

Coefficient of Determination (R²)

R^2 = 1 - \frac{\sum(y_i-\hat{y}_i)^2}{\sum(y_i-\bar{y})^2}

Proportion of variance in the target explained by the model.

Physics Laws & TheoremsAdvanced

Heisenberg Uncertainty Principle

\Delta x \, \Delta p \ge \frac{\hbar}{2}

Position and momentum cannot both be known with unlimited precision.

Thermodynamics & HeatAdvanced

Average KE of a Gas Molecule

\bar{E} = \tfrac{3}{2}k_B T

Mean translational kinetic energy per molecule.

Modern & Quantum PhysicsAdvanced

Uncertainty Principle

\Delta x\,\Delta p \geq \frac{\hbar}{2}

Limit on knowing position and momentum together.

Biology & Life SciencesAdvanced

Hemocytometer Cell Count

C = \bar{n} \times DF \times 10^4

Cells per mL from a counting-chamber average.

Astronomy & SpaceAdvanced

Hawking Temperature

T = \frac{\hbar c^3}{8\pi G M k_B}

Temperature of the radiation a black hole emits.

Aviation & AeronauticsAdvanced

Barometric Altitude

h = \dfrac{RT}{g}\ln\dfrac{P_0}{P}

Altitude derived from the ratio of sea-level to measured pressure.

Geography & EarthAdvanced

Barometric Formula

P = P_0 e^{-\frac{Mgh}{RT}}

Atmospheric pressure as a function of altitude.

Unit ConversionsBasic

Bar to PSI

\text{psi} = \text{bar} \times 14.5038

Convert bar to pounds per square inch.