Science and math questions, solved step by step.
Worked solutions in mathematics, statistics, physics, chemistry, biology, computer science and LaTeX, each checked before publication, plus free calculators that explain every step.
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All topics →Mathematics
Algebra, calculus, analysis, topology, linear algebra and number theory, worked out step by step.
Statistics & Probability
Distributions, estimators, expected values and hypothesis tests with clear derivations.
Physics
Mechanics, electromagnetism, thermodynamics, relativity and quantum physics explained.
Chemistry
Reactions, bonding, equilibrium, thermochemistry and lab calculations.
Biology
Evolution, biochemistry, physiology and ecology, answered with evidence.
Computer Science
Algorithms, complexity, automata, number representation and digital logic.
LaTeX & TikZ
Working solutions for tables, figures, bibliographies, TikZ and pgfplots.
Academia
Publishing, peer review, PhD life and research careers.
Calculators
All calculators →Quartiles
Find Q1, median, Q3 and IQR from a list of values or from histogram bins (grouped data), with every step shown.
Standard deviation
Sample and population standard deviation, variance and mean, with the full sum-of-squares working.
Normal probability
Given the mean and standard deviation, find P(X < b), P(X > a) or P(a < X < b) with z-scores.
Dice probability
Exact distribution, mean and standard deviation for the sum of any number of fair dice.
Matrix
Determinant with cofactor expansion, inverse, symmetric eigen-decomposition and singular values.
Boolean simplifier
Turn minterms or a truth table into a minimal sum-of-products expression using Quine–McCluskey.
Recently updated
- Can the Mass of a Body Change?A body's mass can change when matter or energy enters or leaves it. Learn how classical mass, rest mass, and mass–energy equivalence differ.
- Can the Inner Surface Charge Density Be Negative?Yes. A positive charge inside a spherical conducting cavity induces negative charge on the inner wall; mixed-sign interior charges can make the density change sign.
- Can Composition Increase a Power Series’ Radius of Convergence?Yes. Composition can cancel singularities of the inner function, so the resulting Taylor series may have a larger radius of convergence.
- Can You Change the Capacitance of a Capacitor?You can change a capacitor’s capacitance by changing its plate geometry or dielectric; charge and voltage then respond according to its circuit.
- Can You See a Rainbow on the Moon or Other Worlds?A lunar surface has no ordinary rainbow because it lacks rain and a substantial atmosphere. Learn what a rainbow needs and where one might occur elsewhere.
- Can Conservative Forces Change a System’s Mechanical Energy?A conservative force can change kinetic energy, but its work is offset by an equal change in potential energy when no nonconservative work acts.
- Can You Cite a Workshop or Conference in a Paper?Yes. Cite conference proceedings as publications and cite talks or posters as presentations, using the format required by your journal or citation style.
- Can Electrolysis of Aqueous NaCl Produce Sodium Metal?Aqueous sodium chloride electrolysis produces hydrogen and sodium hydroxide, not sodium metal: water is reduced first, and sodium reacts with water.
- Can a Convergent Sequence Be Unbounded?No: every convergent real sequence is bounded. Bound its tail using the limit, then take a maximum over the finitely many earlier terms.
- How to Add a Short Figure Caption to LyX’s List of FiguresUse LyX’s short-title setting for a caption to control the List of Figures while keeping the full caption beneath the figure.
- How Do You Find the Cardinality of a Set Containing Sets?To find the cardinality of a set containing sets, count its distinct outer elements; do not add the numbers of elements inside those sets.
- What Is the Cartesian Product with the Empty Set?The Cartesian product of any set with the empty set is empty: neither A × ∅ nor ∅ × A contains an ordered pair.