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
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- Does Standard Deviation Change When You Add or Multiply by a Constant?Adding a constant leaves the standard deviation unchanged; multiplying by a scales it by |a| and the variance by a². Proof, examples and the general rule for aX + b.
- Can You Draw This Figure Without Lifting the Pen? (Euler Paths)A figure can be drawn in one stroke exactly when it is connected and has 0 or 2 points where an odd number of lines meet. How to check, and where to start.
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- How to Find Horizontal and Vertical Asymptotes Using LimitsHorizontal asymptotes come from limits as x → ±∞; vertical asymptotes from infinite one-sided limits at a point. Step-by-step method with worked examples.
- Given the Mean and Standard Deviation, How Do You Find a Probability?Convert values to z-scores and use the standard normal table to find P(X < b), P(X > a) and P(a < X < b) from a mean and standard deviation, with examples.
- How to Calculate the Phase Angle (Circuits, Phasors and Waveforms)Phase angle formulas for series RLC circuits (φ = arctan((X_L − X_C)/R)), complex phasors, and two sine waves from a time shift, with worked examples.
- How to Find Quartiles from a Histogram (Grouped Data)Find Q1, the median and Q3 from a histogram using cumulative frequencies and the grouped-data interpolation formula, with a fully worked example.
- How to Read the Potential Energy Graph of the Hydrogen MoleculeThe H₂ potential energy curve explained. The minimum marks the 74 pm bond length, its depth the bond energy, and repulsion dominates at short range.
- Insertion Sort Average Case – Comparisons, Shifts and Θ(n²)Derivation of insertion sort's average case on a random permutation, about n²/4 comparisons and shifts, with the exact formula (n² + 3n)/4 − Hₙ.
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