How to Add and Subtract Fractions With Different Denominators
You cannot add 1/2 and 1/3 by adding the tops and bottoms. This guide shows the common-denominator method, with worked examples and mixed numbers.
Math & Statistics
Percentages, fractions, averages and probability, with the reasoning behind the formulas.
Common Probability Mistakes and How to Avoid Them
Probability is full of results that feel wrong. Here are five classic mistakes, each with a short calculation showing the correct answer.
Converting Between Fractions, Decimals and Percentages
Fractions, decimals and percentages are three ways of writing the same quantity. Learn the six conversions, repeating decimals and the equivalents worth memorising.
Correlation Coefficient: How to Calculate and Interpret r
Pearson's r measures how closely two variables follow a straight line. Here is the hand calculation, how to read the result, and where the number can fool you.
How to Find a Percentage of Any Number by Hand
Percent just means per hundred. Learn the decimal method, the 10% building-block method and the swap trick, then check each answer in seconds.
How to Find the Original Price Before a Discount or Tax
If a coat costs 60 after 25% off, the original price is not 75. Here is the one-step division that undoes a percentage change correctly.
How Big a Sample Do You Need? Sample Size and Margin of Error
Sample size is a trade-off between precision and cost. Here is the survey formula, a worked example, and why bigger is not always better.
How Ratios Work: Simplifying, Comparing and Sharing
A ratio compares two or more quantities. Learn to simplify one, tell part-to-part from part-to-whole, and share a total in a given ratio.
How to Calculate Standard Deviation by Hand, Step by Step
Standard deviation tells you how spread out a set of numbers is. This walkthrough does it by hand on eight values and explains when to divide by n or n minus 1.
How to Read a Confidence Interval (and What 95% Means)
A confidence interval is a range built from sample data. Learn how to calculate one for a mean and a proportion, and how to describe it without overclaiming.
Odds vs Probability: How to Convert Between Them
"1 in 4" and "odds of 1 to 4" are not the same thing. Learn the two conversion formulas and how odds ratios differ from risk ratios.
P-Values Explained: What 0.05 Does and Does Not Prove
A p-value is often read as the chance a result is a fluke, which it is not. Here is what it measures, with a worked coin-flip example and the mistakes to avoid.
Percent Change vs Percentage Points: Why the Difference Matters
Two phrases that sound alike describe very different changes. Here is how to tell them apart, and why +25% then -20% brings you back to the start.
Percent Error Formula: How to Calculate It With Examples
Percent error tells you how far a measurement is from the accepted value, relative to that value. Here is the formula, two worked examples and the usual traps.
Permutations vs Combinations: When Does Order Matter?
Choosing 3 runners for a podium and choosing 3 people for a committee give very different counts. Learn the one question that tells you which formula to use.
Rounding Numbers: Rules, Ties and How Errors Build Up
Rounding looks trivial until a tie, a carry or a long calculation gets involved. Learn the rules, the half-up and half-even methods, and when to round.
Scale Factors: How Lengths, Areas and Volumes Change
A scale factor multiplies every length by the same number, but areas grow by its square and volumes by its cube. Worked examples for maps, models and pans.
How to Write Numbers in Scientific Notation, Step by Step
Scientific notation writes any number as a value between 1 and 10 times a power of ten. Here is how to convert, calculate and read calculator output.
Solving Proportions: Cross-Multiplication, Direct and Inverse
A proportion says two ratios are equal. Learn to solve for the missing number, spot when a relationship is not proportional, and handle inverse cases.
Why the Average Can Mislead: Outliers, Skew and Weights
An average is a summary, and summaries hide things. Here are the three situations where the usual mean gives a misleading answer, with numbers you can check.
Z-Scores Explained: Comparing Values on Different Scales
A z-score re-expresses any value as a number of standard deviations from the mean, which lets you compare results that use different scales.