The average of the squared distances of the values from their mean. It is written σ², and its square root is the standard deviation.
Precisely: σ² = Σ(xᵢ − x̄)² ÷ n, which also equals Σxᵢ²/n − x̄² (the mean of the squares minus the square of the mean).
How it works
Squaring the distances makes them all positive and weights big distances more. The average of the squares is the variance; because the units are squared (cm²), the standard deviation takes the root to get back to cm.
Examples
The variance of 6, 8, 10, …, 24 is 33.
The variance of 2, 4, 6 is 8/3.
Adding the same number to every value does not change the variance.
A variance of 16 means a standard deviation of 4.
Do not confuse: The variance is in squared units; the standard deviation, its root, is in the data's own units.
Used in
the standard deviation, comparing spread, probability distributions
A picture of sets as circles inside a rectangle (the universal set). Where the circles overlap is what the sets share.
Precisely: A diagram in which the universal set is a rectangle and each set is a closed region inside it, so that unions, intersections and complements appear as shaded areas.
How it works
Draw a circle for each set. Fill in the overlap first, then the parts of each circle outside the overlap, then what lies outside all the circles. The counts then add up correctly.
Examples
A Venn diagram of cricket and football players shows the students who play both in the overlap.
In a Venn diagram, A ∪ B is everything inside the two circles.
Filling a Venn diagram from the middle outwards avoids double counting.
The region outside both circles of a Venn diagram is the people in neither group.
Do not confuse: The sizes of the circles do not have to match the sizes of the sets; the counts written inside do the work.
Used in
survey problems, union and intersection, probability
The sequence 1, 2, 3, 5, 8, 13, 21, ... in which each term is the sum of the two before it.
Precisely: The sequence V₁ = 1, V₂ = 2, Vₙ = Vₙ₋₁ + Vₙ₋₂ for n ≥ 3.
How it works
Its rule looks back two steps, not one, so you need two starting terms. Virahānka found it in the 7th century CE while counting the rhythms of Prakrit poetry; Gopāla (c. 1135) and Hemachandra (c. 1150) studied it, and Fibonacci (c. 1200) later.
In the Virahānka-Fibonacci sequence, V₅ = V₄ + V₃ = 5 + 3 = 8.
The number of ways to fill a rhythm of n beats with short (1 beat) and long (2 beats) syllables follows the Virahānka-Fibonacci sequence: 1, 2, 3, 5, ... for n = 1, 2, 3, 4.
Do not confuse: It is neither an AP nor a GP: the differences and the ratios both change. Many books start it 1, 1, 2, 3, 5; the chapter starts it 1, 2, 3, 5.
Used in
patterns in nature (sunflower seeds, pine cones), poetry and music, counting problems, computer science
The amount of space a solid takes up, counted in unit cubes.
Precisely: The measure of the space inside a solid, in cubic units such as cm³ or m³; a 1 by 1 by 1 cube has volume 1 unit³.
How it works
For a cuboid, count the cubes in one layer (length × breadth) and multiply by the number of layers (height). A cylinder is the same idea with a circular base: base area × height. Cones and spheres come from these by clever comparisons.
Examples
A cuboid 3 cm by 4 cm by 12 cm has a volume of 144 cm³.
A cylinder of radius 7 cm and height 10 cm has volume 22/7 × 7² × 10 = 1540 cm³.
A toy made of a hemisphere of radius 2 cm and a cone 2 cm high has volume 8π ≈ 25.12 cm³ (π ≈ 3.14), exactly half of the cylinder that fits round it.
Do not confuse: Volume is space inside (cm³); surface area is the outside covering (cm²). 1 litre is 1000 cm³.
Used in
capacity of tanks and glasses, melting and recasting solids, density in science