The ordinary average: add all the values and divide by how many there are.
Precisely: For values x₁, …, xₙ the mean is x̄ = (x₁ + … + xₙ)/n; for grouped data with class marks xᵢ and frequencies fᵢ, x̄ = Σfᵢxᵢ ÷ Σfᵢ.
How it works
Think of sharing the total out equally: the mean is what each would get. For grouped data each class is represented by its class mark, so the answer is an estimate. The assumed-mean and step-deviation methods give the same answer with smaller numbers.
Examples
The mean of 4, 8, 6, 5, 3 and 8 is 34 ÷ 6 ≈ 5.67.
For 30 students' marks in six classes from 10 to 100, Σfᵢxᵢ = 1860, so the mean is 1860 ÷ 30 = 62.
With assumed mean a = 47.5, the deviations give the same mean: 47.5 + 14.5 = 62.
Do not confuse: The mean is pulled by very large or very small values; the median is not. The mean need not be one of the values (5.67 above).
Used in
averages of marks, heights and runs, grouped data, probability (expected value)
OpenStax: Introductory Statistics 2e, 2.5 Measures of the Center of the Data
mean deviation
meen dee-vee-AY-shunClass 11
The average distance of the values from a centre (usually the mean): add the distances, ignoring signs, and divide by how many values there are.
Precisely: M.D.(a) = Σ|xᵢ − a| ÷ n about a central value a (the mean or the median); for a frequency table, Σfᵢ|xᵢ − a| ÷ Σfᵢ.
How it works
The signed distances from the mean always add to 0, so the signs are dropped (the absolute value) before averaging. The result says how far a typical value is from the centre.
Examples
The mean deviation of 6, 7, 10, 12, 13, 4, 8, 12 about the mean 9 is 2.75.
Values 3, 5, 7, 9, 11 have mean deviation 2.4 about their mean 7.
If every value equals the mean, the mean deviation is 0.
The mean deviation is in the same units as the data.
Do not confuse: The mean deviation averages the distances themselves; the variance averages their squares.
A line from a corner of a triangle to the middle of the opposite side.
Precisely: The line segment joining a vertex of a triangle to the midpoint of the opposite side; every triangle has three.
How it works
The two triangles on either side of a median have equal bases (the two halves of the side) and the same height, so they have equal areas, even though they usually have different shapes.
Examples
AD is a median of triangle ABC when D is the midpoint of BC.
A triangle of area 30 cm² is cut by a median into two triangles of 15 cm² each.
If P is any point on the median AD, triangles ABP and ACP have equal areas.
Do not confuse: In statistics the median is the middle value of ordered data, a different idea. In a triangle, a median goes to a midpoint; a height (altitude) meets the side at a right angle.
Used in
areas of triangles, the centroid (where the three medians meet), coordinate geometry
Precisely: For n ordered values, the ((n + 1)/2)th value if n is odd, or the mean of the (n/2)th and (n/2 + 1)th values if n is even; for grouped data, median = l + ((n/2 − cf) ÷ f) × h, with l, f and h from the median class and cf the cumulative frequency before it.
How it works
Put the values in order and count to the middle. For grouped data, find the class where the cumulative frequency first passes n/2 (the median class), then go into it the right fraction of its width.
Examples
The median of 3, 4, 5, 6, 8, 8 is (5 + 6) ÷ 2 = 5.5.
For the 30 students' marks, n/2 = 15 falls in the class 55 to 70 (cumulative frequencies 2, 5, 12, 18, …), so the median is 55 + (15 − 12)/6 × 15 = 62.5.
In 1, 2, 3, 4, 100 the median is 3, while the mean is 22: one large value moves the mean but not the median.
Do not confuse: The median of data is a middle value; the median of a triangle is a line to the midpoint of a side. Always order the data before finding the middle.
Used in
typical incomes and house prices, grouped data, ogives
OpenStax: Introductory Statistics 2e, 2.5 Measures of the Center of the Data (median)
midpoint
MID-pointClass 9
The point exactly halfway between two points. Its coordinates are the averages of theirs.
Precisely: The point of a line segment that is the same distance from both ends: for (x₁, y₁) and (x₂, y₂) it is ((x₁ + x₂)/2, (y₁ + y₂)/2).
How it works
Average the two x-coordinates, then average the two y-coordinates. Halfway across and halfway up together give the point halfway along.
Examples
The midpoint of (2, 4) and (8, 6) is (5, 5).
A bus stop built at the midpoint of the road between two towns is equally far from both.
In a parallelogram, the two diagonals share the same midpoint.
Do not confuse: The midpoint is a point; half the distance is a length. And the midpoint of a segment is not the midpoint of a curved path between the same ends.
Used in
the midpoint and section formulas, proving shapes are parallelograms, finding a missing end point
OpenStax: College Algebra 2e, 2.1 Key Terms (midpoint formula)
modal class
MOH-dul klahssClass 10
In a grouped table, the class with the largest frequency.
Precisely: The class interval of grouped data with the maximum frequency; the mode of the data is estimated inside it.
How it works
Look down the frequency column for the biggest number: its class is the modal class. Then the mode formula places the mode inside it, nearer to the neighbour with the larger frequency.
Examples
With frequencies 2, 3, 7, 6, 6, 6 for classes 10 to 25, 25 to 40, …, the modal class is 40 to 55.
If most families have 3 to 5 members, that class is the modal class of family sizes.
The mode 52 lies inside the modal class 40 to 55.
Do not confuse: The modal class is an interval (40 to 55); the mode is one value inside it (52).
Precisely: The value with the highest frequency; for grouped data, mode = l + ((f₁ − f₀) ÷ (2f₁ − f₀ − f₂)) × h, where l, f₁ and h belong to the modal class and f₀, f₂ are the frequencies of the classes before and after it.
How it works
Count how often each value occurs and pick the most frequent. Data can have one mode, several modes, or none (when every value appears once). For grouped data the formula leans towards the bigger neighbouring class.
Examples
In 4, 8, 6, 5, 3, 8 the mode is 8: it appears twice.
For the 30 students' marks the modal class is 40 to 55, and the mode is 40 + (7 − 3)/(14 − 3 − 6) × 15 = 52.
The most common shoe size sold in a shop is the mode of its sales.
Do not confuse: The mode is the value that occurs most, not how many times it occurs (the frequency).
Used in
the most popular size, choice or result, grouped data
OpenStax: Introductory Statistics 2e, 2.5 Measures of the Center of the Data (mode)
modulus (of a complex number)
MOD-yoo-lusClass 11
The size of a complex number a + ib: its distance from 0 in the Argand plane, √(a² + b²). It is written |z|.
Precisely: For z = a + ib, the non-negative real number |z| = √(a² + b²); it satisfies |z|² = z z̄ and |z₁z₂| = |z₁| |z₂|.
How it works
Draw a + ib as the point (a, b). The line from 0 to that point is the hypotenuse of a right triangle with sides a and b, so its length is √(a² + b²) by Pythagoras.
Examples
The modulus of 3 + 4i is 5.
The modulus of i is 1, and so is the modulus of −1.
The modulus of −5 is 5: for a real number it is the absolute value.
All the numbers with modulus 2 lie on the circle of radius 2 round 0.
Do not confuse: The modulus is never negative and has no i in it: |3 + 4i| is 5, not 3 + 4i or 25.
Used in
the size of a complex number, dividing complex numbers, the polar form in higher classes
The result of multiplying a number by a whole number: the multiples of 7 are 7, 14, 21, 28, …
Precisely: An integer m is a multiple of n when m = n × k for some integer k; equivalently, n is a factor of m.
How it works
Count up in steps of the number to list its multiples. A number has only a few factors but infinitely many multiples. A common multiple of two numbers is a multiple of both; the smallest is their LCM.
Examples
42 is a multiple of 7, because 42 = 7 × 6.
36 is a common multiple of 12 and 18, and the smallest one.
Every multiple of 10 is also a multiple of 2 and of 5.
Do not confuse: A multiple is at least as big as the number (7, 14, 21); a factor is at most as big (1 and 7). 12 is a multiple of 3, and 3 is a factor of 12.
Used in
times tables, LCM, divisibility rules, arithmetic progressions
OpenStax: Prealgebra 2e, 2.4 Find Multiples and Factors
mutually exclusive events
MYOO-choo-uh-lee ek-SKLOO-siv i-VENTSClass 11
Events that cannot happen together: if one happens, the other cannot. For them, P(A or B) = P(A) + P(B).
Precisely: Events A and B with A ∩ B = ∅, so that P(A ∩ B) = 0 and P(A ∪ B) = P(A) + P(B).
How it works
Mutually exclusive events share no outcomes, so nothing is counted twice when their probabilities are added, and the addition rule loses its overlap term.
Examples
Getting a 2 and getting a 5 on one roll of a die are mutually exclusive events.
Drawing a king and drawing a queen as the same card are mutually exclusive events.
Getting an even number and getting a 4 are not mutually exclusive events: a 4 is both.
An event and its complement are always mutually exclusive events.
Do not confuse: Mutually exclusive is not the same as independent: mutually exclusive events cannot happen together, while independent events do not affect each other's chances.