📊 Math Formulas
24 formulas in 2 sections. For each one you get the formula, what every symbol means, where it is used, an explanation in plain words and a worked example.
Sections: Statistics · Probability
Mean = Sum of observations ÷ Number of observations
| Sum | total of all values |
|---|---|
| Number | how many values |
Where it is used: To find a single representative value of data — average marks, income, temperature.
In simple words: Add everything and share it equally among all the values.
Example: Data 4,6,8,10 → Mean = 28/4 = 7
Median = the middle value when data is arranged in order
| — | for even count, average the two middle values |
|---|
Where it is used: To find the middle of a data set, unaffected by very large or small values — incomes, prices.
In simple words: Line the numbers up in order and pick the one in the very middle.
Example: 3, 5, 7, 9, 11 → median = 7
Mode = the value that occurs most often
| — | the most frequent observation |
|---|
Where it is used: To find the most common value in data — popular shoe size, best-selling item.
In simple words: The mode is simply the number that shows up the most times.
Example: 2, 3, 3, 5, 3, 8 → mode = 3
Range = Highest value − Lowest value
| Highest | largest data value |
|---|---|
| Lowest | smallest data value |
Where it is used: To measure how spread out the data is — temperature swings, score spreads.
In simple words: The gap between the biggest and smallest number tells you how spread out the data is.
Example: Data 4 to 20 → range = 20 − 4 = 16
Mean = Σ(f × x) ÷ Σf
| f | frequency of each value |
|---|---|
| x | the value (or class mark) |
| Σ | sum of |
Where it is used: To find the average when data is given in groups with frequencies — surveys, exam-score tables.
In simple words: Multiply each value by how many times it appears, add those up, and divide by the total count.
Example: x:10,20 with f:2,3 → (20+60)/5 = 16
Value = (Angle ÷ 360°) × Total
| Angle | the slice's angle |
|---|---|
| Total | grand total the pie represents |
Where it is used: To read a quantity from a pie chart slice, or to find the angle for a slice.
In simple words: A full pie is 360°. A slice's share of 360 is its share of the total.
Example: Slice 90°, total 200 → (90/360)×200 = 50
Mode = 3 × Median − 2 × Mean
| Mean | the average |
|---|---|
| Median | the middle value |
| Mode | the most frequent value |
Where it is used: To estimate one average when the other two are known, for moderately skewed data.
In simple words: A handy link between the three 'middle' measures when you know two of them.
Example: Mean=20, Median=22 → Mode = 66 − 40 = 26
Class mark = (Lower limit + Upper limit) ÷ 2
| Lower, Upper | the class interval limits |
|---|
Where it is used: To get a single representative value for a class interval before finding a grouped mean.
In simple words: The middle of a group interval is just the average of its two ends.
Example: Interval 10–20 → class mark = 15
σ² = Σ(x − mean)² ÷ n
| x | each value |
|---|---|
| mean | the average |
| n | number of values |
Where it is used: To measure how spread out data is around the mean — used in statistics and science.
In simple words: The average of the squared distances from the mean; bigger means more spread out.
Example: Data 2,4,6, mean 4 → (4+0+4)/3 ≈ 2.67
σ = √Variance
| σ | standard deviation |
|---|
Where it is used: The most common measure of spread, in the same units as the data.
In simple words: The square root of the variance — the typical distance of values from the average.
Example: Variance 4 → SD = 2
σ = √( Σx²/n − (Σx/n)² )
| Σx² | sum of squares |
|---|---|
| Σx | sum of values |
| n | count |
Where it is used: A faster way to compute standard deviation without first finding each deviation.
In simple words: Uses the sum and the sum of squares directly, avoiding subtracting the mean each time.
Example: Handy for large data sets
CV = (σ ÷ mean) × 100
| σ | standard deviation |
|---|---|
| mean | the average |
Where it is used: To compare the spread of two data sets with different units or scales.
In simple words: Spread as a percentage of the average — lets you compare consistency fairly.
Example: σ=2, mean=40 → CV = 5%
Median = L + [(n/2 − cf) ÷ f] × h
| L | lower limit of median class |
|---|---|
| cf | cumulative frequency before it |
| f | frequency of median class |
| h | class width |
Where it is used: To find the median when data is grouped into class intervals.
In simple words: Locates the middle value inside the class where the halfway count falls.
Example: Used with grouped frequency tables
Mode = L + [(f₁ − f₀) ÷ (2f₁ − f₀ − f₂)] × h
| L | lower limit of modal class |
|---|---|
| f₁ | frequency of modal class |
| f₀, f₂ | frequencies before and after |
| h | class width |
Where it is used: To find the mode when data is grouped into class intervals.
In simple words: Pinpoints the most common value inside the tallest bar of a grouped table.
Example: Used with grouped frequency tables
P(E) = Favourable outcomes ÷ Total outcomes
| P(E) | probability of event E |
|---|---|
| Favourable | outcomes we want |
| Total | all possible outcomes |
Where it is used: To measure how likely something is — dice, cards, weather, games, risk.
In simple words: Chance = the number of ways to win divided by the total number of possible results. It's between 0 and 1.
Example: Rolling a 4 on a die: P = 1/6
P(not E) = 1 − P(E)
| P(E) | probability the event happens |
|---|
Where it is used: To find the chance an event does NOT happen, using the chance that it does.
In simple words: Since something either happens or it doesn't, subtract its chance from 1 to get the opposite chance.
Example: P(rain)=0.3 → P(no rain) = 1 − 0.3 = 0.7
0 ≤ P(E) ≤ 1
| P(E) | probability of event E |
|---|
Where it is used: To check that a probability answer is valid — it can never be negative or more than 1.
In simple words: A chance is always between 0 (impossible) and 1 (certain).
Example: P = 0.5 means a fifty-fifty chance
P(E) + P(not E) = 1
| P(E) | chance it happens |
|---|---|
| P(not E) | chance it doesn't |
Where it is used: The total probability of all outcomes of an event is always 1.
In simple words: The chance something happens plus the chance it doesn't must add up to a full 1.
Example: P(win)=0.4 → P(not win) = 0.6
P(A∪B) = P(A) + P(B) − P(A∩B)
| P(A∩B) | chance of both |
|---|
Where it is used: To find the chance that A or B happens when they can overlap.
In simple words: Add the two chances, then subtract the overlap so it isn't counted twice.
Example: P(A)=0.5,P(B)=0.4,both 0.2 → 0.7
P(A∪B) = P(A) + P(B) (if they can't both happen)
| A, B | events that can't occur together |
|---|
Where it is used: To find the chance of A or B when the two cannot happen at the same time.
In simple words: If the two events never overlap, just add their chances.
Example: Rolling a 2 or a 5: 1/6 + 1/6 = 1/3
P(A∩B) = P(A) × P(B)
| A, B | independent events |
|---|
Where it is used: To find the chance that two independent events both happen.
In simple words: If one event doesn't affect the other, multiply their chances.
Example: Two heads in a row: ½ × ½ = ¼
P(A | B) = P(A∩B) ÷ P(B)
| P(A|B) | chance of A given B happened |
|---|
Where it is used: To find the chance of A when B is already known to have happened.
In simple words: Updates the chance of A once you know B has occurred.
Example: P(both)=0.2, P(B)=0.4 → P(A|B)=0.5
Odds in favour = favourable : unfavourable
| favourable | wanted outcomes |
|---|---|
| unfavourable | the rest |
Where it is used: To express chance as odds rather than a probability — games, betting.
In simple words: Compares the number of winning ways to the number of losing ways.
Example: Rolling a 6: odds in favour 1:5
Mean = np, Variance = npq (q = 1 − p)
| n | number of trials |
|---|---|
| p | success probability |
| q | failure probability |
Where it is used: To find the average and spread of successes in repeated yes/no trials (Class 12).
In simple words: For repeated coin-style trials, the average successes is n×p.
Example: 10 tosses, p=0.5 → mean 5, variance 2.5