Statistics formulas with the conditions kept visible
Scan by topic, distinguish sample notation from population notation, and check the restriction beside a formula before using it.
This reference groups formulas by the question they answer: center and position, variability, event probability, and standardization. Symbols and conditions sit beside the formulas so n versus n minus 1, independence, and nonzero denominators are not hidden.
Summarize the center and spread of a numerical dataset.
xˉ=n1i=1∑nxi
Arithmetic meann is a positive integer. The data contain n numerical observations.
Use the arithmetic mean to describe the center of numerical observations when each observation has equal weight.
xˉw=∑i=1nwi∑i=1nwixi
Weighted meann is a positive integer. Weights are nonnegative and at least one weight is positive. Each weight is paired with its corresponding observation.
Use the weighted mean when observations contribute different known weights.
R=xmax−xmin
RangeThe dataset is nonempty. Maximum and minimum are evaluated in the same units.
Use the range for a quick description of the full spread from the smallest to the largest observation.
IQR=Q3−Q1
Interquartile rangeQ sub one and Q sub three are computed from the same dataset. Use one stated quartile convention consistently because conventions can differ.
Use the interquartile range to describe the spread of the middle half of a dataset.
Variation and standard scores
Distinguish population and sample spread, then standardize observations when appropriate.
σ2=N1i=1∑N(xi−μ)2
Population varianceN is a positive integer. Mu is the arithmetic mean of the N population values.
Use population variance when the values represent the complete population being described.
σ=N1i=1∑N(xi−μ)2
Population standard deviationN is a positive integer. Mu is the population mean. Use the nonnegative square root.
Use population standard deviation to express population spread in the original data units.
s2=n−11i=1∑n(xi−xˉ)2
Sample variancen is at least two. X bar is the arithmetic mean of the n sample observations.
Use this sample-variance definition when estimating spread from a sample with Bessel's correction.
s=n−11i=1∑n(xi−xˉ)2
Sample standard deviationn is at least two. X bar is the sample mean. Use the nonnegative square root.
Use sample standard deviation to express sample spread in the original data units.
z=σx−μ
Population z-scoreSigma is the population standard deviation and is greater than zero. x and mu use the same units.
Use a population z-score to measure how many population standard deviations a value lies above or below the mean.
Probability
Combine events, condition on known information, and model repeated independent trials.
P(Ac)=1−P(A)
Complement ruleA is an event in a probability space. A superscript c contains exactly the outcomes not in A.
Use the complement rule when the probability of an event is easier to find from the probability that it does not occur.
P(A∪B)=P(A)+P(B)−P(A∩B)
General addition ruleA and B are events in the same probability space.
Use the general addition rule to find the probability that at least one of two events occurs.
P(A∣B)=P(B)P(A∩B)
Conditional probabilityA and B are events in the same probability space. The probability of B is greater than zero.
Use conditional probability when the sample space is restricted to outcomes in event B.
P(A∩B)=P(A∣B)P(B)
Multiplication ruleA and B are events in the same probability space. The probability of B is greater than zero for the conditional term shown.
Use the multiplication rule to find a joint probability from a conditional probability and its conditioning event.
P(X=k)=(kn)pk(1−p)n−k
Binomial probabilityn is a nonnegative integer and k is an integer from zero through n. Trials are independent. Each trial has two outcomes and the same success probability p, where p is between zero and one inclusive.
Use the binomial formula for exactly k successes in n independent trials with one constant success probability.
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Notation
Symbols used on this sheet
Interpret each symbol with the conditions attached to its formula.
Symbol
Meaning
xi
The ith observed numerical value.
n
Sample size, the number of observations in a sample.
N
Population size, the number of values in the full population.
xˉ
Sample mean.
μ
Population mean or expected value, depending on context.
s
Sample standard deviation.
σ
Population standard deviation.
P(A∣B)
Probability of event A conditional on event B.
z
Signed distance from a reference mean in standard-deviation units.
Use the reference
Short applications
These examples show when to choose a formula; detailed instruction belongs in the linked guide or calculator.
Choose the sample standard deviation formula
The sample label selects n minus 1. The mean is 3, and the squared deviations total 8.
1,3,5
s=3−18=2
Condition check: Multiplying s squared, which is 4, by n minus 1, which is 2, recovers 8.
Use both independence and the addition rule
Independence finds the intersection. The union formula then removes that overlap from the marginal sum.
P(A)=0.4,P(B)=0.3
P(A∪B)=0.4+0.3−(0.4)(0.3)=0.58
Condition check: The complement of neither is 1 minus (0.6)(0.7), also 0.58.
Interpret a standardized value
The signed deviation is 8, which contains two standard-deviation units of size 4.
x=78,μ=70,σ=4
z=(78−70)/4=2
Condition check: The inverse formula gives x = 70 + (2)(4) = 78.
Explore this collection
Open a focused formula reference
Use the subject sheet for broad review or open the narrower reference that matches the formula you need.
Match not, or, and, given, or exactly k to the correct rule, then check independence, denominator, and equal-likelihood conditions before substituting.
Choose sample or population before selecting a denominator, preserve exact squared deviations through variance, and return to original units with the square root.
Keep the value, mean, and positive standard deviation from one reference group, then choose the population, sample, inverse, comparison, or normal-model relation that matches the task.
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Common questions
Statistics formulas with the conditions kept visible FAQ
When do I use n and when do I use n minus 1?
Use n in the sample mean because it averages the observed values. Use n minus 1 in the usual sample variance and sample standard deviation formulas when estimating population spread from a sample.
What is the difference between s and sigma?
The symbol s denotes sample standard deviation. Sigma denotes population standard deviation. Their formulas use different centers and denominators because they describe different data roles.
Can I multiply two probabilities?
For an intersection, the general rule is P(A given B) times P(B). You may replace P(A given B) with P(A) only when A and B are independent.
Does a z-score give a percentile automatically?
No. A z-score gives standardized distance. Converting that distance to a percentile requires an appropriate distribution model, commonly a normal model when the context justifies it.
Why do some calculators give different quartiles?
Several accepted conventions interpolate or split an ordered data set differently. A trustworthy result should state the quartile convention rather than treating one convention as universal.
Sources and curriculum alignment
This page follows standard introductory mathematics notation and learning sequences. Use these references to continue with a complete course treatment.
