100+ Statistics Formulas and Helpful Explanations

Proportion: p̂ – E < p < p̂ + E
- The actual proportion of the population (p) that has what you’re looking for is somewhere between the sample mean (p̂) minus the margin of error (E) and the sample mean plus the margin of error.
- Where E = z _α/2√[ (p̂ * q-hat) ÷ n ]
- Lower bound: p̂ – E
- Upper bound: p̂ + E
Mean: x̄ – E < µ < x̄ + E
- The actual mean of the population (µ) is somewhere between the sample mean (x̄) minus the margin of error (E) and the sample mean plus the margin of error.
- If σ is known: E = z _α/2σ ÷√n
- If σ is unknown: E = t _α/2s ÷ √n
- Lower bound: x̄ – E
- Upper bound: x̄ + E
Variance: [ (n-1) s² ] ÷ X_R² < σ² < [ (n-1) s² ] ÷ X_L²

Confidence Intervals: Two Populations

Hypothesis

H₀: null hypothesis
- H₀= µ (null hypothesis is the original mean)
H₁: alternate hypothesis
- Choose one: H₁> µ, H₁< µ, or H₁≠ µ
After analyzing the data, you have two choices:
1. “Reject the null hypothesis” (the original mean was not correct)“
2. Fail to reject the null hypothesis” (the original mean was correct)

Linear Correlation/Regression

Means

x̄: mean of sample
- Σx ÷ n: add all the events and divide by the number of eventsµ:
µ: mean of population
- Σx * P(x): sum of all events times probability that each event will happen
n * p: number of events times probability that desired event occurs (binomial distribution)
p̂: mean of sample when event either does or doesn’t happen
- probability of success of sample
- always a decimal
d-bar: mean of the differences

Miscellaneous Letters

α: the significance level
p-bar: pooled estimate (x₁ + x₂) ÷ (n₁ + n₂)
σ²: variance of a binomial distribution
- n*p*q: sample size times probability event will happen times probability event will not happens
s²: also variance
x: the raw number that you actually counted

Multinomial and Contingency Tables

Permutations & Combinations

n = how many total
r = how many people you choose
Permutations
- order matters
- nPr: n! ÷ (n-r)!
Combinations
- order doesn’t matter
- nCr: n! ÷ [ (n-r)! * r! ]

Probability

Definitions
- P(A): probability of A
- P(B): probability of B
- P(B|A): probability of B given A
  - What is the probability that B will happen if you know that A is has happened?
- P(Ā): complement of A
  - 1 – P(A)
- To be or not to be Mutually Exclusive/Disjoint
  - A and B are mutually exclusive or disjoint:
    - A and B cannot both happen at the same time
    - Ex: probability of drawing a black card and probability of drawing a heart
    - P(A or B) = P(A) + P(B)
  - A and B are not mutually exclusive or not disjoint:
    - A and B overlap
    - Ex: probability of drawing a red card and probability of drawing an ace
    - P(A or B) = P(A) + P(B) – P(A and B)Independent vs. Dependent
Independent probability of first event does not affect probability of second event
- Ex: draw a card, put it back, then draw a new card
- Ex: probability of drawing a red card and probability of drawing an ace
- P(A and B) = P(A) * P(B)
Dependent: probability of first event affects probability of second event
- Ex: draw a card, don’t put it back, draw a new card
- Ex: probability of drawing a black card and probability of drawing a heart
- P(A and B) = P(A) * P(B|A)

Probability Distribution/Proportions

p: probability event happens in population
- Always a decimal
- Refers to whole population
- Use test statistic and z- or t-table to find p
- If not given, assume it’s 0.5
- Also called pooled estimate and proportion
p̂: probability event happens in sample
- Always a decimal
x̄: mean number of successes (yeses) divided by total number of trials (people sampled)
q: probability event doesn’t happen
- Always a decimal
- (1 – p)
k: actual number of successes
- Always a whole number

Binomial Probability and Poisson Distribution

Sample Size (n)

Standard Deviation & Standard Error

Test Statistics: Explanation

Statistics Bell Curve — Click to Enlarge

Critical value
- How far x̄ needs to be from µ to reject
- How far p̂ needs to be from p to reject
- The z-score or t-score that alpha (α) indicates
z-score when sample > 30
- 90% confidence: ±1.64
- 95% confidence: ±1.96
- 99% confidence: ±2.575
- If you keep taking samples, 99% of the time the sample mean will be within 2.575 standard deviations of the actual population mean.
- Using sample data: z = (x – x̄) ÷ s
- Using population data: z = (x – µ) ÷ σ
t-score when sample is ≤ 30
- 95% confidence: ±2.08
- 99% confidence: ±2.575
How to use a z-table or t-table
1. If significance is 1%, the p-value is 0.0100
2. Find 0.0100 on the z or t-table (center #s)
3. Use column headings to find z or t-score
4. For a p-value of 0.0100, the z-score is 2.33, so the critical value is 2.33

Test Statistics: One Population

Test Statistics: Two Populations

Test Statistics: Nonparametric Tests

Printable Formula Sheets

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Statistics Formulas and What They Mean 1 — Click to Enlarge Page 1

Statistics Formulas and What They Mean 2 — Click to Enlarge Page 2

Statistics Formulas and What They Mean 3 — Click to Enlarge Page 3

Statistics Formulas and What They Mean 4 — Click to Enlarge Page 4

Beyond Statistics Formulas

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Statistics Formulas and What They Mean

Confidence Intervals: Margin of Error (E)

Confidence Intervals: One Population