📊 Statistics Formulas Cheat Sheet

Quick Reference Guide

This cheat sheet contains all the essential statistical formulas you need, organized by topic with clear explanations and usage guidelines.

📐 Table of Contents

1. Descriptive Statistics
2. Probability & Distributions
3. Sampling & Central Limit Theorem
4. Confidence Intervals
5. Hypothesis Testing Basics
6. Single Sample Tests
7. Two Sample Tests
8. Proportion Tests
9. Effect Sizes
10. Quick Decision Tree

1. Descriptive Statistics

Measures of Central Tendency

Mean (Average)

μ = (Σx) / N          [Population mean]
x̄ = (Σx) / n          [Sample mean]

Where:
- Σx = Sum of all values
- N = Population size
- n = Sample size

When to use: Most common measure, use when data is roughly symmetric without extreme outliers.

Median (Middle Value)

1. Sort data from smallest to largest
2. If n is odd: Median = middle value
3. If n is even: Median = average of two middle values

When to use: Better than mean when data has outliers or is skewed.

Mode (Most Frequent)

Mode = Value that appears most often

When to use: For categorical data or to find most common value.

Measures of Spread

Range

Range = Maximum - Minimum

When to use: Quick measure of spread, but sensitive to outliers.

Variance

σ² = Σ(x - μ)² / N     [Population variance]
s² = Σ(x - x̄)² / (n-1) [Sample variance]

Where:
- σ² (sigma squared) = Population variance
- s² = Sample variance
- (n-1) = Degrees of freedom (Bessel's correction)

When to use: Measure of spread, but in squared units.

Standard Deviation

σ = √[Σ(x - μ)² / N]      [Population std dev]
s = √[Σ(x - x̄)² / (n-1)]  [Sample std dev]

When to use: Most common measure of spread, in original units.

Coefficient of Variation

CV = (s / x̄) × 100%

Where:
- CV = Coefficient of Variation
- s = Standard deviation
- x̄ = Mean

When to use: Compare variability between datasets with different units or scales.

Measures of Position

Percentile

Percentile rank = (Number of values below x / Total values) × 100

When to use: Understand relative position in a distribution.

Quartiles

Q1 = 25th percentile (1st quartile)
Q2 = 50th percentile (median)
Q3 = 75th percentile (3rd quartile)
IQR = Q3 - Q1 (Interquartile Range)

When to use: Identify spread and detect outliers.

Z-Score (Standardization)

z = (x - μ) / σ        [Population]
z = (x - x̄) / s        [Sample]

Where:
- z = Standard score
- x = Individual value
- μ or x̄ = Mean
- σ or s = Standard deviation

Interpretation:

• z = 0: Value is at the mean
• z = 1: Value is 1 std dev above mean
• z = -1: Value is 1 std dev below mean

• z	> 3: Likely an outlier

When to use: Standardize values, compare across different scales, detect outliers.

Correlation

Pearson Correlation Coefficient

r = Σ[(x - x̄)(y - ȳ)] / √[Σ(x - x̄)² × Σ(y - ȳ)²]

Alternative formula:
r = [nΣxy - (Σx)(Σy)] / √{[nΣx² - (Σx)²][nΣy² - (Σy)²]}

Where:
- r = Correlation coefficient (-1 to +1)
- x, y = Individual values
- x̄, ȳ = Means
- n = Sample size

Interpretation:

• r = +1: Perfect positive correlation
• r = +0.7 to +1: Strong positive
• r = +0.3 to +0.7: Moderate positive
• r = 0: No correlation
• r = -0.3 to -0.7: Moderate negative
• r = -0.7 to -1: Strong negative
• r = -1: Perfect negative correlation

When to use: Measure linear relationship between two continuous variables.

2. Probability & Distributions

Basic Probability

Probability Formula

P(A) = (Number of favorable outcomes) / (Total number of outcomes)

Where:
- P(A) = Probability of event A
- 0 ≤ P(A) ≤ 1

Complement Rule

P(not A) = 1 - P(A)

Addition Rule (OR)

P(A or B) = P(A) + P(B) - P(A and B)

If mutually exclusive:
P(A or B) = P(A) + P(B)

Multiplication Rule (AND)

P(A and B) = P(A) × P(B|A)

If independent:
P(A and B) = P(A) × P(B)

Normal Distribution

Standard Normal Distribution

Z ~ N(0, 1)

Where:
- Mean (μ) = 0
- Standard deviation (σ) = 1

Key Properties:

• 68% of data within ±1σ
• 95% of data within ±1.96σ
• 99% of data within ±2.58σ

Converting to Standard Normal

Z = (X - μ) / σ

Where:
- Z = Standard normal variable
- X = Original variable
- μ = Mean
- σ = Standard deviation

When to use: Find probabilities, compare values across different distributions.

3. Sampling & Central Limit Theorem

Sampling Distribution

Mean of Sampling Distribution

μx̄ = μ

Where:
- μx̄ = Mean of sample means
- μ = Population mean

Interpretation: The average of all sample means equals the population mean.

Standard Error (SE)

SE = σ / √n

Where:
- SE = Standard error (std dev of sampling distribution)
- σ = Population standard deviation
- n = Sample size

When to use: Measure variability of sample means, calculate confidence intervals.

Key insight: Larger sample size → Smaller standard error → More precise estimates

Central Limit Theorem

CLT Statement

For n ≥ 30:
x̄ ~ N(μ, σ/√n)

Where:
- x̄ = Sample mean
- N = Normal distribution
- μ = Population mean
- σ/√n = Standard error

What it means: Sample means are approximately normally distributed, regardless of the population’s distribution, when sample size is large enough.

Requirements:

• Random sampling
• Sample size n ≥ 30 (or population is already normal)
• Independent observations

4. Confidence Intervals

Confidence Interval for Mean (σ known)

Z-Interval Formula

CI = x̄ ± Z* × (σ / √n)

Where:
- CI = Confidence interval
- x̄ = Sample mean
- Z* = Critical z-value
- σ = Population standard deviation
- n = Sample size

Common Z* values:

• 90% confidence: Z* = 1.645
• 95% confidence: Z* = 1.96
• 99% confidence: Z* = 2.576

When to use: Population standard deviation is known (rare in practice).

Confidence Interval for Mean (σ unknown)

T-Interval Formula

CI = x̄ ± t* × (s / √n)

Where:
- t* = Critical t-value (from t-table)
- s = Sample standard deviation
- df = n - 1 (degrees of freedom)

When to use: Population standard deviation is unknown (most common case).

Confidence Interval for Proportion

Proportion CI Formula

CI = p̂ ± Z* × √[p̂(1 - p̂) / n]

Where:
- p̂ = Sample proportion
- Z* = Critical z-value
- n = Sample size

Requirements:

• np̂ ≥ 10
• n(1 - p̂) ≥ 10

When to use: Estimate population proportion from sample proportion.

Margin of Error

Margin of Error Formula

ME = Z* × (σ / √n)     [Known σ]
ME = t* × (s / √n)     [Unknown σ]

Where:
- ME = Margin of error
- Larger ME = Less precise estimate
- Smaller ME = More precise estimate

How to reduce margin of error:

• Increase sample size (n)
• Accept lower confidence level
• Reduce population variability (can’t control)

5. Hypothesis Testing Basics

Hypotheses Setup

Null Hypothesis (H₀)

H₀: μ = μ₀     [No difference/change]
H₀: μ₁ = μ₂    [Two groups are equal]
H₀: p = p₀     [Proportion equals claimed value]

Interpretation: Status quo, no effect, no difference.

Alternative Hypothesis (H₁)

Two-tailed:  H₁: μ ≠ μ₀   [Different from]
Right-tailed: H₁: μ > μ₀   [Greater than]
Left-tailed:  H₁: μ < μ₀   [Less than]

Significance Level (α)

α = Probability of Type I Error

Common values:
- α = 0.05 (5%) - Most common
- α = 0.01 (1%) - More strict
- α = 0.10 (10%) - More lenient

Interpretation: Maximum acceptable probability of rejecting H₀ when it’s actually true.

P-Value

P-Value Definition

p-value = P(observing data this extreme | H₀ is true)

Decision Rule:

• If p-value < α: Reject H₀ (statistically significant)
• If p-value ≥ α: Fail to reject H₀ (not significant)

Interpretation:

• p < 0.01: Very strong evidence against H₀
• p < 0.05: Strong evidence against H₀
• p < 0.10: Moderate evidence against H₀
• p ≥ 0.10: Weak or no evidence against H₀

Type I and Type II Errors

Type I Error (α):
- Reject H₀ when H₀ is true
- "False Positive"
- Probability = α

Type II Error (β):
- Fail to reject H₀ when H₀ is false
- "False Negative"
- Probability = β

Power = 1 - β
- Probability of correctly rejecting false H₀

6. Single Sample Tests

One-Sample Z-Test (σ known)

Test Statistic

z = (x̄ - μ₀) / (σ / √n)

Where:
- x̄ = Sample mean
- μ₀ = Claimed population mean
- σ = Population standard deviation
- n = Sample size

Requirements:

• Population standard deviation (σ) is known
• n ≥ 30, OR population is normally distributed
• Random sample

When to use: Testing if sample mean differs from claimed value, σ known.

One-Sample T-Test (σ unknown)

Test Statistic

t = (x̄ - μ₀) / (s / √n)

Where:
- s = Sample standard deviation
- df = n - 1 (degrees of freedom)

Requirements:

• Population standard deviation (σ) is unknown
• n ≥ 30, OR population is normally distributed
• Random sample

When to use: Testing if sample mean differs from claimed value, σ unknown (most common).

Decision:

• Compare

  t

  to critical value t* from t-table

• OR compare p-value to α

7. Two Sample Tests

Independent Two-Sample T-Test

Test Statistic (Equal Variances)

t = (x̄₁ - x̄₂) / √[s²ₚ(1/n₁ + 1/n₂)]

Where:
Pooled variance:
s²ₚ = [(n₁-1)s₁² + (n₂-1)s₂²] / (n₁ + n₂ - 2)

Degrees of freedom:
df = n₁ + n₂ - 2

Requirements:

• Two independent groups
• Both groups approximately normally distributed
• Equal variances (test with F-test or Levene’s test)

Test Statistic (Unequal Variances - Welch’s t-test)

t = (x̄₁ - x̄₂) / √[(s₁²/n₁) + (s₂²/n₂)]

Degrees of freedom (Welch-Satterthwaite):
df = [(s₁²/n₁ + s₂²/n₂)²] / [(s₁²/n₁)²/(n₁-1) + (s₂²/n₂)²/(n₂-1)]

When to use: Comparing means of two independent groups.

Paired T-Test

Test Statistic

t = d̄ / (sᵈ / √n)

Where:
- d̄ = Mean of differences
- sᵈ = Standard deviation of differences
- n = Number of pairs
- df = n - 1

Steps:

1. Calculate differences: d = x₁ - x₂ for each pair
2. Calculate mean of differences: d̄
3. Calculate standard deviation of differences: sᵈ
4. Compute t-statistic

Requirements:

• Same subjects measured twice (paired data)
• Differences are approximately normally distributed
• Random sample

When to use: Before/after comparisons, matched pairs.

8. Proportion Tests

One-Sample Proportion Test

Test Statistic

z = (p̂ - p₀) / √[p₀(1 - p₀) / n]

Where:
- p̂ = Sample proportion (x/n)
- p₀ = Claimed population proportion
- x = Number of successes
- n = Sample size

Requirements:

• np₀ ≥ 10
• n(1 - p₀) ≥ 10
• Random sample

When to use: Testing if sample proportion differs from claimed proportion.

Two-Sample Proportion Test

Test Statistic

z = (p̂₁ - p̂₂) / √[p̂(1 - p̂)(1/n₁ + 1/n₂)]

Where:
Pooled proportion:
p̂ = (x₁ + x₂) / (n₁ + n₂)

- p̂₁ = x₁/n₁ (proportion in group 1)
- p̂₂ = x₂/n₂ (proportion in group 2)

Requirements:

• n₁p̂₁ ≥ 10, n₁(1-p̂₁) ≥ 10
• n₂p̂₂ ≥ 10, n₂(1-p̂₂) ≥ 10
• Independent groups
• Random samples

When to use: Comparing proportions between two independent groups.

9. Effect Sizes

Cohen’s d (Standardized Mean Difference)

For Independent Samples

d = (x̄₁ - x̄₂) / sₚₒₒₗₑᵈ

Where:
sₚₒₒₗₑᵈ = √[(s₁² + s₂²) / 2]

Interpretation:

• d	= 0.2: Small effect

• d	= 0.5: Medium effect

• d	= 0.8: Large effect

For Paired Samples

d = d̄ / sᵈ

Where:
- d̄ = Mean of differences
- sᵈ = Standard deviation of differences

When to use: Measure practical significance (how big is the difference?).

R² (Coefficient of Determination)

R² = r²

Where:
- r = Correlation coefficient
- R² = Proportion of variance explained

Interpretation:

• R² = 0.01: Small effect (1% variance explained)
• R² = 0.09: Medium effect (9% variance explained)
• R² = 0.25: Large effect (25% variance explained)

When to use: Measure strength of relationship in correlation/regression.

10. Quick Decision Tree

Which Test Should I Use?

START: What are you comparing?

├─ ONE sample to a claimed value
│  ├─ Data type: MEAN (continuous)
│  │  ├─ σ known? → Z-test
│  │  └─ σ unknown? → One-sample t-test
│  └─ Data type: PROPORTION (categorical)
│     └─ One-sample proportion test (z-test)
│
├─ TWO groups
│  ├─ Data type: MEANS (continuous)
│  │  ├─ Independent groups? → Independent t-test
│  │  └─ Paired/matched? → Paired t-test
│  └─ Data type: PROPORTIONS (categorical)
│     └─ Two-sample proportion test (z-test)
│
└─ RELATIONSHIP between two variables
   └─ Both continuous? → Pearson correlation (r)

Sample Size Requirements

Test Type          | Minimum Sample Size
-------------------|--------------------
Z-test             | n ≥ 30
T-test             | n ≥ 30 (or normal population)
Proportion test    | np ≥ 10 AND n(1-p) ≥ 10
Paired t-test      | n ≥ 30 pairs (or normal differences)
Independent t-test | n₁, n₂ ≥ 30 (or both normal)

📝 Common Symbols Reference

🎯 Quick Tips

Choosing Confidence Level

• 90%: Quick estimates, exploratory analysis
• 95%: Standard for most research (most common)
• 99%: High-stakes decisions, need more certainty

Interpreting Results

1. Check assumptions first (normality, sample size, independence)
2. Look at p-value (is it < α?)
3. Check effect size (is the difference meaningful?)
4. Consider practical significance (does it matter in real life?)

Common Mistakes to Avoid

• ❌ Using z-test when σ is unknown (use t-test instead)
• ❌ Using independent t-test for paired data (use paired t-test)
• ❌ Confusing statistical significance with practical importance
• ❌ Interpreting p-value as probability that H₀ is true
• ❌ Accepting H₀ (we “fail to reject,” not “accept”)
• ❌ Using proportion test for means or vice versa

📚 Related Resources

• Chapter 1: Exploratory Data Analysis
• Chapter 2: Inferential Statistics and Hypothesis Testing
• Chapter 3: Introduction to Hypothesis Testing
• Chapter 4.1: Single Sample Tests
• Chapter 4.2: Two Sample Tests
• Chapter 4.3: Proportion Tests

Print this page and keep it handy for quick reference! 📄

Last updated: 2026