Essential Statistics Formulas and Concepts

1. Probability

• P(A) = (number of favorable outcomes) / (total number of possible outcomes)
• P(A or B) = P(A) + P(B) - P(A and B)
• P(A|B) = P(A and B) / P(B)
• Expected Value = Σ(x * P(x))

2. Statistical and Practical Significance

• Statistical significance: The likelihood of getting the observed results by chance is very small (usually p < 0.05).
• Practical significance: The results are large enough to be meaningful in real-world applications.

3. Levels of Measurement

• Nominal: Categories with no order.
• Ordinal: Categories with order, but differences are not meaningful.
• Interval: Ordered categories with meaningful differences, no true zero.
• Ratio: Ordered categories with meaningful differences and a true zero.

4. Types of Sampling

• Random: Each member of the population has an equal chance of being selected.
• Systematic: Every kth element is selected from a list.
• Convenience: Subjects are selected based on availability.
• Stratified: Population is divided into subgroups (strata), and random sampling is applied within each stratum.
• Cluster: Population is divided into clusters, and a random sample of clusters is selected.

5. Central Tendency and Variation

• Mean = (Σx) / n
• Median: Middle value when data is ordered.
• Mode: Most frequently occurring value.
• Range = max(x) - min(x)
• Variance = Σ(x - μ)² / (n - 1)
• Standard Deviation = √variance

6. Z-scores and Normal Distribution

• Z-score = (x - μ) / σ
• Standard Normal Distribution: μ = 0, σ = 1.
• Use a Z-table or calculator to find areas under the curve.

7. Confidence Intervals

• Population Mean (normal dist.): x̄ ± z* * (σ / √n)
• Population Proportion: p̂ ± z* * √((p̂ * q̂) / n)
• Difference between two means (independent): (x̄1 - x̄2) ± t* * √((s1² / n1) + (s2² / n2))
• Margin of Error (E): Maximum likely difference between sample estimate and population parameter.

8. Hypothesis Testing

• Null hypothesis (H0): Claim to be tested.
• Alternative hypothesis (Ha): Claim to be supported.
• Type I error: Rejecting H0 when it is true (α).
• Type II error: Failing to reject H0 when it is false (β).
• p-value: Probability of obtaining the observed results or more extreme results, assuming H0 is true.
• One-tailed vs. two-tailed tests.

9. Comparing Two Populations

• Independent Samples: Data from two separate groups or populations.
• Dependent Samples: Paired data, before and after measurements on the same individuals.
• Use appropriate test statistic (z or t) and p-value to make conclusions.

10. Additional Formulas

• Sample size for proportion: n = (z*)² * (p̂ * q̂) / E²
• Confidence Interval for difference in proportions: (p̂1 - p̂2) ± z* * √((p̂1 * q̂1 / n1) + (p̂2 * q̂2 / n2))