π What is Statistics?
Statistics is the science of collecting, analyzing, interpreting, and presenting data. It helps us make informed decisions in fields like science, business, psychology, and more.
π Measures of Central Tendency
Mean (Average)
Definition: Add all values and divide by how many there are.
Example: (10 + 20 + 30) / 3 = 20
Median
Definition: The middle value when data is sorted.
Example: [3, 5, 7] β Median = 5
Mode
Definition: The value(s) that appear most often.
Example: [2, 2, 3, 4] β Mode = 2
π Measures of Spread
Range
Definition: Highest value minus lowest value.
Example: [2, 5, 9] β 9 - 2 = 7
Variance (ΟΒ²)
Definition: The average of squared differences from the mean.
Formula: ΟΒ² = Ξ£(x - ΞΌ)Β² / N
Standard Deviation (Ο)
Definition: The square root of the variance. Shows how spread out data is.
π Z-Scores & The Normal Distribution
Z-Score
Definition: How many standard deviations a value is from the mean.
Formula: z = (x - ΞΌ) / Ο
Example: z = (85 - 70) / 10 = 1.5
Z-Table (Partial)
| Z | Area (Left) | Percentile |
|---|---|---|
| 0.0 | 0.5000 | 50% |
| 1.0 | 0.8413 | 84.13% |
| 1.5 | 0.9332 | 93.32% |
| 2.0 | 0.9772 | 97.72% |
| 3.0 | 0.9987 | 99.87% |
π Distribution Shape
Skewness
Definition: Measures if the data is symmetrical or skewed.
- β‘οΈ Positive skew: longer tail on the right (e.g., income)
- β¬ οΈ Negative skew: longer tail on the left
Kurtosis
Definition: Measures how heavy or light the tails of a distribution are.
π Correlation & Relationships
Correlation (r)
Definition: Measures the linear relationship between two variables.
Range: -1 to +1
Example: r = 0.9 β strong positive correlation
π§ͺ Hypothesis Testing
What is Hypothesis Testing?
Itβs a method to test a claim (hypothesis) about a population using sample data.
- Hβ (Null): No effect or no difference
- Hβ (Alt): There is an effect or difference
P-Value
Definition: Probability of getting your result (or more extreme) if Hβ is true.
Rule: If p < 0.05 β reject Hβ
π Confidence Intervals
Definition: A range likely to contain the true population parameter.
Example: "We are 95% confident the mean is between 48 and 52."