Distribution & Standardization Cheatsheet

Empirical Rule

• For a normal distribution: within 1 standard deviation (SD): ext{P}(|X-BC|\le \sigma) = 0.68

• Within 2 SD: $0.95$

• Within 3 SD: $0.997$

• Example (IQ): mean $\mu=100$, SD $\sigma=11$

  • 1 SD range: $[89, 111]$

  • 2 SD range: $[78, 122]$

  • 3 SD range: $[67, 133]$

  • Score between 67 and 133 corresponds to about $99.7\%$ of the population.

Z-scores and Standardization

• Definition: $z = \frac{x-\mu}{\sigma}$

• Z-score units follow the standard normal: $Z \sim N(0,1)$

• Interpretations:

  • Positive z: above mean; Negative z: below mean

  • Z-scores provide a standardized way to compare different distributions

• Uses:

  • Compare an individual's position within its population

  • Compare across populations (e.g., different exams) by converting to z-scores

  • If exams are the same, you can compare raw scores directly; otherwise use z-scores

From x to z and back

• Given mean and SD: $\mu,\; \sigma$

• To convert: $z = \dfrac{x-\mu}{\sigma}$

• To recover x: $x = \mu + z\,\sigma$

Percentiles and quartiles

• Percentile: the value x such that $P(X\le x) = p$ where p is the percentile expressed as a decimal (e.g., 0.90 for the 90th percentile)

• Area to the left defines the percentile

• Five-number summary (for raw data): min, Q1, median (Q2), Q3, max

• Interquartile range: $\text{IQR} = Q3 - Q1$

• Use quartiles when data may be skewed; quartiles are robust to outliers

Outliers and fences

• Box/whisker interpretation: whiskers indicate spread; long whiskers suggest skewness

• Outlier detection (IQR method):

  • Lower fence: $\text{LFence} = Q1 - 1.5\cdot \text{IQR}$

  • Upper fence: $\text{UFence} = Q3 + 1.5\cdot \text{IQR}$

• Values beyond fences are potential outliers

• Note: some contexts use alternative rules (e.g., 1.5 IQR or 3 IQR) depending on the textbook or software

When to use mean vs median; role of skewness

• If distribution is skewed or has outliers: use median and IQR (robust)

• If roughly symmetric and no major outliers: mean and SD are informative

Practical problem-solving approach

• Step 1: Write down relevant information (mean, SD, sample size, x values)

• Step 2: Decide whether to compare using x (raw) or z (standardized)

• Step 3: If asked for a z-score: use $z = \frac{x-\mu}{\sigma}$

• Step 4: If asked for x from z: use $x = \mu + z\sigma$

• Step 5: For comparing across populations, use z-scores; for same exam, raw x can be compared

Quick notes on interpretation and examples

• Normal assumption matters: many methods assume normality; if X is not normal, consider medians/percentiles or transform data

• Percentiles give position relative to the population: e.g., 5th percentile means 5% are at or below that value; 95th percentile means 95% are at or below that value

• Skewness and outliers affect which measures you report (box plots help visualize)

Five-number summary in practice

• Minimum, Q1, Median, Q3, Maximum

• Box plot interpretation: roughly symmetrical if whiskers are similar length; skew to the right if right whisker longer

• Example workflow for a problem: compute five-number summary from data (min, Q1, median, Q3, max) and check IQR for potential outliers

Quick reference formulas

• Z-score: $z = \dfrac{x-\mu}{\sigma}$

• From Z to x: $x = \mu + z\sigma$

• Standard normal: $Z \sim N(0,1)$

• Within 1/2/3 SD ranges correspond to 68% / 95% / 99.7%

• IQR: $\text{IQR} = Q3 - Q1$

• Outlier fences: $\text{LFence} = Q1 - 1.5\cdot \text{IQR}, \quad \text{UFence} = Q3 + 1.5\cdot \text{IQR}$