Normal Distribution

Normal Distribution - The Bell Curve Basics

• Continuous probability distribution (Gaussian distribution); vital in biostatistics.
• Key features:
  • Symmetrical, bell-shaped curve.
  • Mean = Median = Mode, all at the central peak.
  • Symmetrical around the mean ($\.mu$).
• Total area under curve = 1 (total probability).
• Parameters:
  • Mean ($\.mu$): Determines central location.
  • Standard Deviation ($\.sigma$): Determines spread/width.
• Standard Normal Distribution (SND): Special case with Mean = 0, SD = 1.

⭐ The Normal Distribution, or Gaussian distribution, is foundational in biostatistics largely due to the Central Limit Theorem, which states that the sampling distribution of the mean approaches normal as sample size increases.

Normal Distribution - Slicing The Bell

• Symmetrical, bell-shaped curve where mean = median = mode.
• Total area under curve = 1 (or 100%); 50% of area on each side of mean ($\.mu$).
• Points of inflection (curve changes concavity) are at $\.mu \.pm 1\.sigma$.
• 📌 Empirical Rule (The 68-95-99.7 Rule):
  • $\.mu \.pm 1\.sigma$: Encompasses 68.27% of data.
  • $\.mu \.pm 2\.sigma$: Encompasses 95.45% of data.
  • $\.mu \.pm 3\.sigma$: Encompasses 99.73% of data.
• Specific Z-score areas:
  • $\.mu \.pm 1.96\.sigma$ covers exactly 95% of observations.
  • $\.mu \.pm 2.58\.sigma$ covers exactly 99% of observations.

⭐ Approximately 95.45% of values lie within 2 standard deviations ($\.mu \.pm 2\.sigma$) of the mean, and 5% of values lie outside $\.mu \.pm 1.96\.sigma$.

Normal Distribution - Z-Score Zeroes In

• Z-score (Standard Normal Deviate, SND): Standardizes any normal distribution.
  • Transforms a general normal curve (mean $\mu$, SD $\sigma$) into a Standard Normal Distribution (SND).
  • Formula: $Z = (X - \mu) / \sigma$
    • $X$: observed value
    • $\mu$: population mean
    • $\sigma$: population standard deviation
• SND Characteristics:
  • Mean $\mu = 0$
  • Standard Deviation $\sigma = 1$
• Utility:
  • Compares scores from different distributions.
  • Enables probability calculation via standard Z-tables.
• Key Z-values for Confidence Intervals (CI):
  • 90% CI: $Z = \pm 1.645$
  • 95% CI: $Z = \pm 1.96$ (Most common)
  • 99% CI: $Z = \pm 2.58$

⭐ A Z-score indicates how many standard deviations an observation or data point is from the mean. A positive Z-score means the value is above the mean, and a negative Z-score means it is below the mean.

Normal Distribution - Bell Curve in Action

• Symmetrical, bell-shaped curve; Mean = Median = Mode.
• Area under curve = 1. Defined by mean ($\.mu$) and standard deviation ($\.sigma$).
• Empirical Rule (68-95-99.7 Rule):
  • $\.mu \pm 1\.sigma$: 68% of data.
  • $\.mu \pm 2\.sigma$: 95% of data.
  • $\.mu \pm 3\.sigma$: 99.7% of data.
• Standard Normal Distribution: Special case with $\.mu=0, \.sigma=1$.
• Z-score: $Z = (X - \.mu) / \.sigma$; indicates distance from mean in SDs.
• Deviations from normality:
  • Skewness: Asymmetry. and negative skew (left tail), showing relative positions of mean, median, mode)
  • Kurtosis: Peakedness (e.g., leptokurtic - high peak).

⭐ In a positively skewed distribution, the tail is to the right, and the relationship is Mean > Median > Mode. In a negatively skewed distribution, the tail is to the left, and Mean < Median < Mode.

• Assessing Normality:

High‑Yield Points - ⚡ Biggest Takeaways

• Symmetrical, bell-shaped curve, with tails extending to infinity.
• Mean = Median = Mode are all equal and located at the center of the distribution.
• Total area under the curve equals 1 (or 100%).
• Empirical Rule (for areas under curve): Approx. 68% of data lies within $\pm$1 SD, approx. 95% within $\pm$1.96 SD (commonly rounded to $\pm$2 SD), and approx. 99.7% within $\pm$3 SD of the mean.
• The Standard Normal Distribution (SND or Z-distribution) has a mean of 0 and a standard deviation of 1.
• A Z-score quantifies the number of standard deviations a particular data point is from the mean.
• Characterized by zero skewness (perfectly symmetrical) and zero excess kurtosis (mesokurtic).