Normal Distribution – Comprehensive Study Notes

Abraham de Moivre (French mathematician & astronomer)
- Authored The Doctrine of Chances (1718).
- Showed how probabilities from repeated binary events (e.g.
  coin-tosses) gradually take on a bell-shaped (normal) form.
- Book became popular with gamblers because it let them estimate odds.
Carl Friedrich Gauss (German mathematician)
- Derived the exact mathematical function for the normal curve once the population mean (\mu) and standard deviation (\sigma) are known.
- Hence the normal curve is often called the Gaussian distribution.
Adolphe Quetelet (Belgian mathematician & statistician)
- Transferred astronomical/physical uses of the normal distribution into the social sciences (crime rates, marriage rates, etc.).
- Argued that all people possess “average social traits,” positioned along a normal curve.
Sir Francis Galton (English polymath, cousin of Darwin)
- Applied normal-curve reasoning to human intelligence.
- Asserted intelligence ranges from very low to very high with most people in the middle (bell-shaped).
- Proposed controversial eugenic ideas (only the “more intelligent” should reproduce).
- Famous quote from Natural Inheritance praising the “cosmic order” of the normal law.

Single coin toss
- Outcomes: Heads or Tails; probability P=0.5 for each.
Two tosses
- Four ordered sequences (HH, HT, TH, TT).
- Distribution of heads counts:
  • 0 heads: 1/4 • 1 head: 2/4 • 2 heads: 1/4.
Four tosses
- Enumerating all 2^4=16 sequences and plotting “number of heads” produces a histogram that visibly approaches a bell shape.
General principle
- As the number of independent, identically distributed (i.i.d.) binary events grows, the sampling distribution of the count/mean tends toward a normal curve (early intuitive glimpse of the Central Limit Theorem).

Wechsler Intelligence Scale (IQ)
- Mean (M) =100, SD =15.
- Score \ge 130 places an individual above 98\% of the population (top 2\%).
Depression scores (DASS subscale)
- Skewed, not normal: most people show very low depression, minority show high scores (positive skew).
- Illustrates that not all real-world data are normal; checking distributional shape is essential.
Student height example
- Raw histogram not perfectly symmetrical, yet overlaying a fitted normal curve estimates the underlying population distribution.

Probability-density function (pdf):
f(x)=\frac{1}{\sigma\sqrt{2\pi}}\,e^{-\frac12\left(\frac{x-\mu}{\sigma}\right)^2}
Inputs: only the population mean (\mu) and population standard deviation (\sigma).
Constants: \pi and the factor \sqrt{2\pi} are fixed.
Area under the curve = 1 (total probability).

Fix \mu, vary \sigma (red, blue, yellow curves):
- Larger \sigma ⇒ curve spreads horizontally, peak lowers, fatter tails (reduces kurtosis).
- Smaller \sigma ⇒ curve narrows, peak height increases.
Fix \sigma, vary \mu (green curve demo):
- Entire curve shifts left/right without shape change.

Graphs/derivations assume true population parameters (\mu,\sigma).
Empirical data give sample statistics (\bar{x}, s), which estimate the population values.
Many inferential tests (t-test, ANOVA, regression, etc.) rely on an assumption of normality (either of raw data or of residuals).
- If assumption holds → tests are more powerful & accurate.
- Non-normal data can be handled with transforms or non-parametric tests.

Symmetrical about the mean.
Unimodal (single highest peak).
Bell-shaped with tails extending to \pm\infty (the pdf never actually reaches zero).
Measures of central tendency coincide:
\text{mean} = \text{median} = \text{mode}.

Histogram + fitted curve shows approximate symmetry.
Box-and-whisker plot: equal whisker lengths left/right of median implies symmetry.
Q-Q plot can further verify linearity against theoretical quantiles (mentioned implicitly by “three different ways”).

In any normal distribution:
- 68\% of observations lie within \pm1\,\sigma of \mu.
- 95\% lie within \pm2\,\sigma.
- 99.7\% lie within \pm3\,\sigma.
Provides quick probability estimates & aids in outlier detection.

Given: \mu=34, \sigma=8.5.
\pm1\sigma band:
- Lower bound =34-8.5=24.5.
- Upper bound =34+8.5=42.5.
- Contains 68\% of students.
\pm2\sigma band:
- Lower bound =34-2(8.5)=17.0.
- Upper bound =34+2(8.5)=51.0.
- Contains 95\% of students.
\pm3\sigma band:
- Lower bound =34-3(8.5)=8.5.
- Upper bound =34+3(8.5)=59.5.
- Contains 99.7\% of students; only 0.3\% \;(≈3 in 1000) fall outside → classify as outliers.

Comparing individuals to reference populations (e.g.
IQ, height): enables percentile ranks, cut-scores for diagnostics, giftedness, etc.
Policy & ethics: misuse of “average” or “extreme” labels (e.g.
Galton’s eugenic ideas) highlights need for ethical caution.
Real-world nuance: Many psychological traits (e.g.
depression) are skewed or multi-modal; always check empirical shape rather than assume normality.

Normal distributions are common yet not universal; always verify shape.
The curve is fully determined by two parameters \mu,\sigma.
Historical development links gambling, astronomy, social science & psychology.
Gauss’s formula plus the 68\,95\,99.7 rule give powerful shortcuts for probability & outlier assessment.
Understanding normality underpins many classical statistical tests and interpretations.