Normal distribution Knowledge-Based Study

Study notes

1. The Normal Distribution

   The normal distribution, often called the Gaussian distribution or bell curve, is a symmetric, continuous probability distribution. It is completely defined by two parameters: the mean, denoted by $\mu$, and the standard deviation, denoted by $\sigma$.

   • Symmetry: The curve is symmetric around its center, meaning that the mean, median, and mode are all equal.
   • Total Area: The total area under the probability density curve is exactly 1 (or 100%).
   • Asymptotic Tails: The tails approach the horizontal axis indefinitely in both directions without ever touching it.
   • Spread: The width of the curve is determined by $\sigma$; larger values produce a flatter, wider curve, whereas smaller values yield a taller, narrower peak.

   The 68–95–99.7 Rule

   For any normally distributed variable, the spread of data relative to the mean follows the Empirical Rule:

   • Approximately 68% of the observations lie within 1 standard deviation of the mean: $[\mu - \sigma, \mu + \sigma]$.
   • Approximately 95% of the observations lie within 2 standard deviations of the mean: $[\mu - 2\sigma, \mu + 2\sigma]$.
   • Approximately 99.7% of the observations lie within 3 standard deviations of the mean: $[\mu - 3\sigma, \mu + 3\sigma]$.

   For example, if test scores follow a normal distribution with $\mu = 70$ and $\sigma = 10$, roughly 68% of students score between 60 and 80, and roughly 95% score between 50 and 90.

   The Standard Normal Distribution

   To compare values across different normal distributions or compute specific probabilities, raw scores are converted into standard scores, known as z-scores. The standard normal distribution has a mean of 0 and a standard deviation of 1, written as $Z \sim N(0, 1)$.

   The transformation formula is:

   $z = \frac{x - \mu}{\sigma}$

   • A positive z-score indicates a value above the population mean.
   • A negative z-score indicates a value below the population mean.
   • A z-score of 0 represents a value exactly equal to the mean.

   Standard normal distribution tables or statistical functions can then determine the cumulative probability $P(Z \le z)$ for any standardized value.

   Sampling Distribution of the Mean

   In statistical inference, we examine how sample statistics behave across repeated random samples taken from a population. By the Central Limit Theorem (CLT), if a random sample of size $n$ is sufficiently large (typically $n \ge 30$), the sampling distribution of the sample mean $\bar{X}$ will be approximately normal, regardless of the population shape.

   The parameters of this sampling distribution are:

   • Mean of Sample Means: $\mu_{\bar{X}} = \mu$
   • Standard Error: $\sigma_{\bar{X}} = \frac{\sigma}{\sqrt{n}}$

   The standard error decreases as sample size increases, meaning larger samples yield sample means that cluster more tightly around the true population mean. Standardizing a sample mean uses the formula:

   $z = \frac{\bar{x} - \mu}{\frac{\sigma}{\sqrt{n}}}$

   This principle forms the foundation for hypothesis testing and constructing confidence intervals.