The Normal Distribution

The famous bell curve

• Many real variables — heights, measurement errors, test scores — follow a bell-shaped pattern.
• The normal distribution 正态分布 is the idealized bell curve, symmetric about its center.
• It's completely described by two numbers: the mean $\mu$ (center) and standard deviation $\sigma$ (spread).
• Because so much falls into this shape, the normal model is the workhorse of statistics.

The empirical rule (68–95–99.7)

• For a normal distribution, the empirical rule 经验法则 tells you how much data falls within a few SDs of the mean:
• About $68\%$ within $1$ standard deviation of the mean.
• About $95\%$ within $2$ SDs; about $99.7\%$ within $3$ SDs.
• These three numbers let you estimate proportions without a calculator.

The normal curve: a probability is the area under it, centred on the mean

z-scores and the standard normal

• Convert any value to a $z$-score $z=\frac{x-\mu}{\sigma}$ — its distance from the mean in SDs.
• The standard normal 标准正态 distribution has mean $0$ and SD $1$; every normal curve becomes it after standardizing.
• Use a $z$-table (or technology) to turn a $z$-score into a proportion or percentile.
• This is how you find "what percent scored below $x$."

Is a normal model reasonable?

• Not everything is normal — check before assuming it.
• A roughly symmetric, bell-shaped, single-peaked graph supports a normal model.
• Strong skew, multiple peaks, or heavy outliers mean a normal model is a poor fit.
• Assess the data's shape first; the empirical rule and $z$-tables only apply if it's approximately normal.

Write from a z-score of 3

• A test has mean 100 and standard deviation 10.
• One score is 130. The lesson z-score of 2 is a different score.