We represent the shape of a continuous variable using a density curve. This is like a histogram, but with a smooth curve:

Properties:

1. The curve is always above the horizontal axis (because probabilities are always nonnegative).
2. The total area under the curve equals 1.

The proportion of all possible observations that lie within a specified range equals the corresponding area under the density curve.

• A normal curve is a special type of density curve that has a “bell-shaped” distribution.
• A variable is normally distributed or has a normal distribution if its distribution has the shape of a normal curve.

Why “normal”? Because it appears so often in practice!

Normal distributions…

• are fully determined by parameters mean \(\mu\) and standard deviation \(\sigma\).
• are symmetric and centered at \(\mu\).
• have spreads that depend on \(\sigma\).

To check whether a variable is (approximately) normally distributed,

1. Check the histogram to see if it is symmetric and bell-shaped.
2. Estimate the parameters: \(\mu\) using \(\bar{x}\) and \(\sigma\) using \(s\).

• We standardize a variable using \[z = \frac{x-\mu}{\sigma}.\]
• This is also called a z-score.
• Standardizing using this formula will always result in a variable with mean 0 and standard deviation 1.
• These are useful for comparing values which are originally on very different scales.

A z-score tells us how many standard deviations an observation is from the mean.

• A positive z-score \(z>0\) is above the mean.
• A negative z-score \(z<0\) is below the mean.

Example: \(z=-0.23\) is 0.23 standard deviations below the mean.

• ACT scores have mean 20.8 and standard deviation 5.8.
• SAT scores have mean 1500 and standard deviation 300.
• If Jose scored a 27 on his ACT and Alice scored a 1850 on her SAT, who would you say got the better score?

For any (approximately) normally distributed variable,

1. Approximately 68% of all possible observations lie within one standard deviation of the mean: \(\mu \pm \sigma.\)
2. Approximately 95% of all possible observations lie within two standard deviations of the mean: \(\mu \pm 2\sigma.\)
3. Approximately 99.7% of all possible observations lie within three standard deviations of the mean: \(\mu \pm 3\sigma.\)

• A standard normal distribution is a normal distribution with mean \(\mu=0\) and standard deviation \(\sigma=1\).
• If \(X\) is approximately normal, then the standardized variable \(Z\) will have a standard normal distribution.

Note: when we z-score a variable, we preserve the area under the curve properties!

• If \(X\) is Normal\((\mu,\sigma)\), then \[P(X < c) = P\left(Z < \frac{c - \mu}{\sigma}\right) = P(Z < z).\]