Math 132A

Normal Distributions

Probability Distribution

• A mathematical model for a random variable.

• Describes how likely different random values are.

  • Discrete variables:

    • Probability distribution table or function
    • Tells us what is \(\operatorname{P}(X = x_i)\)

  • Continuous variables:

    • Given by a density curve, or density function
    • Probability is the area under the curve

Normal Distributions

• Special shape of the density curve (kind of like a bell)
• Perfectly symmetric, centered around a specific number (called the mean, denoted \(\mu\))
• The “width” of the “bell” depends on the spread of the distribution (the amount of uncertainty), and is usually determined by the standard deviation, denoted by \(\sigma\).
• The \(\mu\) and \(\sigma\) are the parameters of the model.
• Variable \(X\) having a normal distribution with given \(\mu\) and \(\sigma\) is denoted at \(X \sim N(\mu, \sigma)\).

Example (\(\mu = 4\), \(\sigma = 2\))

Example (\(\mu = 4\), \(\sigma = 3\))

Meaning of Standard Deviation

Empirical Rule for normal distribution:

• approximately 68% of the values are within 1 SD of the mean

• approximately 95% of the values are within 2 SDs of the mean

• approximately 99.7% of the values are within 3 SDs of the mean

68-95-99.7

A Normal Example (a picture)

The distribution of test scores on the SAT and the ACT are both nearly normal.

Suppose that one student scores an 1800 on the SAT (Student A) and another student scores a 24 on the ACT (Student B). Which student performed better?

A Normal Example

Standard Normal Distribution

The standard normal distribution is defined as a normal distribution with mean 0 and standard deviation 1. It is often denoted as \(Z \sim N(0, 1)\).

Any normal random variable \(X \sim N(\mu, \sigma)\) can be transformed into a standard normal random variable \(Z\).

\[Z = \dfrac{X - \mu}{\sigma}\]

\[X = \mu + Z\sigma\]

A Normal Example (the calculation)

• SAT scores are \(N(1500, 300)\). ACT scores are \(N(21,5)\).

• \(x_A\) represents the score of Student A; \(x_B\) represents the score of Student B.

\[z_{A} = \frac{x_{A} - \mu_{SAT}}{\sigma_{SAT}} = \frac{1800-1500}{300} = 1\]

\[z_{B} = \frac{x_{B} - \mu_{ACT}}{\sigma_{ACT}} = \frac{24 - 21}{5} = 0.6\]

A Normal Example

Calculating Normal Probabilities

What is the percentile rank for a student who scores an 1800 on the SAT for a year in which the scores are \(N(1500, 300)\)?

What is the probability that, if we randomly select a student that took the exam that year, their score would be less than 1800?

Using the Table

Standard normal distribution table

1. Calculate a \(Z\)-score.

   \[Z = \frac{X - \mu}{\sigma}\]

2. The table gives you the area to the left of that \(z\)-score.

Negative \(z\)-scores

Positive \(z\)-scores

Back to our student

What is the percentile rank for a student who scores an 1800 on the SAT for a year in which the scores are \(N(1500, 300)\)?

The other student

What is the percentile rank for a student who scores an 24 on the ACT for a year in which the scores are \(N(21, 5)\)?

Another example

Find the probability that \(X {\color{red}\ge} 17.5\) if \(X \sim N(13, 2.7)\)

• Calculate the \(z\)-score:

  \[z = \frac{17.5 - 13}{2.7} = 1.6666667\]

• Now we need to find the area to the right of 1.6666667 under the standard normal curve.

The original curve

Area to the right of 17.5.

The standard normal curve

Area to the right of 1.6666667.

Wrong area

• The tables give you area to the left of a given \(z\)-score!

Like this:

Subtraction

• The total area under the normal curve is always 1.

• All we need to do is subtract:

  area to the right = 1 - area to the left

• Finally, we can find the area to the right of 1.67.