The Normal Distribution

Reading

• Section 3.1
• Section 3.2

Practice Problems

3.6.1 (p. 158)

3.1, 3.2, 3.3, 3.5, 3.7, 3.9, 3.13, 3.16

3.6.2 (p. 161)

3.17, 3.18

Notes

Normal Distribution Theory

• The Normal Distribution is a bell-shaped curve.
• Complicated equation, but a lot of data follow the distribution, so we have to work with it.
• Its parameters are easy to relate to the data (mean and standard deviation).
• Its equation depends on two parameters:
  • \(\mu\) (the mean) controls the center
  • \(\sigma\) (the standard deviation) controls the width. More specifically, it is the distance between the center and the “inflection point”.
• Denoted \(N(\mu, \sigma)\).
• We use Table A or calculator/computer for computing values. We will explain shortly.
• Key step: \(z\)-scores. \[z = \frac{x-\mu}{\sigma}\]
• They are a simple rescaling of the \(x\) values.

• Can also write: \[x = \mu + \sigma \times z\]

  \(z\) scores measure “number of standard deviations away from the mean” that the corresponding \(x\) value is.

• \(z\)-scores follow Standard Normal Distribution. With mean \(0\) and standard deviation \(1\).

• It is those \(z\) values we can look up in the table.

In general we deal with two kinds of problems:

Standard

We are given \(x\), and need to find corresponding \(p\).

1. Turn the \(x\) into \(z\): \(z=\frac{x-\mu}{\sigma}\)
2. Look \(z\) up in table to find a \(p\).
3. Possibly adjust the \(p\) based on the problem.

Reverse

We are given some sort of \(p\), need to find corresponding \(x\).

1. Possibly adjust the \(p\) based on the problem.
2. “Reverse Look” in table for the entry with that \(p\), then get corresponding \(z\).
3. Convert the \(z\) into an \(x\): \(x=\mu + \sigma z\)

Standard Direction

• Compute \(z\) from \(x\) if need be.
• Look \(z\) up in the table. For example say \(z=2.31\):
  • Find \(2.3\) on the left column.
  • Find \(0.01\) at the top row.
  • Their intersection is the “p-value”.
• p-value represents “the percent of data points below the given \(z\) (or \(x\))”.
• Use that to compute the answer to the actual question.
• For values with more decimals, round to closest end, or average results in two ends.

Practice questions:

• What percent of the data is below \(z=1.23\)?
• What percent of the data is below \(z=-1.5\)?
• What percent of the data is below \(z=-1.555\)?
• What percent of the data is above \(z=2.1\)?
• What percent of the data is between \(z=1.56\) and \(z=2.1\)?
• If we had a normal distribution with mean \(2.5\) and standard deviation \(1\), how much data is there between \(x=1.2\) and \(x=2.7\)?

Inverse Lookup in Table A

• Do this if you know a \(p\) and want to find a \(z\).
• Make sure the \(p\) represents “data below a point”. If not convert it.
• Look for the \(p\) INSIDE Table A. You will probably find one value bigger than it, right next to a value smaller than it.
• If your \(p\) is closer to one of these values, just use the \(z\) from that value.
• If it’s closer to the middle between them, use the average of the two \(z\) values.

Practice questions:

• At what \(z\) is the first quartile?
• At what \(z\) is the third quartile?
• What is the IQR for the standard normal distribution?
• Find the \(z\) range where the middle 20% of the data lies.
• If we had a normal distribution with mean \(2.5\) and standard deviation \(1\), find the range where the middle 20% of the data lies.