AppliedStatsCourse

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.