# Chapter 6.1 & 6.2

## Normal Curves and Sampling Distribution

Important Properties of a Normal Curve:

1. The curve is bell-shaped, with the highest point over the mean u.
2. The curve is symmetrical about a vertical line through u.
3. The curve approaches the horizontal axis but never touches or crosses it.
4. The inflection (transition) points between cupping upward and downward occur above u + o and u - o.
5. The area under the entire curve is 1.

## M & O

The parameters that control the shape of a normal curve are the mean u and the standard deviation o. When both u and o are specified, a specific curve is determined. In brief, u locates the balance point and o determines the extent of the spread

## Empirical Rule

For a distribution that is symmetrical and bell-shaped (in particular, for a normal distribution):

• Approximately 68% of the data values will lie within 1 standard deviation on each side of the mean.
• Approximately 95% of the data values will lie within 2 standard deviations on each side of the mean.
• Approximately 99.7% (or almost all) of the data values will lie within 3 standard deviations on each side of the mean

## Control Chart

How to make a control chart for the random variable x

A control chart for a random variable x is a plot of observed x values in time sequence order

1. Find the mean u and standard deviation o of the x distribution by using past data from a period during which the process was "in control" or using specified "target" values for u and o
2. Create a graph in which the vertical axis represents x values and the horizontal axis represents time
3. Draw a horizontal line at height u and horizontal, dashed control-limit lines at u +- 2o and u +- 3o
4. Plot the variable x on the graph in time sequence order. Use line segments to connect the points in time sequence order

## Out-of-Control Signals

1. Out-of-Control Signal I: One point falls beyond the 3o level
2. Out-of-Control Signal II: A run of nine consecutive points on one side of the center line (the line at target value u)
3. Out-of-Control Signal III: At least two of three consecutive points lie beyond the 2o level on the same side of the center line

## Standard Scores

A standard score or z score of a measurement tells us the number of standard deviations the measurements is from the mean.

• A standard score close to zero tells us the measurement is near the mean of the distribution
• A positive standard score tells us the measurement is above the mean.
• A negative standard score tells us the measurement is below the mean.

## Appendix II

Convention for Using Table 5 of Appendix II

1. Treat any area to the left of a z value smaller than -3.49 as 0.000.
2. Treat any area to the left of a z value greater than 3.49 as 1.000.