# 6.1 and 6.2

### Jacob LaCour

## Properties or Normal Curve

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

## 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 plot of observed x values in time sequence order

1. Find the mean µ and standard deviation ∂ of the x distribution by

- Using past data from a period during which the process was "in control" or
- Using specified "target" values for µ and ∂

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 µ and horizontal, dash control-limit lines at µ +/- 2∂ and µ +/- 3∂

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

Signal 1- the point falls beyond the 3∂ level

Signal 2-a run of nine consecutive points on one side of the center line (the line at target value µ)

Signal 3-at least two of three consecutive points lie beyond the 2∂ 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 measurement is from the mean
- A standard score is 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

## Z Score-Calculation

- The Z value or Z score (also known as standard score) gives the number of standard deviation between the original measurement X and the mean µ of the X distribution.
- Z=X-µ/∂