Unit 6 STAT

Chapter 6: Normal Curves and Sampling Distributions

The normal distribution is a continuous probability distribution foundational to statistics, also known as the Gaussian distribution.
It is described by its mean (µ) and standard deviation (σ), though direct use of its formula is not needed for practical applications.
The graph of a normal distribution is called a normal curve, resembling a bell (bell curve).

The normal curve is smooth and symmetric about its mean (µ).
The highest point of the curve occurs at µ, and if balanced on a knife edge, it would remain upright.
As the curve extends towards the tails, it approaches the horizontal axis but never touches it, reflecting that it theoretically runs infinitely in both directions.
The spread of the curve is determined by the standard deviation (σ):
- A larger σ results in a more spread out curve.
- A smaller σ produces a peakier curve.

The curve transitions between concave down and concave up at points called inflection points, which occur at μ - σ and μ + σ.
Important Properties of Normal Curve:
- Bell-shaped and symmetric about its mean (µ).
- Curve never touches the horizontal axis.
- Total area under the curve equals 1.
- Area represents probabilities for distribution.

For data values in a normal distribution, the empirical rule states:
- Approximately 68% of the data lies within ±1 standard deviation from the mean (μ ± σ).
- About 95% lies within ±2 standard deviations (μ ± 2σ).
- Nearly 99.7% lies within ±3 standard deviations (μ ± 3σ).

Control charts are used in quality control to monitor a process over time:
- They combine graphical and numerical data descriptions with probability distributions.
- A control chart signals if a system is operating within defined limits or indicates potential issues.
- They display the variable being measured over time, alongside the mean and control limits (mu ± 3 sigma).

The chapter sets the stage for applying the normal distribution in practical statistical applications, including analysis and decision-making based on sample means and proportions. Control charts and empirical rules provide the foundation for statistical quality control.