6.5

Normal Probability Distributions

Assessing Normality

Assessing whether a dataset follows a normal distribution is a crucial step in many statistical analyses. The normality assumption underpins various inferential techniques, including hypothesis testing and confidence interval estimation. This section outlines practical methods for evaluating normality in sample data.

• Key Point 1: Construct a histogram and determine whether it is roughly bell-shaped.

• Key Point 2: Construct a normal quantile plot and use the criteria for normality.

Histograms

A histogram is a graphical representation of the distribution of a dataset. When assessing normality, look for a bell-shaped curve that is symmetric about the mean.

• Bell-shaped histogram: Suggests data may be normally distributed.

• Flat or uniform histogram: Indicates a uniform distribution, not normal.

• Skewed histogram: Indicates the data are not normally distributed.

Normal Quantile Plot (Normal Probability Plot)

A normal quantile plot (or normal probability plot) is a graph of points (x, y) where each x value is from the original set of sample data, and each y value is the corresponding z-score expected from the standard normal distribution.

• Straight-line pattern: Indicates the data are approximately normal.

• Systematic deviations from a straight line: Suggest the data are not normal.

Formula for z-score:

$z = \frac{x - \mu}{\sigma}$

where $x$ is a data value, $\mu$ is the mean, and $\sigma$ is the standard deviation.

Examples of Assessing Normality

Distribution Type	Histogram Shape	Normal Quantile Plot Pattern	Conclusion
Normal	Bell-shaped	Points close to a straight line	Data likely from a normal distribution
Uniform	Flat	Systematic pattern not a straight line	Data not from a normal distribution
Skewed	Skewed right	Points not close to a straight line	Data not from a normal distribution

Outliers

Outliers are data points that are significantly different from the rest of the dataset. They can have dramatic effects on statistical results and may indicate important characteristics or errors in the data.

• Investigate outliers: They may reveal critical information about the data.

• Discard outliers only if: They are identified as errors.

Example: If a dataset of IQ scores contains a value far outside the typical range, check whether it is a recording error or a genuine observation before deciding to exclude it.

Summary Table: Steps for Assessing Normality

Step	Description
1. Histogram	Check for bell-shaped, symmetric distribution
2. Normal Quantile Plot	Check if points follow a straight-line pattern
3. Outlier Analysis	Investigate and address outliers appropriately

Additional info: Normality assessment is foundational for parametric statistical methods. If data are not normal, consider nonparametric alternatives or data transformation techniques.