Chapter 5: The Standard Deviation as a Ruler and the Normal Model

5.1 Using the Standard Deviation to Standardize Values

The standard deviation is a fundamental measure of variation in statistics. It allows us to compare values from different distributions by standardizing them, making it possible to assess how unusual a value is relative to its group.

• Standard Deviation: Measures the average distance of data points from the mean.

• Standardizing Values: By expressing values in terms of standard deviations from the mean, we can compare across different groups or units.

• Example: Comparing the heights of the tallest living man and woman using z-scores to determine which is more unusual.

Standardizing with z-scores

Standardizing a value means converting it to a z-score, which tells us how many standard deviations a value is from the mean. This process is essential for comparing values from different distributions.

• z-score Formula:

• Interpretation: A z-score of 0 means the value is at the mean; positive z-scores are above the mean, negative are below.

• Example: Calculating z-scores for Sultan Kösen (99.0 inches) and Yao Defen (92.0 inches) relative to their respective population means and standard deviations.

5.2 Shifting and Scaling

When we convert an entire dataset to z-scores, we are standardizing the data. This process shifts the mean to 0 and rescales the standard deviation to 1, but does not change the shape of the distribution.

• Standardizing: Makes data unitless and comparable across different scales.

• Shifting: Adding or subtracting a constant to all values shifts the mean but does not affect the standard deviation.

• Scaling: Multiplying or dividing all values by a constant changes both the mean and standard deviation proportionally.

• Example: Converting daily high temperatures from Fahrenheit to Celsius and comparing z-scores.

5.3 Normal Models

The Normal Model is a statistical model that describes distributions that are unimodal and symmetric. It is used extensively in statistics to model real-world phenomena.

• Normal Distribution: Bell-shaped curve characterized by mean (μ) and standard deviation (σ).

• Notation: represents a normal model with mean μ and standard deviation σ.

• Standard Normal Model: , where the mean is 0 and the standard deviation is 1.

• When is a z-score big? z-scores greater than 3 (in absolute value) are considered unusual or outliers.

The 68-95-99.7 Rule

This rule describes the percentage of values that fall within 1, 2, and 3 standard deviations of the mean in a normal distribution:

• 68% within 1 standard deviation

• 95% within 2 standard deviations

• 99.7% within 3 standard deviations

Checking Normality

To use the Normal model, the data should be nearly normal (unimodal and symmetric). Histograms are useful for checking this condition.

5.4 Working with Normal Percentiles

Percentiles indicate the value below which a given percentage of observations fall. They are useful for interpreting z-scores and normal distributions.

• Median: 50th percentile

• Quartiles: Q1 (25th percentile), Q3 (75th percentile)

• Example: Calculating the percentile for a z-score of 1.80 in SAT scores.

Verifying the 68-95-99.7 Rule with Technology

Statistical software can be used to verify the rule and calculate exact percentages for any z-score interval.

Body Temperature Example

Using the Normal model to analyze body temperature data:

• Mean: 98.2°F, Standard deviation: 0.7°F

• Calculate z-scores and percentiles for specific temperatures.

• Find the interval for the vast majority (using the 68-95-99.7 rule).

Finding Quartiles

To find the value corresponding to a specific percentile (e.g., the first quartile), use the normal distribution calculator.

5.5 Normal Probability Plots

Normal Probability Plots (also called Normal Quantile Plots) are graphical tools used to assess whether a dataset is approximately normal. If the points on the plot form a straight line, the data is likely normal.

• Interpretation: Straightness indicates normality; curvature suggests deviation from normality.

• Application: Used before applying normal models to real data.

Chapter 5 Supplement: Goodness of Fit Test for Normality

The Goodness of Fit Test provides an objective criterion for assessing normality, especially when graphical methods are inconclusive. The Shapiro-Wilk test is commonly used, and the p-value determines whether the data is nearly normal.

• Threshold: 0.05 is a common cutoff; p-value above 0.05 means data is nearly normal.

• Example: Gasoline consumption data for states in 2004 and 2000.

Chapter Example: Tire Company

A tire manufacturer models tread life using a normal distribution with mean 32,000 miles and standard deviation 2,500 miles. Questions include:

• Probability of lasting 40,000 miles: Calculate z-score and percentile.

• Percent between 30,000 and 35,000 miles: Use z-scores and normal model.

• Warranty for 2% of tires: Find the mileage corresponding to the 98th percentile.

Key Formulas:

• z-score:

• Normal Model Notation:

• Percentile Calculation: Use normal distribution tables or technology.