Numerical Summaries of Center and Variation

The Empirical Rule

The Empirical Rule is a guideline used in statistics to describe the spread of data in a unimodal and symmetric distribution. It provides a quick way to estimate the proportion of data within certain distances from the mean, measured in standard deviations.

• Approximately 68% of the observations are within 1 standard deviation of the mean.

• Approximately 95% of the observations are within 2 standard deviations of the mean.

• Nearly all of the observations are within 3 standard deviations of the mean.

This rule is especially useful for quickly assessing what values are typical or unusual in a dataset, provided the distribution is roughly symmetric and unimodal.

Example: Smog Levels Suppose the mean particulate level in a sample of cities is 7.9 micrograms per cubic meter, with a standard deviation of 2.2 micrograms per cubic meter. Applying the Empirical Rule:

• About 68% of cities have smog levels between 5.7 and 10.1 (mean ± 1 SD).

• About 95% of cities have smog levels between 3.5 and 12.3 (mean ± 2 SD).

• Nearly all cities have smog levels between 1.3 and 14.5 (mean ± 3 SD).

Key Point: The Empirical Rule is an approximation and works best for large, unimodal, and symmetric datasets. Actual percentages may vary slightly in real data.

Determining If a Data Value Is Unusual

Values that fall outside the range covered by 95% of the data (i.e., more than 2 standard deviations from the mean) are considered unusual. For symmetric distributions, about 2.5% of values are unusually low, and 2.5% are unusually high.

Example: Temperatures in San Francisco If 95% of daily high temperatures are between certain values, then only 5% of days are outside this range. Thus, a temperature colder than the lower bound is fairly unusual.

z-Scores

A z-score is a standardized measure that indicates how many standard deviations a data value is from the mean. It allows for comparison across different datasets or variables, even if they have different units or scales.

• A z-score of 1.5 means the value is 1.5 standard deviations above the mean.

• A z-score of -1.5 means the value is 1.5 standard deviations below the mean.

The formula for the z-score is:

where: = observed value = sample mean = sample standard deviation

z-Scores in Context

z-scores are useful for:

• Comparing values from different groups (e.g., exam scores from different tests).

• Comparing values measured in different units (e.g., inches and pounds).

• Identifying unusual values (those with z-scores greater than 2 or less than -2).

Example: In a sample of men's heights, two men have z-scores greater than 2, and two men have z-scores less than -2, indicating their heights are unusually high or low compared to the group.

Comparing z-Scores: Example

Maria scored 80 on her first statistics exam (mean = 70, SD = 10) and 85 on her second exam (mean = 80, SD = 5). Her z-scores are:

• First exam:

• Second exam:

Maria performed equally well on both exams relative to her classmates, as her z-score was 1 in both cases.

Snapshot: The z-Score