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

Standard Deviation as a Ruler and the Normal Model

Section 5.1: Using the Standard Deviation to Standardize Values

The standard deviation is a key measure of spread in statistics, allowing us to compare values from different distributions by standardizing them. This section introduces the concept of using standard deviation as a 'ruler' to measure how far a value is from the mean, and introduces the z-score as a standardized measure.

Comparing Athletes: An Example

• Context: Comparing performances in different Olympic events (e.g., long jump and 200m run) requires a common scale.

• Key Idea: Standard deviation allows us to compare how far each athlete's performance is from the average in their event.

Standard Deviations Above and Below the Mean

To determine how unusual a value is, we calculate how many standard deviations it is from the mean.

	Long Jump	200 m Run
Mean	6.17 m	24.58 s
Standard Deviation (SD)	0.247 m	0.654 s
Individual Performance	6.58 m	23.26 s

• 1 SD above mean (Long Jump): $6.17 + 0.247 = 6.417$

• 2 SD above mean (Long Jump): $6.17 + 2 \times 0.247 = 6.664$

• 1 SD below mean (200m Run): $24.58 - 0.654 = 23.926$

• 2 SD below mean (200m Run): $24.58 - 2 \times 0.654 = 23.272$

Definition: z-Score

• z-score: The number of standard deviations a value is from the mean.

• Formula:

$z = \frac{y - \bar{y}}{s}$

• Interpretation:

  • Positive z-score: Value is above the mean.

  • Negative z-score: Value is below the mean.

  • Small |z|: Value is close to the mean.

  • Large |z|: Value is far from the mean.

Example: Calculating z-Scores for Athletes

• Long Jump: $z = \frac{6.58 - 6.17}{0.247} \approx 1.66$

• 200 m Run: $z = \frac{23.26 - 24.58}{0.654} \approx -2.02$

• Interpretation: The 200 m run performance is more impressive (further from the mean in standard deviations) than the long jump.

Summary Table: Standard Deviation Comparisons

Event	Mean	SD	Individual	z-score
Long Jump	6.17 m	0.247 m	6.58 m	1.66
200 m Run	24.58 s	0.654 s	23.26 s	-2.02

Additional info: z-scores allow for direct comparison of performances across different units and scales by standardizing the values.