Standard Deviation as a Ruler and the Normal Model

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

Section 5.1: Using the Standard Deviation to Standardize Values

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 and the calculation and interpretation of z-scores. "

Comparing Athletes: An Example

• Scenario: Two athletes excel in different events: a long jump and a 200 m run. Their performances are compared to the average using standard deviation.

• Key Point: Standard deviation allows for comparison across different units and scales.

How Many Standard Deviations Above or 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 (200 m Run): $24.58 - 0.654 = 23.926$

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

The z-Score

The z-score standardizes values by expressing them in terms of standard deviations 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-score: value is close to the mean.

  • Large z-score: value is far from the mean.

Example: Calculating z-Scores

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

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

• Conclusion: The 200 m run performance is more impressive because it is further from the mean in standard deviation units.

Section 5.2: Shifting and Scaling

Shifting and scaling are two types of linear transformations that affect the center and spread of a data set in predictable ways.

Shifting Data

• Definition: Adding or subtracting the same constant to every data value.

• Effect:

  • Measures of position (mean, median, mode) are shifted by the constant.

  • Measures of spread (standard deviation, range, IQR) are unchanged.

• Example: If all weights are reduced by 74 kg, the mean decreases by 74 kg, but the standard deviation remains the same.

Scaling Data

• Definition: Multiplying or dividing all data values by the same constant.

• Effect:

  • Measures of position and spread are multiplied (or divided) by the constant.

• Example: Converting kilograms to pounds (multiply by 2.2): both the mean and standard deviation are multiplied by 2.2.

Rescaling and z-Scores

• When converting data to z-scores:

  • Subtract the mean: $y - \bar{y}$ (shifts the center to 0)

  • Divide by the standard deviation: $\frac{y - \bar{y}}{s}$ (scales the spread to 1)

  • The shape of the distribution does not change.

Example: Rescaling Combined Times

• Given: Mean = 168.93 seconds, SD = 2.90 seconds.

• Convert to minutes:

  • Mean: $\frac{168.93}{60} = 2.816$ minutes

  • SD: $\frac{2.90}{60} = 0.048$ minutes

Summary Table: Effects of Shifting and Scaling

Transformation	Effect on Center (mean, median, mode)	Effect on Spread (SD, range, IQR)
Add/Subtract constant	Shifted by constant	Unchanged
Multiply/Divide by constant	Multiplied/Divided by constant	Multiplied/Divided by constant

Additional info: These principles are foundational for understanding how data transformations affect statistical summaries and for interpreting standardized scores in various contexts.