Descriptive Statistics: Percentiles, Z-Scores, and the Empirical Rule

Measures of Center

• Mean (average): $\bar{x} = \frac{1}{n} \sum<em>{i=1}^{n} x</em>i$
• Median: Middle value when data are ordered; average of two middle values if even count.
• Mode: Most frequently occurring value.

Measures of Variation (Spread)

• Range: $\text{Range} = x<em>{\max} - x</em>{\min}$
• Interquartile Range (IQR): $\text{IQR} = Q3 - Q1$ (Q1 = median of lower half, Q3 = median of upper half).
• Standard Deviation (spread about the center):
  • Sample: $s = \sqrt{\frac{1}{n-1} \sum<em>{i=1}^{n} (x</em>i - \bar{x})^2}$
  • Population: $\sigma = \sqrt{\frac{1}{N} \sum<em>{i=1}^{N} (x</em>i - \mu)^2}$

Percentile

• Definition: A value below which a given percentage of observations fall. E.g., 20th percentile means better than 20% of data.
• Computation: Count dots smaller than target ($x$), total dots ($n$). Percentile = $\frac{x}{n} \times 100$. Round down to nearest integer.

Z-Score (Standard Score)

• Definition: How many standard deviations an observation is from the mean.
• Formula: $z = \frac{x - \mu}{\sigma}$
• Interpretation: Positive z means above mean, negative z means below mean. Magnitude indicates distance in standard deviations.
• Why use: Enables comparison of values from different distributions (different scales/units).

Empirical Rule (68-95-99.7)

• Applies to approximately normal distributions.
• About 68% of data within $\mu \pm \sigma$.
• About 95% within $\mu \pm 2\sigma$.
• About 99.7% within $\mu \pm 3\sigma$.

Summary Takeaways

• Descriptive statistics include measures of center (mean, median, mode) and variation (range, IQR, standard deviation).
• Percentiles describe relative standing; z-scores describe distance from the mean in standard deviations.
• Use z-scores or percentiles for fair comparisons across different distributions.
• Excel (AVERAGE, MEDIAN, STDEV.S, QUARTILE.EXC, STANDARDIZE) automates these calculations.