Descriptive Statistics: Percentiles, Quartiles, Outliers, and Z-Scores

Percentiles and Quartiles

• Percentile Definition: A percentile is a numerical value representing the threshold at or below which a specified percentage ($P\%$) of observations in a dataset fall.
• Interpretation of Percentiles:
  • Morgan's Chemistry Exam Example:
  • Score ($x$): $80$ on a chemistry exam.
  • Percentile Rank: $62^{\text{nd}}$ percentile.
  • Interpretation: A score of $80$ is higher than or equal to approximately $62\%$ of the class (approaching two-thirds of the class).
  • Complementary Percentage: $38\%$ of the class achieved an exam score greater than $80$, while $62\%$ achieved a score at or below $80$.
  • Pediatric Growth Checkup Example:
  • A 3-year-old male child is evaluated at a checkup and placed in the $75^{\text{th}}$ percentile for weight.
  • Interpretation: The child is heavier than $75\%$ of all boys his age, while $25\%$ of boys his age weigh more than him.
  • Standardized Testing (ACT / SAT) Example:
  • A score in the $90^{\text{th}}$ or $95^{\text{th}}$ percentile in mathematics indicates performance superior to $90\%$ or $95\%$ of all test-takers, with only $10\%$ or $5\%$ scoring higher.
  • Fiftieth Percentile ($50^{\text{th}}$ Percentile / Median):
  • A student scoring at the $50^{\text{th}}$ percentile performed better than $50\%$ of the class, while $50\%$ performed better than the student.
  • The $50^{\text{th}}$ percentile represents the exact center of the dataset, dividing the lower half of scores from the upper half.

Quartiles and the Five-Number Summary

• Quartiles: Values that partition an ordered dataset into four equal segments, each containing $25\%$ of the total observations.
  • First Quartile ($Q_1$): Equivalent to the $25^{\text{th}}$ percentile. Exactly $25\%$ of the data values are less than or equal to $Q_1$, and $75\%$ are greater.
  • Second Quartile ($Q_2$): Equivalent to the $50^{\text{th}}$ percentile or Median. Exactly $50\%$ of the data values are less than or equal to $Q_2$, and $50\%$ are greater.
  • Third Quartile ($Q_3$): Equivalent to the $75^{\text{th}}$ percentile. Exactly $75\%$ of the data values are less than or equal to $Q_3$, and $25\%$ are greater.
• Five-Number Summary: A collection of five descriptive statistics summarizing the center, spread, and range of a dataset:
  1. Minimum value ($\text{Minimum}$)
  2. First Quartile ($Q_1$)
  3. Second Quartile / Median ($Q_2$)
  4. Third Quartile ($Q_3$)
  5. Maximum value ($\text{Maximum}$)

Calculating Quartiles and Summary Statistics by Hand

• Step-by-Step Method for Ordered Data:
  • Sample Dataset: Weekly screen time (in hours) collected from $n = 15$ college students.
  • Step 1: Determine the Median ($Q_2$):
  • For an odd sample size $n$, the position of the median is calculated using the formula:       $\text{Position of } Q_2 = \frac{n + 1}{2}$
  • For $n = 15$:       $\text{Position of } Q_2 = \frac{15 + 1}{2} = 8$
  • The $8^{\text{th}}$ ordered value is $30\,\text{hours}$. Thus, $Q_2 = 30\,\text{hours}$.
  • The median splits the dataset into a lower half ($7$ observations) and an upper half ($7$ observations).
  • Step 2: Determine the First Quartile ($Q_1$):
  • $Q_1$ is the median of the lower half of the data ($n_{\text{lower}} = 7$).
  • Position formula:       $\text{Position of } Q_1 = \frac{7 + 1}{2} = 4$
  • The $4^{\text{th}}$ observation in the lower half is $20\,\text{hours}$. Thus, $Q_1 = 20\,\text{hours}$.
  • Step 3: Determine the Third Quartile ($Q_3$):
  • $Q_3$ is the median of the upper half of the data ($n_{\text{upper}} = 7$).
  • Position formula:       $\text{Position of } Q_3 = \frac{7 + 1}{2} = 4$
  • The $4^{\text{th}}$ observation in the upper half is $39\,\text{hours}$. Thus, $Q_3 = 39\,\text{hours}$.
• Interpretation of Results:
  • $50\%$ of students logged less than $30\,\text{hours}$ of screen time, and $50\%$ logged more than $30\,\text{hours}$.
  • $25\%$ of students logged less than $20\,\text{hours}$ of screen time.
  • $75\%$ of students logged less than or equal to $39\,\text{hours}$ of screen time ($25\%$ logged more than $39\,\text{hours}$).

Interquartile Range, Fences, and Outliers

• Interquartile Range (IQR):
  • Measures the variability or spread of the middle $50\%$ of the data.
  • Formula:     $\text{IQR} = Q_3 - Q_1$
  • Calculation for Screen Time Data:     $\text{IQR} = 39 - 20 = 19\,\text{hours}$
• Outliers:
  • Definition: Extreme observations that deviate significantly from the overall pattern of the dataset.
  • Causes of Outliers: Machine malfunctions (e.g., automated candy bar packaging equipment dispensing an abnormal amount of peanut filling), or exceptional human characteristics (e.g., elite Kenyan distance runners training at high elevations outperforming general populations).
• Fence Calculations for Outlier Detection:
  • Standard Lower Fence Formula:     $\text{Lower Fence} = Q_1 - 1.5 \times \text{IQR}$$\text{Lower Fence} = 20 - 1.5 \times 19 = 20 - 28.5 = -8.5\,\text{hours}$
  • Standard Upper Fence Formula:     $\text{Upper Fence} = Q_3 + 1.5 \times \text{IQR}$$\text{Upper Fence} = 39 + 1.5 \times 19 = 39 + 28.5 = 67.5\,\text{hours}$
• Identifying Outliers in Data:
  • Lower bound test: Minimum value $12\,\text{hours} > -8.5\,\text{hours}$. No lower-end outliers exist.
  • Upper bound test: The dataset contains an observation of $83\,\text{hours}$. Because $83\,\text{hours} > 67.5\,\text{hours}$, $83\,\text{hours}$ is identified as an outlier.
• Box Plot (Box-and-Whisker Plot) Representation:
  • The whiskers of a box plot terminate at the most extreme non-outlier values in the dataset.
  • Outliers (such as $83\,\text{hours}$) are displayed as distinct points outside the fences.
  • In statistical software, identified outliers are excluded from the whisker endpoints of the box-and-whisker structure so they do not distort the visualization of typical data spread.

Z-Scores and Standardized Comparisons

• Definition and Function of Z-Scores:
  • A $Z$-score (or standard score) quantifies how many standard deviations a raw score ($x$) lies above or below the mean of a distribution.
  • Serves as a method to compare individual values from entirely different distributions or units ("comparing apples to oranges"), such as comparing salaries across distinct occupational classes (e.g., janitors vs. corporate CEOs/managers).
• Mathematical Formulas:
  • Population Z-Score Formula:     $Z = \frac{x - \mu}{\sigma}$
  • Sample Z-Score Formula:     $Z = \frac{x - \bar{x}}{s}$
  • Where $x$ represents the raw observation value, $\mu$ or $\bar{x}$ represents the mean, and $\sigma$ or $s$ represents the standard deviation.
• Direction and Interpretation:
  • A positive $Z$-score ($Z > 0$) indicates an observation above the mean.
  • A negative $Z$-score ($Z < 0$) indicates an observation below the mean.
  • A $Z$-score of zero ($Z = 0$) indicates an observation exactly equal to the mean.
• Contextual Evaluation:
  • For metrics like exam scores or annual salaries, a higher positive $Z$-score is typically desirable.
  • For metrics like race completion times, a lower negative $Z$-score (indicating a faster time below the average) is desirable.
  • In general analytical settings, $Z$-scores determine relative extremity (how far an observation lies from the center).

Comparative Z-Score Analysis: Exam Performance Example

• Scenario Overview:
  • Evaluating relative performance between two students taking different academic exams.
  • Anna's Statistics Exam:
  • Student Score ($x$): $88$
  • Class Mean ($\mu$ or $\bar{x}$): $80$
  • Class Standard Deviation ($\sigma$ or $s$): $4$
  • Z-Score Calculation:       $Z_{\text{Anna}} = \frac{88 - 80}{4} = \frac{8}{4} = 2.00$
  • Blake's English Exam:
  • Student Score ($x$): $90$
  • Class Mean ($\mu$ or $\bar{x}$): $80$
  • Class Standard Deviation ($\sigma$ or $s$): $7.5$
  • Z-Score Calculation:       $Z_{\text{Blake}} = \frac{90 - 80}{7.5} = \frac{10}{7.5} = 1.33$
• Comparison & Conclusion:
  • Both Anna and Blake performed above their respective class averages ($Z_{\text{Anna}} > 0$ and $Z_{\text{Blake}} > 0$).
  • Anna performed relatively better compared to her peer group than Blake did compared to his peer group, because Anna scored $2.00$ standard deviations above the statistics mean, whereas Blake scored $1.33$ standard deviations above the English mean.