Measures of Relative Standing: Z-Scores, Percentiles, Quartiles, and Boxplots

Measures of Relative Standing

Introduction

Measures of relative standing are statistical tools used to describe the position of a data value within a dataset. They help compare values from different datasets or assess how unusual a value is within its own dataset. Common measures include Z-scores, percentiles, quartiles, and boxplots.

Z-Scores

Definition and Calculation

• Z-score is a standardized value representing the number of standard deviations a data value is from the mean.

• Z-scores can be negative, zero, or positive:

  • If a value is smaller than the mean, the Z-score is negative.

  • If a value equals the mean, the Z-score is zero.

  • If a value is larger than the mean, the Z-score is positive.

• To calculate a Z-score from sample data, use: $z = \frac{x - \bar{x}}{s}$

• To calculate a Z-score from population data, use: $z = \frac{x - \mu}{\sigma}$

• Always round Z-scores to two decimal places.

Ordinary vs. Unusual Values

• Values with Z-scores between -2 and 2 are considered ordinary.

• Values with Z-scores outside this range are considered unusual.

Example

Given data: 20, 39, 46, 46, 47, 49, 49, 50, 52, 52, 54, 58

• Mean ($\bar{x}$) = 47.3

• Standard deviation ($s$) = 8.8

• Is the value 20 considered unusual? $z = \frac{20 - 47.3}{8.8} = -3.10$ (YES, unusual)

• Is the value 39 considered unusual? $z = \frac{39 - 47.3}{8.8} = -0.95$ (NO, ordinary)

Percentiles

Definition and Calculation

• Percentiles divide a dataset into 100 equal parts, each representing 1% of the data.

• Percentiles help determine what percentage of data falls below a certain value.

• Notation: $P_1, P_2, \ldots, P_{99}$

• To calculate the location of a percentile value: $L = \frac{k}{100} \times n$ where $k$ is the percentile, $n$ is the sample size, and $L$ is the location in the ordered data.

Example

• Find the 88th percentile ($P_{88}$) in a dataset of 15 values: $L = \frac{88}{100} \times 15 = 13.2$ The 13th value in the ordered dataset is $P_{88}$.

Quartiles

Definition and Calculation

• Quartiles divide a dataset into four equal parts, each containing 25% of the data.

• Q1: First quartile, separates the lowest 25% from the rest.

• Q2: Second quartile, same as the median, separates the bottom 50% from the top 50%.

• Q3: Third quartile, separates the lowest 75% from the top 25%.

Other Commonly Calculated Values

• Interquartile Range (IQR): $IQR = Q_3 - Q_1$

• Semi-quartile Range: $\frac{Q_3 - Q_1}{2}$

• Midquartile: $\frac{Q_1 + Q_3}{2}$

• 10-90 Percentile Range: $P_{90} - P_{10}$

Boxplots

Using Quartiles to Create Boxplots

Boxplots (box-and-whisker plots) visually display the distribution of a dataset using the five-number summary.

• Five-number summary includes: minimum, Q1, median (Q2), Q3, maximum.

• Steps to construct a boxplot:

  1. Find the five-number summary.

  2. Create a horizontal scale including the minimum and maximum.

  3. Create a box between Q1 and Q3 with a line at the median.

  4. Draw lines (whiskers) from the box to the minimum and maximum values.

• Boxplots can be symmetric, skewed right, or skewed left.

Modified Boxplots and Outliers

• Outliers are values that fall far outside the majority of the data.

• Outliers can affect the mean, standard deviation, and the scale of histograms.

• To identify outliers, use the formulas: Lower outlier threshold: $Q_1 - 1.5 \times IQR$ Upper outlier threshold: $Q_3 + 1.5 \times IQR$ Values outside these thresholds are considered outliers.

Example

Given data: 20, 39, 46, 46, 47, 49, 49, 50, 52, 52, 54, 58

• Five-number summary:

  • Minimum = 20

  • Q1 = 46

  • Median = 49

  • Q3 = 52

  • Maximum = 58

• IQR = 52 - 46 = 6

• Lower outlier threshold: 46 - 1.5(6) = 37

• Upper outlier threshold: 52 + 1.5(6) = 61

• Value 20 is an outlier (less than 37), should be marked with an asterisk in the boxplot.

Summary Table: Measures of Relative Standing

Measure	Definition	Formula	Application
Z-score	Standardized value showing distance from mean in standard deviations	$z = \frac{x - \bar{x}}{s}$ or $z = \frac{x - \mu}{\sigma}$	Identifying ordinary/unusual values
Percentile	Value below which a given percentage of data falls	$L = \frac{k}{100} \times n$	Comparing relative standing
Quartile	Divides data into four equal parts	Q1, Q2, Q3	Summarizing data distribution
Boxplot	Graphical summary using five-number summary	Minimum, Q1, Median, Q3, Maximum	Visualizing spread and outliers