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:

• To calculate a Z-score from population data, use:

• 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 () = 47.3

• Standard deviation () = 8.8

• Is the value 20 considered unusual? (YES, unusual)

• Is the value 39 considered unusual? (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:

• To calculate the location of a percentile value: where is the percentile, is the sample size, and is the location in the ordered data.

Example

• Find the 88th percentile () in a dataset of 15 values: The 13th value in the ordered dataset is .

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):

• Semi-quartile Range:

• Midquartile:

• 10-90 Percentile Range:

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: Upper outlier threshold: 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