Section 2.5: Measures of Position

Introduction

Measures of position are statistical tools that describe the relative standing of a data value within a data set. They help to identify how a particular value compares to the rest of the data, using concepts such as quartiles, percentiles, and standard scores (z-scores).

Quartiles

• Definition: Quartiles are values that divide an ordered data set into four equal parts, each containing approximately 25% of the data.

• First Quartile (Q1): About one quarter of the data fall on or below this value.

• Second Quartile (Q2): The median; about one half of the data fall on or below this value.

• Third Quartile (Q3): About three quarters of the data fall on or below this value.

Example: In a data set of fuel wasted by commuters in 15 large U.S. urban areas, Q1 = 26 gallons, Q2 = 31 gallons, Q3 = 35 gallons. This means 25% of cities waste 26 gallons or less, 50% waste 31 gallons or less, and 75% waste 35 gallons or less.

Interquartile Range (IQR)

• Definition: The interquartile range (IQR) measures the spread of the middle 50% of the data. It is calculated as the difference between the third and first quartiles.

• Identifying Outliers: Outliers are data values that fall significantly outside the range of the rest of the data. To identify outliers using the IQR:

  • Calculate 1.5 × IQR.

  • Any data value less than or greater than is considered an outlier.

Example: For tuition data with Q1 = 47, Q3 = 58.5, IQR = 11.5. Outlier boundaries: less than 29.75 or greater than 75.75. The value 19 is an outlier.

Box-and-Whisker Plot

• Definition: A box-and-whisker plot is a graphical summary of data based on the five-number summary: minimum, Q1, median, Q3, and maximum.

• Construction Steps:

  1. Find the five-number summary.

  2. Draw a horizontal scale covering the data range.

  3. Plot the five summary values above the scale.

  4. Draw a box from Q1 to Q3 with a line at the median.

  5. Draw whiskers from the box to the minimum and maximum values.

Interpretation: The box represents the middle 50% of the data. Whiskers show the spread of the lower and upper quarters. Asymmetry in whisker length may indicate skewness or outliers.

Percentiles and Other Fractiles

• Definition: Percentiles divide an ordered data set into 100 equal parts. The nth percentile is the value below which n% of the data fall.

• Finding a Percentile for a Data Value:

• Round to the nearest whole number.

Example: If 15 out of 25 tuition values are less than $57,000, then $57,000 is at the 60th percentile.

• Interpreting Percentiles: For example, a 90th percentile SAT score of 1350 means 90% of students scored 1350 or less.

Standard Score (z-score)

• Definition: The standard score or z-score indicates how many standard deviations a value x is from the mean μ.

• Interpretation: A positive z-score means the value is above the mean; a negative z-score means it is below the mean. Values with z-scores greater than 2 or less than -2 are often considered unusual.

Example: For a mean speed of 56 mph and standard deviation of 4 mph, a speed of 62 mph has a z-score of 1.5; 47 mph has a z-score of -2.25; 56 mph has a z-score of 0.

• Comparing z-scores from Different Data Sets: Standard scores allow comparison of values from different distributions. For example, a 6-foot-tall man and a 6-foot-tall woman have different z-scores relative to their respective population means and standard deviations.

Example: If the mean height for men is 69.9 inches (σ = 2.9) and for women is 64.3 inches (σ = 2.8), a 72-inch man has a z-score of 0.72, while a 72-inch woman has a z-score of 2.75. The woman's height is much more unusual in her population.