Descriptive Statistics: Measures of Position (Quartiles, Percentiles, and z-Scores)

Measures of Position

Introduction

Measures of position are statistical tools used to describe the relative standing of a data value within a data set. They help to partition data into equal parts, identify outliers, and compare values across different data sets. Common measures include quartiles, percentiles, and standard scores (z-scores).

Quartiles and Fractiles

Quartiles

• Fractiles are numbers that partition an ordered data set into equal parts.

• Quartiles divide an ordered data set into four equal parts:

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

  • Second quartile (Q2): About one half of the data fall on or below Q2 (the median).

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

Example: Finding Quartiles

Given the data set (gallons of fuel wasted): 11, 20, 22, 23, 24, 25, 25, 25, 28, 29, 29, 33, 35, 35

• Q1 = 23

• Q2 = 25

• Q3 = 30

Interpretation: About one-quarter of the areas waste 23 gallons or less, one-half waste 25 gallons or less, and three-quarters waste 30 gallons or less.

Using Technology to Find Quartiles

• Statistical software and calculators (e.g., Minitab, TI-84 Plus) can compute quartiles efficiently.

• Example (tuition costs in thousands): Q1 = 28.5, Q2 = 36, Q3 = 48

Interquartile Range (IQR)

Definition and Calculation

• The interquartile range (IQR) measures the spread of the middle 50% of the data.

• Formula: $IQR = Q_3 - Q_1$

Identifying Outliers Using IQR

1. Find Q1 and Q3.

2. Compute IQR: $IQR = Q_3 - Q_1$

3. Multiply IQR by 1.5: $1.5 imes IQR$

4. Any data entry less than $Q_1 - 1.5 imes IQR$ or greater than $Q_3 + 1.5 imes IQR$ is an outlier.

Example: Finding IQR and Outliers

• Given Q1 = 23, Q3 = 30, IQR = 7

• $1.5 imes IQR = 10.5$

• Lower bound: $23 - 10.5 = 12.5$

• Upper bound: $30 + 10.5 = 40.5$

• Any value below 12.5 or above 40.5 is an outlier. In this data set, 11 is an outlier.

Box-and-Whisker Plot

Definition and Construction

• A box-and-whisker plot is a graphical tool that displays the distribution of a data set using a five-number summary:

  1. Minimum entry

  2. First quartile (Q1)

  3. Median (Q2)

  4. Third quartile (Q3)

  5. Maximum entry

Steps to Draw a Box-and-Whisker Plot

1. Find the five-number summary.

2. Construct a horizontal scale spanning the data range.

3. Plot the five numbers above the scale.

4. Draw a box from Q1 to Q3 with a line at Q2.

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

Example: Box-and-Whisker Plot

• Min = 11, Q1 = 23, Q2 = 25, Q3 = 30, Max = 35

• The box represents the middle 50% of the data (23 to 30).

• Longer whiskers may indicate skewness or outliers.

Percentiles and Other Fractiles

Definitions

Fractile	Summary	Symbols
Quartiles	Divides data into 4 equal parts	Q1, Q2, Q3
Deciles	Divides data into 10 equal parts	D1, D2, ..., D9
Percentiles	Divides data into 100 equal parts	P1, P2, ..., P99

Interpreting Percentiles

• The p-th percentile is the value below which p% of the data fall.

• Example: If the 80th percentile of SAT scores is 1250, then 80% of students scored 1250 or less.

Finding the Percentile for a Data Entry

• Formula: $Percentile ext{ of }x = rac{ ext{number of values less than }x}{ ext{total number of values}} imes 100$

• Round to the nearest whole number.

Example: Finding Percentiles

• For a tuition cost of $34,000 (data entry 34) in a set of 25 values, with 8 values less than 34:

• $Percentile = rac{8}{25} imes 100 = 32$

• So, $34,000 is at the 32nd percentile.

Standard Score (z-Score)

Definition

• The standard score (z-score) indicates how many standard deviations a value x is from the mean μ.

• Formula: $z = rac{x - ext{mean}}{ ext{standard deviation}} = rac{x - oldsymbol{ ext{ extmu}}}{oldsymbol{ ext{ extsigma}}}$

Interpreting z-Scores

• A z-score of 0 means the value is equal to the mean.

• Positive z-scores are above the mean; negative z-scores are below the mean.

• Values with |z| > 2 are often considered unusual or outliers.

Example: Calculating z-Scores

• Mean speed = 56 mph, standard deviation = 4 mph

• For x = 62 mph: $z = rac{62 - 56}{4} = 1.5$

• For x = 47 mph: $z = rac{47 - 56}{4} = -2.25$

• For x = 56 mph: $z = rac{56 - 56}{4} = 0$

• Interpretation: 62 mph is 1.5 standard deviations above the mean; 47 mph is 2.25 below (unusually slow); 56 mph is at the mean.

Comparing z-Scores from Different Data Sets

• z-scores allow comparison of values from different distributions.

• Example: Heights of men (μ = 69.9 in, σ = 3.0 in) and women (μ = 64.3 in, σ = 2.6 in):

  • 6-foot-tall man (72 in): $z = rac{72 - 69.9}{3.0} = 0.7$

  • 6-foot-tall woman (72 in): $z = rac{72 - 64.3}{2.6} = 3.0$

  • Interpretation: 6 feet is typical for men but very unusual for women.

Additional info: The notes provide a comprehensive overview of measures of position, including practical examples, formulas, and interpretation guidelines, suitable for exam preparation in a college-level statistics course.