Measures of Position in Descriptive Statistics

Measures of Position

Introduction

Measures of position are statistical tools used to describe the relative standing of data values within a data set. They help to identify how individual data points compare to the rest of the data, and are essential for understanding the distribution and spread of data. Common measures include quartiles, percentiles, and standard scores (z-scores).

Quartiles

Quartiles are a type of fractile that divide an ordered data set into four equal parts. Each quartile represents a specific portion of the data:

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

• Second quartile (Q2): About one half of the data falls on or below Q2 (also known as the median).

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

Definition: Fractiles are numbers that partition (divide) an ordered data set into equal parts.

Example: Finding Quartiles

Given the following data set representing amounts of fuel wasted (in gallons per year) by automobile commuters in 15 large urban areas:

30, 31, 31, 25, 31, 25, 24, 38, 26, 39, 26, 38, 31

After ordering the data and identifying quartiles:

• Q1 = 26

• Q2 = 31

• Q3 = 35

Interpretation: About one-quarter of the areas waste 26 gallons or less, half waste 31 gallons or less, and three-quarters waste 35 gallons or less.

Example: Using Technology to Find Quartiles

Tuition costs (in thousands of dollars) for 25 liberal arts colleges:

55, 59, 55, 56, 57, 55, 60, 59, 61, 58, 57, 61, 59, 48, 44, 30, 39, 58, 48, 46, 19, 55, 45, 48, 48

Using statistical software or calculators, the quartiles are:

• Q1 = 47

• Q2 = 55

• Q3 = 58.5

Interpretation: One-quarter of colleges charge $47,000 or less, half charge $55,000 or less, and three-quarters charge $58,500 or less.

Interquartile Range (IQR)

The interquartile range (IQR) is a measure of variation that describes the spread of the middle 50% of the data. It is calculated as the difference between the third and first quartiles:

• Formula:

$IQR = Q_3 - Q_1$

Identifying Outliers Using IQR

1. Find Q1 and Q3.

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

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

4. Subtract $1.5 imes IQR$ from Q1. Any data entry less than this value is an outlier.

5. Add $1.5 imes IQR$ to Q3. Any data entry greater than this value is an outlier.

Example: Finding the Interquartile Range

Given Q1 = 47 and Q3 = 58.5:

$IQR = 58.5 - 47 = 11.5$

$1.5 imes IQR = 1.5 imes 11.5 = 17.25$

Lower bound: $47 - 17.25 = 29.75$ Upper bound: $58.5 + 17.25 = 75.75$

Any data entry less than 29.75 or greater than 75.75 is an outlier. In this example, 19 is an outlier.

Box-and-Whisker Plot

A box-and-whisker plot is a graphical tool for exploratory data analysis that highlights important features of a data set. It is constructed using the five-number summary:

• Minimum entry

• First quartile (Q1)

• Median (Q2)

• Third quartile (Q3)

• 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 vertical line at Q2.

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

Interpretation: The box represents the middle 50% of the data. The whiskers show the spread of the lower and upper 25% of the data. The length of the whiskers can indicate skewness in the data.

Percentiles and Other Fractiles

Percentiles and other fractiles divide data into equal parts:

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

Interpreting Percentiles

Percentiles indicate the percentage of data entries below a specific value. For example, if a test score is at the 90th percentile, 90% of scores are below that value.

Finding the Percentile for a Specific Data Entry

To find the percentile corresponding to a data entry $x$:

$\text{Percentile of } x = \frac{\text{number of values less than } x}{\text{total number of values}} \times 100$

Round to the nearest whole number.

Example: Finding a Percentile

Given 25 tuition costs, and 15 entries are less than $57,000$:

$\text{Percentile of } 57 = \frac{15}{25} \times 100 = 60$

Thus, $57,000$ is at the 60th percentile, meaning it is greater than 60% of the tuition costs.

Standard Score (z-score)

The standard score or z-score indicates how many standard deviations a data value $x$ is from the mean $\mu$:

$z = \frac{x - \mu}{\sigma}$

Where $\sigma$ is the standard deviation.

• A positive z-score means the value is above the mean.

• A negative z-score means the value is below the mean.

• A z-score of 0 means the value equals the mean.

Example: Finding z-Scores

Mean speed $\mu = 56$ mph, standard deviation $\sigma = 4$ mph. For $x = 62$ mph:

$z = \frac{62 - 56}{4} = 1.5$

For $x = 47$ mph:

$z = \frac{47 - 56}{4} = -2.25$

For $x = 56$ mph:

$z = \frac{56 - 56}{4} = 0$

Interpretation: 62 mph is 1.5 standard deviations above the mean, 47 mph is 2.25 below, and 56 mph is at the mean.

Example: Comparing z-Scores from Different Data Sets

Population	Mean ($\mu$)	Standard Deviation ($\sigma$)
Men's heights	69.9 in	3.0 in
Women's heights	64.3 in	2.6 in

For a 6-foot (72 in) tall man:

$z = \frac{72 - 69.9}{3.0} = 0.7$

For a 6-foot (72 in) tall woman:

$z = \frac{72 - 64.3}{2.6} = 3.0$

Interpretation: The 6-foot-tall man is within 1 standard deviation of the mean (typical), while the 6-foot-tall woman is 3 standard deviations above the mean (unusually tall).

Additional info: The notes above are expanded and clarified for academic completeness, including definitions, formulas, and examples for each concept.