Descriptive Statistics: Measures of Position

Descriptive Statistics

Measures of Position

Measures of position are statistical tools used to describe the location of a data value within a data set. They help to understand how individual values relate to the rest of the data, especially in terms of quartiles, percentiles, and z-scores.

Quartiles

Quartiles are values 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 fall on or below this value.

• Second Quartile (Q2): About one half of the data fall on or below this value; this is also known as the median.

• 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, Q1 = 26 gallons, Q2 = 31 gallons, Q3 = 35 gallons. This means 25% of cities waste ≤26 gallons, 50% waste ≤31 gallons, and 75% waste ≤35 gallons.

Interquartile Range (IQR)

The Interquartile Range (IQR) measures the spread of the middle 50% of the data. It is calculated as:

$IQR = Q_3 - Q_1$

• Identifying Outliers: Multiply IQR by 1.5. Any data entry less than $Q_1 - 1.5 \times IQR$ or greater than $Q_3 + 1.5 \times IQR$ is considered an outlier.

Example: If Q1 = 47, Q3 = 58.5, then IQR = 11.5. Outliers are values less than 29.75 or greater than 75.75. In the tuition data, 19 is an outlier.

Box-and-Whisker Plot

A box-and-whisker plot is a graphical tool for exploratory data analysis. It visually displays the five-number summary:

• Minimum

• First Quartile (Q1)

• Median (Q2)

• Third Quartile (Q3)

• Maximum

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 the median.

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

Example: In the tuition data, the box represents values between 47 and 58.5, with whiskers extending to the minimum and maximum. The left whisker is longer, indicating a possible outlier.

Percentiles and Other Fractiles

Percentiles are values that divide a data set into 100 equal parts. The nth percentile is the value below which n% of the data fall.

• Interpreting Percentiles: For example, the 90th percentile SAT score is 1350, meaning 90% of students scored ≤1350.

• Finding the Percentile for a Data Entry: Use the formula:

$Percentile = \frac{Number\ of\ values\ less\ than\ x}{Total\ number\ of\ values} \times 100$

Round to the nearest whole number.

Example: In the tuition data, $57,000 corresponds to the 60th percentile, meaning it is greater than 60% of the other tuition costs.

Standard Score (z-score)

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

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

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

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

• Values with z-scores greater than 2 or less than -2 are considered unusual.

Example: For vehicle speeds with mean 56 mph and standard deviation 4 mph: 62 mph has z = 1.5, 47 mph has z = -2.25, 56 mph has z = 0.

Comparing z-scores: For a 6-foot-tall man (mean = 69.9 in, σ = 2.8 in) and woman (mean = 64.3 in, σ = 2.3 in):

• Man: $z = \frac{72 - 69.9}{2.8} \approx 0.75$ (typical height)

• Woman: $z = \frac{72 - 64.3}{2.3} \approx 3.35$ (unusual height)