Measures of Position: Z-Scores, Percentiles, Quartiles, and Boxplots

Measures of Position

Z-Scores

The z-score (also called the standard score or standardized value) is a statistical measure that describes a value's position relative to the mean of a data set, expressed in terms of standard deviations. Z-scores are useful for comparing values from different data sets or distributions.

• Definition: The z-score for a value x is the number of standard deviations x is above or below the mean.

• Formula: $z = \frac{x - \bar{x}}{s}$ where $x$ is the value, $\bar{x}$ is the sample mean, and $s$ is the sample standard deviation.

• Properties:

  • Z-scores are unitless (no units of measurement).

  • A z-score less than or equal to -2 indicates a significantly low value.

  • A z-score greater than or equal to +2 indicates a significantly high value.

  • If a value is less than the mean, its z-score is negative.

• Example: Comparing extremeness of two values:

  • 99°F temperature among adults ($\bar{x} = 98.20$°F, $s = 0.62$°F)

  • 5.7790 g weight of a quarter ($\bar{x} = 5.63930$ g, $s = 0.06194$ g)

  • Calculate z-scores for each to determine which is more extreme relative to its data set.

Percentiles

Percentiles are measures of location that divide a data set into 100 equal groups, with about 1% of the values in each group. They are denoted as $P_1, P_2, ..., P_{99}$.

• Notation:

  • n: Total number of values in the data set

  • k: Percentile being used (e.g., for the 25th percentile, $k=25$)

  • L: Locator that gives the position of a value (e.g., for the 12th value in the sorted list, $L=12$)

  • $P_k$: The kth percentile (e.g., $P_{25}$ is the 25th percentile)

• Application Example: Determining what percentile a wait time of 45 minutes represents in a data set of Space Mountain wait times.

Finding a Percentile in a Data Set

To find the value corresponding to a given percentile in a data set, follow these steps:

1. Sort the data in ascending order.

2. Compute the locator $L$ using the formula: $L = \frac{k}{100} \times n$ where $k$ is the desired percentile and $n$ is the number of data values.

3. If $L$ is a whole number, the percentile is the average of the $L$th and $(L+1)$th values.

4. If $L$ is not a whole number, round up to the next integer; the percentile is the value at that position.

Quartiles

Quartiles are special percentiles that divide a data set into four equal groups, each containing about 25% of the values. The three quartiles are:

• Q1 (First Quartile): Same as $P_{25}$. Separates the lowest 25% from the highest 75% of values.

• Q2 (Second Quartile): Same as $P_{50}$ and the median. Separates the lowest 50% from the highest 50% of values.

• Q3 (Third Quartile): Same as $P_{75}$. Separates the lowest 75% from the highest 25% of values.

5-Number Summary

The 5-number summary provides a concise description of a data set using five key values:

• Minimum

• First quartile ($Q_1$)

• Second quartile ($Q_2$ or median)

• Third quartile ($Q_3$)

• Maximum

This summary is especially useful for understanding the spread and center of the data, and for constructing boxplots.

Boxplot (Box-and-Whisker Diagram)

A boxplot is a graphical representation of the 5-number summary. It displays the distribution of a data set by showing the minimum, first quartile, median, third quartile, and maximum. The box represents the interquartile range (IQR), and the 'whiskers' extend to the minimum and maximum values.

• Purpose: To visualize the spread, center, and skewness of the data, and to identify potential outliers.

• Example: Boxplot of Space Mountain wait times at 10AM.

Interpretation: The boxplot shows the minimum (10), $Q_1$ (25), median ($Q_2$, 35), $Q_3$ (50), and maximum (110) wait times, allowing for quick assessment of data distribution and variability.