Measures of Relative Standing and Boxplots: Chapter 3 Study Notes

Describing, Exploring, and Comparing Data

Measures of Relative Standing and Boxplots

This section covers statistical methods for describing the position of data values within a data set. Key concepts include z scores, percentiles, quartiles, the 5-number summary, and boxplots. These tools help compare data values, identify outliers, and visualize data distribution.

z Scores

Definition and Calculation

A z score (also called standard score or standardized value) indicates how many standard deviations a data value x is above or below the mean. It allows for comparison across different data sets.

• Sample z score: $z = \frac{X - \overline{X}}{s}$

• Population z score: $z = \frac{X - \mu}{\sigma}$

Round-off Rule: Round z scores to two decimal places (e.g., 2.31).

Properties of z Scores

• A z score is the number of standard deviations a value is above or below the mean.

• z scores have no units of measurement.

• A value is significantly low if $z \leq -2$; significantly high if $z \geq +2$.

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

Example: Comparing Data Values

Given two data values from different sets:

• 99°F body temperature (mean = 98.20°F, s = 0.62°F): $z = \frac{99 - 98.20}{0.62} = 1.29$

• 5.7790 g quarter (mean = 5.63930 g, s = 0.06194 g): $z = \frac{5.7790 - 5.63930}{0.06194} = 2.26$

The quarter's weight is more extreme (farther above the mean) than the body temperature.

Using z Scores to Identify Significant Values

• Significantly low: $z \leq -2$

• Significantly high: $z \geq 2$

• Not significant: $-2 < z < 2$

Example: Earthquake Magnitude

Given mean = 2.572, s = 0.651, magnitude = 4.01:

• $z = \frac{4.01 - 2.572}{0.651} = 2.21$

Since $z = 2.21 \geq 2$, the magnitude is significantly high.

Percentiles

Definition

Percentiles are measures of location, denoted $P_1, P_2, \ldots, P_{99}$, dividing data into 100 groups with about 1% of values in each group.

Finding the Percentile of a Data Value

To find the percentile for a value x:

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

Example: Percentile Calculation

For a wait time of 45 minutes among 50 sorted values, with 36 values less than 45:

• $\text{Percentile of 45} = \frac{36}{50} \times 100 = 72$

Interpretation: 45 minutes is the 72nd percentile ($P_{72}$).

Notation

• n: total number of values

• k: percentile (e.g., $k = 25$ for 25th percentile)

• L: locator for position in sorted list ($L = 12$ means 12th value)

• $P_k$: kth percentile

Converting a Percentile to a Data Value

• Compute $L = \frac{k}{100} \times n$

• If $L$ is not a whole number, round up to next whole number

• The $L$th value in the sorted list is $P_k$

Example: 25th Percentile

• $k = 25$, $n = 50$

• $L = \frac{25}{100} \times 50 = 12.5$ (round up to 13)

• 13th value is 25 minutes ($P_{25} = 25$)

Quartiles

Definition

Quartiles are measures of location, denoted $Q_1$, $Q_2$, $Q_3$, dividing data into four groups with about 25% of values in each group.

Descriptions of Quartiles

• $Q_1$ (First quartile): Same as $P_{25}$; separates bottom 25% from top 75%.

• $Q_2$ (Second quartile): Same as $P_{50}$ and the median; separates bottom 50% from top 50%.

• $Q_3$ (Third quartile): Same as $P_{75}$; separates bottom 75% from top 25%.

Caution: Procedures for finding percentiles and quartiles may vary between technologies.

Statistics Defined Using Quartiles and Percentiles

• Interquartile range (IQR): $Q_3 - Q_1$

• Semi-interquartile range: $\frac{Q_3 - Q_1}{2}$

• Midquartile: $\frac{Q_3 + Q_1}{2}$

• 10–90 percentile range: $P_{90} - P_{10}$

5-Number Summary

Definition

The 5-number summary for a data set consists of:

1. Minimum

2. First quartile ($Q_1$)

3. Second quartile ($Q_2$, median)

4. Third quartile ($Q_3$)

5. Maximum

Example: Finding a 5-Number Summary

Minimum	Q1	Median (Q2)	Q3	Maximum
10	25	35	50	110

All values are in minutes (from the "Space Mountain" wait times example).

Boxplot (Box-and-Whisker Diagram)

Definition

A boxplot (or box-and-whisker diagram) is a graphical representation of a data set, showing the minimum, $Q_1$, median ($Q_2$), $Q_3$, and maximum. It helps visualize the spread and skewness of data.

Procedure for Constructing a Boxplot

1. Find the 5-number summary.

2. Draw a line from the minimum to the maximum value.

3. Draw a box from $Q_1$ to $Q_3$, with a line at the median ($Q_2$).

Additional info:

• Boxplots are useful for identifying skewness and outliers.

• Skewness is present if the boxplot is not symmetric and extends more to one side.

• Outliers can be identified using modified boxplots, where values beyond $1.5 \times \text{IQR}$ from $Q_1$ or $Q_3$ are marked as outliers.