Describing, Exploring, and Comparing Data: Measures of Relative Standing and Boxplots

Measures of Relative Standing and Boxplots

Introduction

This section explores statistical tools used to describe the position of individual data values within a data set. Key concepts include z scores, percentiles, quartiles, and the construction and interpretation of boxplots. These measures help compare values from different data sets and identify significant or unusual observations.

z Scores

Definition and Calculation

• z Score: The number of standard deviations a data value (x) is above or below the mean.

• For a sample: $z = \frac{x - \bar{x}}{s}$

• For a population: $z = \frac{x - \mu}{\sigma}$

• Round z scores to two decimal places.

Properties of z Scores

• z scores are unitless.

• 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.

• Negative z scores indicate values below the mean.

Using z Scores to Identify Significant Values

• Significantly low values: $z \leq -2$

• Significantly high values: $z \geq 2$

• Values not significant: $-2 < z < 2$

Example: Comparing Data Values

• To compare values from different data sets, convert each to a z score.

• Example: A body temperature of 99°F has a z score of 1.29; a quarter weighing 5.7790 g has a z score of 2.26. The quarter's weight is more extreme relative to its data set.

Example: Identifying Significant Earthquake Magnitude

• Given: Mean = 2.572, Standard deviation = 0.651, Value = 4.01

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

• Interpretation: Since 2.21 ≥ 2, the magnitude is significantly high.

Percentiles

Definition and Interpretation

• Percentiles divide a data set into 100 groups, each containing about 1% of the values.

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

Finding the Percentile of a Data Value

• Count the number of values less than the given value, divide by the total number of values, and multiply by 100. Round to the nearest whole number.

• Example: If 36 out of 50 wait times are less than 45 minutes, then $\frac{36}{50} \times 100 = 72$. Thus, 45 minutes is the 72nd percentile.

Converting a Percentile to a Data Value

• Calculate the locator $L = \frac{k}{100} \times n$ (where $k$ is the percentile and $n$ is the number of values).

• If $L$ is not a whole number, round up to the next whole number. The value at this position in the sorted data is the percentile value.

Quartiles

Definition and Description

• Quartiles divide data into four groups, each containing about 25% of the values.

• $Q_1$ (first quartile): Same as $P_{25}$; separates the lowest 25% from the rest.

• $Q_2$ (second quartile): Same as $P_{50}$ and the median; separates the lowest 50% from the highest 50%.

• $Q_3$ (third quartile): Same as $P_{75}$; separates the lowest 75% from the highest 25%.

Note: There is not universal agreement on the exact procedure for calculating quartiles; results may vary by method or technology.

Statistics Defined Using Quartiles and Percentiles

• Interquartile Range (IQR): $IQR = Q_3 - Q_1$

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

• Midquartile: $\frac{Q_1 + Q_3}{2}$

• 10–90 Percentile Range: $P_{90} - P_{10}$

5-Number Summary

Definition and Example

• The 5-number summary consists of: Minimum, $Q_1$, Median ($Q_2$), $Q_3$, Maximum.

• Example (Space Mountain wait times): 10, 25, 35, 50, 110 (all in minutes).

Boxplots (Box-and-Whisker Diagrams)

Definition and Construction

• A boxplot is a graphical representation of the 5-number summary.

• It consists of a box from $Q_1$ to $Q_3$, a line at the median ($Q_2$), and "whiskers" extending to the minimum and maximum values.

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.

Skewness

Identifying Skewness with Boxplots

• A distribution is skewed if it is not symmetric and extends more to one side.

• Boxplots can help visually identify skewness in data distributions.

Identifying Outliers and Modified Boxplots

Procedure for Identifying Outliers

1. Find $Q_1$, $Q_2$, and $Q_3$.

2. Calculate $IQR = Q_3 - Q_1$.

3. Compute $1.5 \times IQR$.

4. A value is an outlier if it is below $Q_1 - 1.5 \times IQR$ or above $Q_3 + 1.5 \times IQR$.

Modified Boxplots

• Modified boxplots use special symbols (e.g., asterisks) to mark outliers.

• The whiskers extend only to the most extreme non-outlier values.