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

Measures of Relative Standing and Boxplots

Introduction

This chapter focuses on statistical methods for describing, exploring, and comparing data, with an emphasis on measures of relative standing such as z scores, percentiles, quartiles, and graphical summaries like boxplots. These tools help to understand the position of individual data values within a dataset and to identify significant or unusual observations.

Z Scores

Definition and Calculation

• Z score (also called standard score or standardized value) measures how many standard deviations a data value $x$ is above or below the mean.

• For a sample: $z = \frac{x - \overline{x}}{s}$ where $\overline{x}$ is the sample mean and $s$ is the sample standard deviation.

• For a population: $z = \frac{x - \mu}{\sigma}$ where $\mu$ is the population mean and $\sigma$ is the population standard deviation.

• Z scores are typically rounded to two decimal places (e.g., 2.31).

Identifying Significant Values

• Values with $z \leq -2.00$ or $z \geq 2.00$ are considered significantly low or significantly high, respectively.

• Values with $-2.00 < z < 2.00$ are not considered significant.

Example: Comparing Relative Standing

• To compare which of two data values is more extreme, calculate their z scores relative to their respective datasets.

• Example: - Newborn baby weight: $x = 4000$ g, $\overline{x} = 3152.0$ g, $s = 693.4$ g - Adult body temperature: $x = 99$°F, $\overline{x} = 98.20$°F, $s = 0.62$°F Calculate z scores for each to determine which is more extreme.

Quartiles

Definition and Calculation

• Quartiles are measures of location that divide a dataset into four groups, each containing approximately 25% of the data.

• Denoted as $Q_1$ (first quartile), $Q_2$ (second quartile, or median), and $Q_3$ (third quartile).

• Procedure:

  1. Arrange data in increasing order.

  2. Find the median ($Q_2$).

  3. Divide the data into two halves. If the number of observations is odd, include the median in both halves.

  4. $Q_1$ is the median of the lower half; $Q_3$ is the median of the upper half.

Quartiles in Different Distributions

• Quartiles divide data into four equal parts regardless of the distribution's shape (uniform, bell-shaped, right-skewed, left-skewed).

Statistics Defined Using Quartiles and Percentiles

Key Measures

• Interquartile Range (IQR): Measures the spread of the middle 50% of data. $IQR = Q_3 - Q_1$

• Semi-interquartile Range: Half the IQR. $\text{Semi-interquartile range} = \frac{Q_3 - Q_1}{2}$

• Midquartile Range: Average of $Q_1$ and $Q_3$. $\text{Midquartile range} = \frac{Q_3 + Q_1}{2}$

• 10–90 Quartile Range: Difference between the 90th and 10th percentiles. $\text{10–90 quartile range} = P_{90} - P_{10}$

5-Number Summary

Definition

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

Boxplots (Box-and-Whisker Diagrams)

Definition and Construction

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

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

• Procedure:

  1. Find the five-number summary.

  2. Draw a box from $Q_1$ to $Q_3$.

  3. Draw a line at the median inside the box.

  4. Extend lines (whiskers) from the box to the minimum and maximum values.

Determining Outliers

Outlier Detection Using Quartiles

• Lower limit: $Q_1 - 1.5 \times IQR$

• Upper limit: $Q_3 + 1.5 \times IQR$

• Any data value less than the lower limit or greater than the upper limit is considered an outlier.

Skewness

Types of Distribution Shapes

• Right-skewed: Longer tail on the right; most data are concentrated on the left.

• Symmetric: Data are evenly distributed around the center.

• Left-skewed: Longer tail on the left; most data are concentrated on the right.

Percentiles

Definition and Calculation

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

• There is no universal method for calculating percentiles, but a common approach is: $\text{Percentile of value } x = \frac{\text{number of values less than } x}{\text{total number of values}} \times 100$

• Percentiles are denoted as $P_1, P_2, ..., P_{99}$.

Finding the Percentile of a Data Value

• Arrange the data in increasing order.

• Count the number of values less than the target value $x$.

• Apply the formula above and round to the nearest whole number.

• Example: If 27 out of 50 data speeds are less than 14.7 Mbps, then $\text{Percentile} = \frac{27}{50} \times 100 = 54$ So, 14.7 Mbps is at the 54th percentile.

Converting Percentile to a Data Value

• Given a percentile $k$, the position in the ordered data is: $\text{Locator} = \frac{k}{100} \times n$ where $n$ is the total number of values.

• The value at this position is the $k$th percentile.

Example: Comparing Percentile Standing

• If Jack is ranked 70th in a class of 200, then 130 students are ranked below him: $130/200 = 65$ percentile.

• If Jill is at the 70th percentile, then 70% of the class is ranked below her.

• Therefore, Jill has a higher standing than Jack.

Summary Table: Key Measures of Relative Standing

Measure	Definition	Formula
Z Score	Standardized value indicating number of standard deviations from mean	$z = \frac{x - \overline{x}}{s}$ (sample) $z = \frac{x - \mu}{\sigma}$ (population)
Quartile	Divides data into four equal parts	$Q_1$, $Q_2$ (median), $Q_3$
Interquartile Range (IQR)	Spread of middle 50% of data	$IQR = Q_3 - Q_1$
Percentile	Divides data into 100 equal parts	$\frac{\text{number of values less than } x}{\text{total number of values}} \times 100$
Five-number summary	Describes data spread and center	Min, $Q_1$, $Q_2$, $Q_3$, Max
Boxplot	Graphical summary of five-number summary	Box from $Q_1$ to $Q_3$, line at $Q_2$, whiskers to Min/Max
Outlier	Extreme value outside expected range	Below $Q_1 - 1.5 \times IQR$ or above $Q_3 + 1.5 \times IQR$

Additional info: Academic context and examples have been expanded for clarity and completeness.