Question

A report from the U.S. Department of Justice (www.ojp.usdoj.gov/bjs/) reported the percent changes in federal prison populations in 21 northeastern and midwestern states during 2005. Using appropriate graphical displays and summary statistics, write a report on the changes in prison populations. \(\begin{array}{l|c|l|c} \text { State } & \begin{array}{l} \text { Percent } \\ \text { Change } \end{array} & \text { State } & \begin{array}{c} \text { Percent } \\ \text { Change } \end{array} \\ \hline \text { Connecticut } & -0.3 & \text { Iowa } & 2.5 \\ \text { Maine } & 0.0 & \text { Kansas } & 1.1 \\ \text { Massachusetts } & 5.5 & \text { Michigan } & 1.4 \\ \text { New Hampshire } & 3.3 & \text { Minnesota } & 6.0 \\ \text { New Jersey } & 2.2 & \text { Missouri } & -0.8 \\ \text { New York } & -1.6 & \text { Nebraska } & 7.9 \\ \text { Pennsylvania } & 3.5 & \text { North Dakota } & 4.4 \\ \text { Rhode Island } & 6.5 & \text { Ohio } & 2.3 \\ \text { Vermont } & 5.6 & \text { South Dakota } & 11.9 \\ \text { Illinois } & 2.0 & \text { Wisconsin } & -1.0 \\ \text { Indiana } & 1.9 & & \\ \hline \end{array}\)

Short answer

The states show varied changes in federal prison populations, with significant increases in South Dakota (+11.9%) and decreases in states like New York (-1.6%). The average change reflects a slight overall increase.

Step-by-step solution

Organize Data

List all the states along with their corresponding percent changes for an easier overview. Collect values for analysis and to create graphs or charts as necessary.

Calculate Summary Statistics

Find the mean, median, and mode of the percent changes to understand the central tendency of the data. Also, find the range, variance, and standard deviation to understand the variability.

Create a Histogram

Plot a histogram of the percent changes to visualize the distribution of the data. This will help in identifying any patterns or outliers in the data set.

Create a Box Plot (Box-and-Whisker Plot)

Use a box plot to depict the five-number summary (minimum, first quartile, median, third quartile, and maximum) of the percent changes. This will help highlight the median, spread, and outliers.

Analyze Findings

Based on the summary statistics and graphical displays, describe the overall trend in federal prison population changes. Discuss any noticeable increases or decreases, and highlight states that deviate significantly from the average.

Key concepts

Statistical Summary

When exploring data, a statistical summary provides a quick overview of the key figures that describe the data set. It answers questions like what the average of the data is, how spread out the data is, and if there are any outliers which may skew the data. In this context, we look at the federal prison population changes.

• Mean: This is the average of the percentages listed. You calculate it by adding all the percent changes and dividing by the number of states.
• Median: This is the middle point of the data when ordered from smallest to largest. If the data set has an even number of observations, the median is the average of the two middle numbers.
• Mode: This is the number that appears most frequently in the data set; it helps show the most common percent change.
• Range: The difference between the maximum and minimum percent change, indicating the spread of the data.
• Variance and Standard Deviation: These measure data dispersion around the mean. A higher value implies more variability.

Understanding these values gives insight into the central tendency and variability of the data, setting a basis for further analysis using graphical displays.

Graphical Displays

Graphical displays are visual tools used to analyze and interpret data more easily. They can provide insights that numbers alone may not readily reveal.
The two common graphical displays used in data analysis are histograms and box plots.

• Histogram: This is a bar graph depicting the distribution of the data set. Each bar represents a range of percent changes, showing how frequently each range appears. Patterns such as skewness or peaks (modes) can be easily identified, making it simpler to spot trends.
• Box Plot: Also known as a box-and-whisker plot, this displays the five-number summary visually. It highlights the median, quartiles, and potential outliers. The box represents the interquartile range, and lines (whiskers) extend to the minimum and maximum values.

These graphical tools make data analysis more efficient by summarizing complex data sets, emphasizing trends, and spotting outliers at a glance.

Central Tendency

Central tendency refers to the way we can describe the "center" of a data set. It tells us where most of the values are clustered.

• Mean: As the statistical summary highlights, the mean is the arithmetic average and gives a quick overview of the data's general trend. However, it can be affected by extreme values, known as outliers.
• Median: This is often considered more representative in skewed data or when outliers are present. It's the exact middle value, providing a better central location for asymmetric data.
  For example, if the percent change data is skewed by a few states with extremely high or low changes.
• Mode: This helps identify the most frequent percent change, reflecting what might be considered a common trend among the states.

These measures give a robust picture of central tendency, helping to understand the overall pattern of changes in the prison populations.

Variability

Variability in data analysis refers to how spread out the data points are. By analyzing variability, we determine if data points are close to the mean or dispersed across a wider range.

• Range: This quick measure shows the span of the data, highlighting the gap between the smallest and largest percent changes.
• Variance: This provides a numerical value of how much the data points diverge from the mean. Larger variance implies greater spread, indicating that states had diverse percent changes in prison populations.
• Standard Deviation: The square root of variance, standard deviation is more interpretable, measuring how much individual observations deviate from the mean on average.
  A low standard deviation means most percent changes are near the average, suggesting consistent trends across states. Conversely, a high value indicates significant differences.

Understanding variability is crucial for emphasizing stability or highlighting inconsistency in prison population changes.

Five-number Summary

The five-number summary is a concise way of presenting a data set's dispersion, giving a quick snapshot of its distribution.
It consists of:

• Minimum: The smallest percent change, showing the lowest decline or smallest growth in prison population.
• First Quartile (Q1): The median of the lower half of the data. It divides the lowest 25% of data from the rest.
• Median: The central value, as explained in central tendency.
• Third Quartile (Q3): The median of the upper half, separating the highest 25% from the remainder.
• Maximum: The largest percent change, displaying the highest growth.

By reviewing these five statistics, you gain insight into the data's center, spread, and overall shape, identifying outliers and supporting further trend analysis. Explorations of quartiles and extremes provide a foundation for assessing distribution patterns.