Question

The report "Who Moves? Who Stays Put? Where's Home?" (Pew Social and Demographic Trends, December 17,2008 ) gave the accompanying data on the percentage of the population in a state that was born in the state and is still living there for each of the 50 U.S. states. \(\begin{array}{llllllll}75.8 & 71.4 & 69.6 & 69.0 & 68.6 & 67.5 & 66.7 & 66.3 \\\ 66.1 & 66.0 & 66.0 & 65.1 & 64.4 & 64.3 & 63.8 & 63.7 \\ 62.8 & 62.6 & 61.9 & 61.9 & 61.5 & 61.1 & 59.2 & 59.0 \\ 58.7 & 57.3 & 57.1 & 55.6 & 55.6 & 55.5 & 55.3 & 54.9 \\ 54.7 & 54.5 & 54.0 & 54.0 & 53.9 & 53.5 & 52.8 & 52.5 \\\ 50.2 & 50.2 & 48.9 & 48.7 & 48.6 & 47.1 & 43.4 & 40.4 \\ 35.7 & 28.2 & & & & & & \end{array}\) a. Find the values of the median, the lower quartile, and the upper quartile. b. The two smallest values in the data set are 28.2 (Alaska) and 35.7 (Wyoming). Are these two states outliers? c. Construct a modified boxplot for this data set and comment on the interesting features of the plot.

Short answer

a. Once the data is organized from smallest to largest, we can identify positions for Median, Q1 and Q3. Median is the middle value, while Q1 is the median of first half data and Q3 is the median of the second half data. b. Whether the smallest values 28.2 and 35.7 are outliers is determined through the Interquartile Range(IQR) and using formulae to calculate limits for considered outliers. c. A boxplot gives a graphical representation of these datasets helping to analyze trends and outliers.

Step-by-step solution

Organize and Identify

First, organize the data from smallest to largest. This will facilitate the computation of quartiles and the median. Once the data is organized, we can find the position of Q1, the median, and Q3. When the data set has an odd number of observations, the median is the middle value. When the data set has an even number of observations, the median is the average of the two middle values. Q1 is the median of the first half of the data, and Q3 is the median of the second half.

Calculate Median, Q1 and Q3

Now we calculate the Median, Q1 (Lower quartile) and Q3 (Upper quartile) using the positions identified in Step 1. If any of these positions are not whole numbers, we take the average of the two numbers that would 'bracket' that position.

Identify outliers

Identify outliers by calculating the interquartile range (IQR), which is Q3 - Q1. Any value lower than Q1 - 1.5 * IQR or higher than Q3 + 1.5 * IQR is considered an outlier.

Construct boxplot

The boxplot should show Q1, median, Q3, and any potential outliers. The data from the minimum value to Q1 represents the first 25% of the data, from Q1 to the median represents the second 25%, from the median to Q3 represents the third 25% and from Q3 to the maximum values represents the last 25%.

Analyze the boxplot

The boxplot gives a visual representation of the distribution of the data. Features to comment on include the range, interquartile range, potential outliers, skewness and whether the data is symmetric or not. Make sure to specifically comment on any observed outliers.

Key concepts

Median Calculation

When working with a data set, finding the median provides a valuable insight into the central tendency of the distribution. Simply put, the median is the middle value that divides a sorted data set in half. If you have an odd number of observations, the median is the one in the center. For an even number, it is the average of the two central figures.

To compute the median in our exercise, we first arrange the state percentages in ascending order. With an even number of states, we find the median by identifying the two middle values. Then, we calculate their average using the formula: \( \text{median} = \frac{\text{value at position} (n+1)/2 + \text{value at position} n/2}{2} \), where \(n\) is the total number of values in the data set. Applying this process delivers a precise median, depicting the 'middle ground' of the population's inclination to stay in their birth state.

Interquartile Range (IQR)

Understanding the spread of a data set is as crucial as its central value. The interquartile range (IQR) is a measurement that captures the range within which the middle 50% of the data resides. It goes one step further than the median, by partitioning the data into quarters. The lower quartile, or Q1, is the median of the data's lower half, while the upper quartile, or Q3, is the median of the upper half.

To calculate the IQR, we subtract Q1 from Q3: \( IQR = Q3 - Q1 \). Then, we can utilize the IQR to identify potential outliers. These are values so far from the central tendency that they stand out. Specifically, an observation is considered an outlier if it falls below \( Q1 - 1.5 \times IQR \) or above \( Q3 + 1.5 \times IQR \). This method pinpoints statistical anomalies, offering insights into the characteristics of the data that may warrant further investigation.

Boxplot Construction

A boxplot, or box-and-whisker plot, is a graphical summary of a data set's distribution. In constructing a boxplot, key components include the median (marking the center of the data), Q1 and Q3 (the edges of the box), and the 'whiskers' — lines extending from the box to the maximum and minimum values, excluding outliers.

We start by drawing a box from Q1 to Q3, placing a line inside it at the median. Outliers, if any, are plotted as individual points. Lastly, the whiskers are drawn from the edges of the box to the highest and lowest values within the bounds calculated by the IQR method, without including outliers. This diagram is invaluable for quickly observing the data's spread, symmetry, and any unusual observations. For the data in our exercise, the boxplot thus visually explicates the variation and central tendencies in the states population's mobility, helping us interpret complex data with ease.