91Ó°ÊÓ

The report "Who Moves? Who Stays Put? Where's Home?" (Pew Social and Demographic Trends, December 17,2008 ) gave the accompanying data for the 50 U.S. states on the percentage of the population that was born in the state and is still living there. The data values have been arranged in order from largest to smallest. \(\begin{array}{lllllllllll}75.8 & 71.4 & 69.6 & 69.0 & 68.6 & 67.5 & 66.7 & 66.3 & 66.1 & 66.0 & 66.0\end{array}\) \(\begin{array}{lllllllllll}65.1 & 64.4 & 64.3 & 63.8 & 63.7 & 62.8 & 62.6 & 61.9 & 61.9 & 61.5 & 61.1\end{array}\) \(\begin{array}{lllllllllll}59.2 & 59.0 & 58.7 & 57.3 & 57.1 & 55.6 & 55.6 & 55.5 & 55.3 & 54.9 & 54.7\end{array}\) \(\begin{array}{lllllllllll}54.5 & 54.0 & 54.0 & 53.9 & 53.5 & 52.8 & 52.5 & 50.2 & 50.2 & 48.9 & 48.7\end{array}\) \(\begin{array}{llllll}48.6 & 47.1 & 43.4 & 40.4 & 35.7 & 28.2\end{array}\) 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 boxplot for this data set and comment on the interesting features of the plot.

Step by step solution

01

Organize Data

Arrange the provided percentages in ascending order.

02

Calculating Median

Find the center data. If the data set has an odd number of observations, the middle value is used as the median. If the data set has an even number of observations, the average of the two middle numbers is the median.

03

Calculating Lower Quartile

The lower quartile Q1 is the median of the lower half of the data - it is the middle number between the smallest number and the median of data set.

04

Calculating Upper Quartile

The upper quartile, Q3, is the median of the upper half of the data -it is the middle value between the median and the largest number in the data set.

05

Determining Outliers

Now, calculate the interquartile range (IQR = Q3 - Q1), and identify any values that are less than Q1 - 1.5*(IQR) or more than Q3 + 1.5*(IQR) as outliers.

06

Construct a Boxplot

Now that we have all the five number summary(minimum, Q1, median, Q3, maximum), outliers if any, we can construct a Boxplot.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Median

The median is a key measure in descriptive statistics. It represents the middle value of a data set when the values are ordered from least to greatest. In this case, we have 50 data points representing percentages.

To find the median, first ensure the data is arranged in ascending order, which has already been done for you. With 50 values, this makes the median the average of the 25th and 26th values in the sequence.

This placement ensures the median divides the data into two equal halves, offering a robust central measure that’s less affected by extremely high or low values compared to the mean. This makes it particularly useful in this scenario, where we're gauging centrism in population retention.

Quartiles

Quartiles break the data into four equal parts, each containing 25% of the data points. This division helps better understand the distribution of the data.

The first quartile (Q1), also known as the lower quartile, is the median of the first 25% of data. Similarly, the third quartile (Q3) is the median of the last 25% of the ordered data.

Calculating these quartiles involves:

• Finding Q1 by determining the median of the first half of values, up to the middle of the data set.
• Finding Q3 by determining the median of the second half of values, from the median to the highest data point.

This quartile information aids in determining the spread and symmetry of the data set, offering critical insights beyond just averages.

Outliers

Outliers are data points that deviate significantly from other observations in the data set. They can dramatically affect statistical analyses and interpretations.

To detect outliers, we use the Interquartile Range (IQR), calculated as Q3 - Q1.

Data points more than 1.5 times the IQR below Q1 or above Q3 are typically considered outliers. Using this method, we can objectively decide if values such as 28.2 and 35.7 are outliers.

By identifying outliers, we assess their impact and decide whether they indicate variability, errors, or important insights.

Boxplot

A boxplot, or box-and-whisker plot, visually represents the five-number summary: minimum, Q1, median, Q3, and maximum. It's an efficient way to convey key data insights at a glance.

In constructing a boxplot:

• Draw a box from Q1 to Q3.
• Mark the median inside the box.
• Extend "whiskers" from the box to the minimum and maximum data points that are not outliers.
• Indicate any outliers with individual points, using a different symbol if desired.

The boxplot provides invaluable information about the data’s central tendency, variability, and symmetry, allowing you to spot trends and make comparisons more effectively.