Question

The following data values are the 2009 per capita expenditures on public libraries for each of the \(50 \mathrm{U} . \mathrm{S}\). states and Washington, D.C. (from www.statemaster.com): \(\begin{array}{rrrrrrr}16.84 & 16.17 & 11.74 & 11.11 & 8.65 & 7.69 & 7.48 \\\ 7.03 & 6.20 & 6.20 & 5.95 & 5.72 & 5.61 & 5.47 \\ 5.43 & 5.33 & 4.84 & 4.63 & 4.59 & 4.58 & 3.92 \\ 3.81 & 3.75 & 3.74 & 3.67 & 3.40 & 3.35 & 3.29 \\ 3.18 & 3.16 & 2.91 & 2.78 & 2.61 & 2.58 & 2.45 \\ 2.30 & 2.19 & 2.06 & 1.78 & 1.54 & 1.31 & 1.26 \\ 1.20 & 1.19 & 1.09 & 0.70 & 0.66 & 0.54 & 0.49 \\ 0.30 & 0.01 & & & & & \end{array}\) a. Summarize this data set with a frequency distribution. Construct the corresponding histogram. b. Use the histogram from Part (a) to find the approximate values of the following percentiles: i. 50 th iv. 90 th ii. 70 th v. 40 th iii. 10 th

Short answer

The frequency distribution and histogram provide a summary and visual representation of the data set. The percentiles indicated that a certain percentage of the data falls below those values. For example, the 50th percentile, also known as the median, is the value below which 50% of the data falls, and similarly for the other percentiles. The exact values can't be determined without the actual dataset, but could be estimated from the histogram.

Step-by-step solution

Organize the Data into a Frequency Distribution

Start by defining the range and class intervals for the data. This will involve determining the highest and lowest values, and then dividing the range (highest value - lowest value) by an appropriate number of classes. For example, one might choose 5 as a starting point. Normalize the data to make it easier to work with. Then tally the number of data points that fall into each class interval and record these as the 'frequency' of each class in a frequency table.

Construct the Histogram

Plot the data from the frequency table onto a graph. The class intervals or ranges should be on the x-axis and the frequencies on the y-axis. Each bar in the histogram represents a class interval, with its height corresponding to the frequency of that class.

Estimate the Percentiles

Percentiles tell us where a particular value is in relation to the rest of the data set. The nth percentile is the value below which n% of the data falls. To find the approximated value of specific percentiles, use the histogram, which is the graphical representation of the frequency distribution table. The 50th, 70th, 10th, 90th and 40th percentiles can then be estimated by counting bars in the histogram from left to right until you reach the bar under which the respective percent of total data falls.

Key concepts

Frequency Distribution

Understanding the frequency distribution is vital to analyzing data sets effectively. This procedure allows us to summarize large data groups into manageable categories. To create a frequency distribution, we start by arranging data points into a table, divided into class intervals or 'bins'. Each interval corresponds to a range of values, and we count how many data points fall into each range.

For the per capita expenditures on public libraries, we'd first identify the minimum and maximum expenditures, then determine the range by subtracting the minimum from the maximum. Next, we choose an appropriate number of intervals, ensuring they are mutually exclusive and cover the entire range of data. Finally, we tally up the number of expenditures that fit into each interval to get the frequencies. The result is a structured overview that highlights how the data is distributed across different expenditure levels.

Constructing Histograms

A histogram is a visual representation of frequency distribution and is an excellent tool to grasp the shape of your data distribution. Once we have our frequency distribution table ready from the per capita expenditures, constructing a histogram becomes straightforward.

The x-axis represents the class intervals, also known as bins, while the y-axis denotes frequency. Each bar on the histogram corresponds to a particular range and its height mirrors the number of data points within that interval. In our case, we would have bars representing ranges of expenditures, offering a quick visual way to assess which expenditure levels are most or least common among U.S. states and territories.

Estimating Percentiles

Percentiles are useful tools in statistics, enabling us to understand the relative standing of a value within a data set. By definition, the nth percentile is the value below which a given percentage of observations in a group falls. To estimate percentiles using a histogram, we perform an approximate graphical analysis.

In practice, with our expenditures histogram, we would look at the cumulative height of the bars to estimate the value below which, for instance, 50% (median), 70%, or 90% of the observations lie. It's like taking a vertical slice of the histogram and determining what proportion of the total area is to the left of that slice. This helps locate the point at which the desired percentage of data points falls below, without requiring exact calculations.

Data Organization

Good data organization is the backbone of any quantitative analysis. It transforms raw numbers into meaningful insights. Before we even begin to calculate frequency distributions or construct histograms, data should be neatly organized. In our exercise, data were first listed in descending order to simplify further processing.

To keep data organized, we often use spreadsheets or database tables where columns and rows denote different variables and observations, respectively. Consistent formatting, categorization, and the use of metadata are also integral to data organization, allowing for more efficient sorting, retrieval, and analysis. It makes subsequent tasks, like constructing histograms or estimating percentiles, much more straightforward and less error-prone.

Quantitative Data Analysis

Quantitative data analysis involves applying statistical methods to understand the characteristics and patterns of the data. It starts with data organization followed by summarization through measures like mean, median, and mode. Next, we use frequency distributions to categorize numerical data, then construct histograms to visualize the frequency of different categories.

Following visualization, we often perform inference using methods such as percentile estimation. This allows us to draw conclusions about the probability of a value falling within a certain range. Through this process, we can extract meaningful patterns and trends from the per capita expenditure data, aiding decision-makers in understanding how different states allocate funds to public libraries and how those amounts compare across the country.