The following data represent the per capital (average)...

The following data represent the per capital (average) disposable income (income after taxes) for the 50 states and the District of Columbia in 2009. $$\begin{array}{llllll} 30,103 & 33,096 & 35,507 & 37,916 & 41,344 & 43,874 \\ \hline 30,875 & 33,725 & 35,667 & 38,081 & 41,411 & 45,705 \\ \hline 31,632 & 33,786 & 35,676 & 38,503 & 41,552 & 46,957 \\ \hline 31,799 & 34,004 & 36,484 & 38,578 & 41,751 & 48,285 \\ \hline 31,883 & 34,025 & 36,745 & 39,530 & 42,009 & 49,875 \\ \hline 31,946 & 34,089 & 36,751 & 39,578 & 42,325 & 50,313 \\ \hline 32,219 & 34,453 & 36,822 & 39,817 & 42,603 & 54,397 \\ \hline 32,935 & 35,268 & 36,935 & 41,003 & 42,831 & 66,000 \\ \hline 32,992 & 35,381 & 37,780 & & & \\\hline\end{array}$$ With the first class having a lower class limit of 30,000 and a class width of 6000: (a) Construct a frequency distribution. (b) Construct a relative frequency distribution. (c) Construct a frequency histogram of the data. (d) Construct a relative frequency histogram of the data. (e) Describe the shape of the distribution. (f) Repeat parts (a)-(e) using a class width of 3000 . (g) Does one frequency distribution provide a better summary of the data than the other? Explain.

The following data represent the per capital (average) disposable income (income after taxes) for the 50 states and the District of Columbia in 2009. $$\begin{array}{llllll}
30,103 & 33,096 & 35,507 & 37,916 & 41,344 & 43,874 \\
\hline 30,875 & 33,725 & 35,667 & 38,081 & 41,411 & 45,705 \\
\hline 31,632 & 33,786 & 35,676 & 38,503 & 41,552 & 46,957 \\
\hline 31,799 & 34,004 & 36,484 & 38,578 & 41,751 & 48,285 \\
\hline 31,883 & 34,025 & 36,745 & 39,530 & 42,009 & 49,875 \\
\hline 31,946 & 34,089 & 36,751 & 39,578 & 42,325 & 50,313 \\
\hline 32,219 & 34,453 & 36,822 & 39,817 & 42,603 & 54,397 \\
\hline 32,935 & 35,268 & 36,935 & 41,003 & 42,831 & 66,000 \\
\hline 32,992 & 35,381 & 37,780 & & & \\\hline\end{array}$$
With the first class having a lower class limit of 30,000 and a class width of 6000:
(a) Construct a frequency distribution.
(b) Construct a relative frequency distribution.
(c) Construct a frequency histogram of the data.
(d) Construct a relative frequency histogram of the data.
(e) Describe the shape of the distribution.
(f) Repeat parts (a)-(e) using a class width of 3000 .
(g) Does one frequency distribution provide a better summary of the data than the other? Explain.

Key Concepts

Frequency Distribution

A frequency distribution is a tabular summary that displays the number of observations within equal intervals, known as classes. It is used to organize raw data into a manageable form, facilitating an understanding of the data’s overall distribution and highlighting patterns or clusters within the dataset.

Relative Frequency Distribution

A relative frequency distribution is similar to a frequency distribution, but it expresses the frequency of each class as a proportion or percentage of the total number of observations. This allows for easier comparison between groups, especially when the total sample sizes differ or when probabilities are being considered.

Histogram

A histogram is a graphical representation of a frequency distribution. It displays the frequencies of data within each class using contiguous bars, where the area of each bar (or height, depending on the scale) corresponds to the frequency or relative frequency. Histograms are particularly useful for visualizing the shape and spread of a dataset.

Class Intervals and Class Width

Class intervals are the divisions or groups into which data are sorted in a frequency distribution. The class width is the difference between the upper and lower boundaries of a class, and it determines how detailed the grouping is. The choice of class width affects the detail and clarity of the resulting analysis, with narrower widths providing more detail and wider widths offering a broader overview.

Shape of the Distribution

The shape of a distribution describes the overall pattern of the data when plotted, including aspects such as symmetry, skewness, and modality. Common descriptions include symmetric, skewed (to the left or right), unimodal, or bimodal. Understanding the shape helps in identifying trends, making inferences, and choosing appropriate statistical methods.

Effect of Varying Class Width

Altering the class width in a frequency distribution can significantly affect how the data are summarized and interpreted. A smaller class width provides a more detailed view and can reveal more subtle distribution patterns, while a larger class width may smooth over irregularities and provide a more general overview. The choice of class width thus plays a crucial role in data presentation and analysis.

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