Income Distribution up to 100,000 The following table shows...

Income Distribution up to $\$ 100,000$ The following table shows the distribution of household incomes for a sample of 1,000 households in the United States with incomes up to $\$ 100,000 .^{28}$ $$ \begin{array}{|c|l|c|c|c|c|} \hline \multirow{2}{*} {\text { Income Bracket }} & 0- & 20,000- & 40,000- & 60,000- & 80,000- \\ & 19,999 & 39,999 & 59,999 & 79,999 & 99,999 \\ \hline \text { Households } & 270 & 280 & 200 & 150 & 100 \\ \hline \end{array} $$ Use this information to estimate, to the nearest $\$ 1,000$, the average household income for such households.

Income Distribution up to $\$ 100,000$ The following table shows the distribution of household incomes for a sample of 1,000 households in the United States with incomes up to $\$ 100,000 .^{28}$
$$
\begin{array}{|c|l|c|c|c|c|}
\hline \multirow{2}{*} {\text { Income Bracket }} & 0- & 20,000- & 40,000- & 60,000- & 80,000- \\
& 19,999 & 39,999 & 59,999 & 79,999 & 99,999 \\
\hline \text { Households } & 270 & 280 & 200 & 150 & 100 \\
\hline
\end{array}
$$
Use this information to estimate, to the nearest $\$ 1,000$, the average household income for such households.

Key Concepts

Grouped Data

Grouped data refers to data that has been organized into intervals or classes rather than individual data points. This approach simplifies large data sets by grouping values into ranges, making it easier to analyze the overall distribution of the data without getting lost in every specific detail.

Frequency Distribution

A frequency distribution is a summary that shows the number or percentage of observations falling within each interval or category. It allows one to see patterns such as concentration or spread in the data, which is particularly useful when dealing with grouped data as it presents the count of records that fall into each defined group.

Midpoint Calculation

Midpoint calculation involves finding the value that lies in the middle of a given interval. In the context of grouped data, the midpoint of each class is used as a representative value for all data points in that interval, which simplifies the process of estimating measures such as the mean.

Weighted Average

A weighted average is a mean where each value in the data set is multiplied by a weight indicating its importance or frequency. When estimating the mean from grouped data, each group’s midpoint is multiplied by the number of observations in that group, and the sum of these products is divided by the total number of observations to derive the weighted average.

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