Distribution of Wealth If we model after-tax household...

Distribution of Wealth If we model after-tax household income by a normal distribution, then the figures of a 1995 study imply the information in the following table, which should be used for Exercises $49-60 .^{55}$ Assume that the distribution of incomes in each country is bell shaped and symmetric. $$ \begin{array}{|r|c|c|c|c|c|} \hline \text { Country } & \text { United States } & \text { Canada } & \text { Switzerland } & \text { Germany } & \text { Sweden } \\ \hline \begin{array}{r} \text { Mean Household } \\ \text { Income } \end{array} & \$ 38,000 & \$ 35,000 & \$ 39,000 & \$ 34,000 & \$ 32,000 \\ \hline \begin{array}{r} \text { Standard } \\ \text { Deviation } \end{array} & \$ 21,000 & \$ 17,000 & \$ 16,000 & \$ 14,000 & \$ 11,000 \\ \hline \end{array} $$ Refer to Exercise $49 .$ Which of the five countries listed has the poorest households (i.e., the lowest cutoff for considering a household poor)?

Distribution of Wealth If we model after-tax household income by a normal distribution, then the figures of a 1995 study imply the information in the following table, which should be used for Exercises $49-60 .^{55}$ Assume that the distribution of incomes in each country is bell shaped and symmetric.
$$
\begin{array}{|r|c|c|c|c|c|}
\hline \text { Country } & \text { United States } & \text { Canada } & \text { Switzerland } & \text { Germany } & \text { Sweden } \\
\hline \begin{array}{r}
\text { Mean Household } \\
\text { Income }
\end{array} & \$ 38,000 & \$ 35,000 & \$ 39,000 & \$ 34,000 & \$ 32,000 \\
\hline \begin{array}{r}
\text { Standard } \\
\text { Deviation }
\end{array} & \$ 21,000 & \$ 17,000 & \$ 16,000 & \$ 14,000 & \$ 11,000 \\
\hline
\end{array}
$$
Refer to Exercise $49 .$ Which of the five countries listed has the poorest households (i.e., the lowest cutoff for considering a household poor)?

Key Concepts

Standard Deviation

The standard deviation quantifies the amount of variation or dispersion in a set of data values. Within a normal distribution, it measures how spread out the incomes are around the mean. A larger standard deviation indicates a greater spread of incomes, which is important for understanding the diversity in income levels and determining significant thresholds such as the poverty cutoff.

Percentiles and Cutoff Points

Percentiles rank values within a distribution by indicating the percentage of observations below a given value. In analyzing household income, specific percentiles can be used to set cutoff points for poverty or other income classifications. This concept allows statisticians to determine which households fall below a certain income threshold relative to the overall income distribution.

Normal Distribution

A normal distribution is a continuous probability distribution characterized by its symmetric, bell-shaped curve. It serves as a model for many natural phenomena and is defined by its mean and standard deviation, which determine its center and spread respectively. This distribution is crucial in statistics because many real-world data sets approximate this form, allowing for the application of standard analysis techniques.

Mean

The mean is a measure of central tendency representing the average value in a data set. In the context of income distributions, it provides an overall indication of the typical household income. Its value is central to determining the shape and location of a normal distribution, influencing decisions about cutoff points for various classifications such as poverty.

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