The percentage of foreign-born population for each of the 50...

The percentage of foreign-born population for each of the 50 states is represented here. Find the mean and modal class for the data. Do you think the mean is the best average for this set of data? Explain. Percentage$\quad$ Frequency $\begin{array}{ccc}{0.8-4.4} & {} {26} \\ {4.5-8.1} & {} {11} \\ {8.2-11.8} & {4} & {} \\ {11.9-15.5} & {5} & {} \\ {15.6-19.2} & {2} & {} \\ {19.3-22.9} & {1} \\ {23.0-26.6} & {1}\end{array}$

The percentage of foreign-born population for each of the 50 states is represented here. Find the mean and modal class for the data. Do you think the mean is the best average for this set of data? Explain.
Percentage$\quad$ Frequency
$\begin{array}{ccc}{0.8-4.4} & {} {26} \\ {4.5-8.1} & {} {11} \\ {8.2-11.8} & {4} & {} \\ {11.9-15.5} & {5} & {} \\ {15.6-19.2} & {2} & {} \\ {19.3-22.9} & {1} \\ {23.0-26.6} & {1}\end{array}$

Key Concepts

Frequency Distribution

A frequency distribution is a tabular or graphical representation that shows how often each different value in a set of data occurs. It organizes raw data into classes or intervals, making it easier to understand the overall pattern and distribution of the data.

Grouped Data

Grouped data refers to data that have been organized into intervals or classes rather than listed as individual data points. This helps in summarizing large data sets, but it requires using class midpoints and frequencies for further calculations, such as estimating the mean.

Mean (Arithmetic Average) for Grouped Data

The arithmetic mean for grouped data is estimated by multiplying the midpoint of each class interval by the frequency of that interval, summing these products, and then dividing by the total number of observations. This estimate serves as a measure of central tendency and provides an overall summary statistic for the data.

Modal Class

The modal class is the class interval in a frequency distribution that contains the highest frequency. It represents the most frequently occurring range of values in the dataset and indicates where the peak of the distribution is located.

Measures of Central Tendency and Their Appropriateness

Measures of central tendency, such as the mean, median, and mode, summarize a set of data by identifying the central position within that dataset. Depending on the distribution of the data, some measures may better represent the dataset than others. For instance, if the data are skewed or have outliers, the mean may be less representative than the median or mode, which are more robust to extreme values.