Question

A mobile phone company examines the ages of 150 customers to start special plans for them. Consider frequency table shown in Table \(3.15\). 1\. Draw the histogram for the data. 2\. Estimate the mean age for these policyholders. 3\. Estimate the median age for these policyholders. $$ \begin{array}{l} \text { Table 3.15 Frequency table for Problem } 3.6\\\ \begin{array}{l|l|l|l|l|l|l} \hline \text { Age(years) } & 0-14 & 15-19 & 20-29 & 30-39 & 40-49 & 50-79 \\ \hline \text { Frequency } & 14 & 40 & 28 & 27 & 24 & 17 \\ \hline \end{array} \end{array} $$

Short answer

The histogram can be drawn where the x-axis represents the age groups, and the y-axis represents the frequencies.The estimated mean age is the sum of the product of mid-point of each age group and their corresponding frequency divided by total number of observations.The estimated median age is the age group where we reach the middle when frequencies are counted from the lowest age group.

Step-by-step solution

Draw the Histogram

A histogram can be used to graphically summarize and display the distribution of a data set. The x-axis represents the age groups, and the y-axis represents the frequencies. The histogram can be drawn using any data visualization tool where each age group is a bar and the height of the bar equals its frequency.

Estimate the Mean Age

Mean (average) can be estimated by summing the product of mid-point of each age group and their corresponding frequency divided by total number of observations. The mid-point of each age group can be calculated by averaging the lower and upper bounds of each group. For example, the mid-point of 0-14 age group is (0+14)/2=7.

Estimate the Median Age

Median is the value separating the higher half from the lower half of a data sample. In other words, half of the observations are below it and half are above it. The median of a frequency table can be found by counting the frequencies starting from lowest age group until we reach the middle of the total, which is the sum of the frequencies. The age group where we reach the middle is the median group.

Key concepts

Histogram

A histogram is a type of graph used to represent the distribution of numerical data. It differs from a bar chart in that it shows the frequency of data points in consecutive, non-overlapping intervals. To create a histogram, one needs to divide the entire range of values into a series of intervals and then count how many values fall into each interval.

The intervals are known as 'bins' and they are usually of equal size. Each bin is represented as a bar on the graph where the height of the bar is the frequency - the number of data points that fall within the bin's range. When dealing with continuous data like age, histograms are particularly useful as they can easily show the data distribution and give a clear visual of the central tendency, variability, and the shape of the data distribution.

Estimating Mean Age

The mean, commonly referred to as the average, is a measure of the center of a data set. Estimating the mean age from a frequency table involves calculating a weighted average, where each age's impact on the mean is proportional to its frequency.

This is done by determining the midpoint of each age group (the average of the lower and upper bounds) and multiplying it by the frequency of the age group. These products are summed and then divided by the total number of observations. Although this is an estimate, as it assumes the ages are evenly distributed within each group, it gives a reasonable approximation of the true mean when precise age data is unavailable.

Estimating Median Age

The median age is another measure of central tendency and is defined as the value which divides the data set into two equal parts; half the data points are above the median and half are below. To estimate the median age from a histogram or frequency table, the cumulative frequency is used.

One needs to add the frequencies from the lowest age group up, until reaching the middle point of the total number of observations (which is half of the sum of the frequencies). The age group where this middle value falls is the median age group. The exact age within the group is estimated by interpolation or assuming the middle of that range.

Frequency Distribution

A frequency distribution is a summary of how often different values occur within a data set. A frequency table is a simple way to display the frequency distribution and lists the number of occurrences for each category or group of data.

For example, a frequency distribution of ages would list various age groups alongside the number of people who fall into each group. This data representation is crucial for statistical analysis as it helps in understanding how data is dispersed across various categories and is often the precursor to further statistical calculations and visualizations such as histograms.

Data Visualization

Data visualization encompasses a broad range of techniques used to communicate data clearly and effectively through graphical means. This includes tools like charts, graphs, maps, and more sophisticated interactive visualizations.

For statistics education, visualizing data helps to convey complex information in a way that can be understood quickly and intuitively. Effective data visualization aids in the analysis of data, revealing patterns, trends, outliers, and unexpected insights that may not be apparent from raw data or tables. It is often used in tandem with statistical measures, such as the mean and median, as part of a comprehensive approach to data analysis.