Question

The accompanying data on number of cell phone minutes used in one month are consistent with summary statistics from a marketing study of San Diego residents (Tele-Truth, March, 2009).$$ \begin{array}{rrrrrrrrrr} 189 & 0 & 189 & 177 & 106 & 201 & 0 & 212 & 0 & 306 \\ 0 & 0 & 59 & 224 & 0 & 189 & 142 & 83 & 71 & 165 \\ 236 & 0 & 142 & 236 & 130 & & & & & \end{array} $$ Calculate and interpret the values of the median and the interquartile range.

Short answer

The Median is 142, indicating that 50% of people used 142 minutes or less, while the Interquartile Range is 201, showing that the middle 50% of the data set ranges from 0 minutes to 201 minutes.

Step-by-step solution

Arrange the data in ascending order

The first step involves arranging the data from lowest to highest. It should look like this: 0, 0, 0, 0, 0, 59, 71, 83, 106, 130, 142, 142, 165, 177, 189, 189, 201, 212, 224, 236, 236, 306.

Find the Median

The median is the middle number of the data set when arranged in ascending order. As there are 22 numbers in our data set, an even number, the median is the average of the 11th and 12th numbers. Looking at our arranged data set, those are both 142. So, \( \frac{142 + 142}{2} = 142 \). The median is 142.

Find the First Quartile (Q1) and Third Quartile (Q3)

The First Quartile is the median of the lower half of the data (excluding the median if the number of data points is odd), and the Third Quartile is the median of the upper half of the data. Since we have 22 data points, we take the first 11 for the lower half and the remaining 11 for the upper half for the First and Third Quartiles. The Q1 is the median of the lower half, which is 0 (the sixth data point). The Q3 is the median of the upper half, which is 201 (the 6th data point from the end).

Calculate the Interquartile Range

The interquartile range (IQR) is the range between Q1 and Q3, it measures the statistical spread, being equal to Q3 minus Q1. So, we subtract Q1 from Q3 to get the IQR: \( IQR = Q3 - Q1 = 201 - 0 = 201 \).

Key concepts

Descriptive Statistics

When we talk about descriptive statistics, we focus on summarizing and describing the features of a specific dataset. Consider it as a way to give a quick and simple overview of the data at hand without diving into complex analyses. It's essential for understanding the basic attributes and tendencies in data, which can include measures like the mean (average), median (middle value), mode (most frequent), and range, among others. These measures help to grasp the general pattern within a data set and can involve both graphical representations like charts and numeric summaries, which provide a clear picture of the data for everyone, from researchers to students.

For students starting in data analysis, recognizing these basic statistical measures is the first step in being able to assess data critically. By mastering these concepts, students can move on to more complex analyses, using other statistical tools and methods as their understanding deepens.

Data Analysis

The process of inspecting, cleaning, transforming, and modeling data with the goal of discovering useful information, forming conclusions, and supporting decision-making is known as data analysis. It is a cornerstone of any field that relies on data, which, in our digital era, means pretty much every field you can think of. From businesses using data to better understand their customers to scientists analyzing research results, the techniques and skills of data analysis are widely applied.

For students, learning data analysis involves understanding how to organize data correctly, selecting the appropriate techniques to summarize and interpret it, and, critically, questioning the results to avoid incorrect conclusions. Students learn to spot trends, test hypotheses, and draw actionable insights, which are invaluable skills in virtually any career path they might choose.

Statistical Measures

Delving into statistical measures means looking at the tools we use to describe the characteristics of datasets. These measures are divided into two categories: measures of central tendency and measures of variability (also called measures of spread).

Measures of central tendency include the mean, median, and mode, which tell us about the typical value in a dataset. On the other hand, measures of variability include the range, variance, and standard deviation, which reveal the spread and dispersion of our data.

Understanding the Median

The median is a robust measure of central tendency that is not affected by outliers (unusually high or low values). In the provided exercise, the median is calculated by finding the middle number(s) once the data is ordered. With an even number of data points, the median is the average of the middle two numbers, like in the step-by-step solution, which avoids distortion by outliers.

Importance of Variability

Variability measures are just as important; they help us understand how diverse or uniform the data is, which can impact conclusions and decision-making. By anchoring to these concepts, students can interpret data sets more critically, and understand not just 'average' behavior, but the spread of data which often matters just as much.

Interquartile Range (IQR)

The interquartile range (IQR) is a measure of statistical spread and is part of the more comprehensive set of tools used in descriptive statistics. It is specifically designed to focus on the middle portion of the dataset. IQR is the difference between the third quartile (Q3) and the first quartile (Q1) of a dataset and thus represents the range within which the central 50% of the data lies.

Calculating the IQR helps to mitigate the effect of outliers or extreme values, as it focuses only on the data in the middle. The larger the IQR, the more spread out the middle values are; conversely, a smaller IQR indicates a more tightly clustered set of middle values. This is significant because these measures can drastically change the interpretation of the data's spread and variability.

Applying IQR in Exercises

In the example given, the IQR calculation was straightforward: subtract Q1 from Q3. The resulting IQR tells us that although the data includes users with zero minutes, the typical user (falling within the middle 50%) uses between 0 and 201 minutes on their cell phone. Understanding how to calculate and interpret the IQR can give students a critical advantage in analyzing data sets, allowing them to communicate their findings with greater clarity and precision.