Question

Use the table that lists the weights (in ounces) and operating times (in minutes) for 10 cellular telephones. $$\begin{array}{|c|c|}\hline \text { Wompor } & \text { Caresesum } \\\\\hline 8.8 & 140 \\\\\hline 7.7 & 80 \\\\\hline 4.8 & 90 \\\\\hline 7.0 & 90 \\\\\hline 7.1 & 90 \\\\\hline 8.4 & 70 \\\\\hline 3.8 & 60 \\\\\hline 3.6 & 260 \\\\\hline 6.7 & 75 \\\\\hline 6.2 & 135 \\\\\hline\end{array}$$ Draw a box-and-whisker plot of the operating times of the phones.

Short answer

The box-and-whisker plot will have in the following sequence from left to right: the minimum value of 60, the Q1 value of 75, the median value of 90, the Q3 value of 135, and the maximum value of 260. The whiskers are drawn from the Q1 box to 60 and from the Q3 box to 260.

Step-by-step solution

Identify and arrange the values

The first step is to identify all the operating times given in the table and arrange them in ascending order. The operating times are 140, 80, 90, 90, 90, 70, 60, 260, 75, and 135. In ascending order, they are 60, 70, 75, 80, 90, 90, 90, 135, 140, and 260.

Calculate the median

The median is the middle value when all values are arranged in ascending order. This is also known as the second quartile or Q2. For this data, since there are 10 observations (an even number), the median would be the average of the 5th and 6th observations. i.e. (90+90)/2 = 90.

Calculate the first and third quartiles

The first quartile Q1 is the median of the lower half of the data and the third quartile Q3 is the median of the upper half of the data (do not include the overall median (Q2) in these calculations, if it is one of the data points). For Q1, the lower half of the data includes the points 60, 70, 75, 80, and 90 (first 5 points). The median of these is 75. For Q3, the upper half of the data includes the points 90, 90, 135, 140, and 260 (last 5 points). The median of these is 135.

Identify the minimum and maximum values

The minimum value in the data is the smallest operating time, which is 60. The maximum value in the data is the largest operating time, which is 260.

Draw the box-and-whisker plot

First, draw a number line that fits all the data points. Next, place a vertical line at each of the five values calculated: the minimum, Q1, median (Q2), Q3, and maximum. Draw a box that stretches from Q1 to Q3, and draw 'whiskers' or lines that extend from the box to the minimum and maximum.

Key concepts

Data Representation

Data representation is crucial for understanding complex information easily. In the case of the operating times for cellular telephones, a box-and-whisker plot provides a visual summary that showcases the data's distribution at a glance. Instead of sifting through a table of numerical values, a box-and-whisker plot translates these numbers into a graphic consisting of a 'box' and 'whiskers'. The box encapsulates the central 50% of the data – from the first quartile (Q1) to the third quartile (Q3) – indicating where the bulk of the values lie. The 'whiskers' stretch out to the extreme data points – in this exercise, the minimum and maximum operating times – offering a visual representation of the range of the dataset. This kind of plot is particularly useful when you want to identify outliers or understand the spread and skewness of the data.

Quartiles

Quartiles divide a dataset into four equal parts, providing a deeper understanding of the distribution. If you picture the data spread across a number line, quartiles segment it into quarters:

• The first quartile (Q1) is the median of the lower half and marks the 25th percentile.
• The second quartile (Q2) is the median of the entire set and is synonymous with the 50th percentile.
• The third quartile (Q3) is the median of the upper half and corresponds to the 75th percentile.

In our exercise, after arranging the operating times in ascending order, we calculated Q1 to be 75 and Q3 to be 135. These values indicate where the lower 25% and upper 25% of our data points reside, respectively, with Q1 at the lower edge of the 'box' and Q3 at the upper edge in the box-and-whisker plot.

Median

The median is a measure of central tendency that divides a dataset in half and is unaffected by extreme values, hence considered a robust summary statistic. To find the median, data points must be ordered from smallest to largest. If the count of data points is odd, the median is the middle value. Otherwise, it's the average of the two middle values.In our example with 10 data points, the median (Q2) is the average of the fifth and sixth values after sorting the data, resulting in a median operating time of 90 minutes. The median can be particularly telling in skewed datasets, offering a truer center point than the mean, which can be distorted by outliers.

Statistics in Algebra

The field of algebra frequently incorporates statistics to solve problems involving data analysis. Calculating measures like median and quartiles involves basic algebraic concepts such as order of operations, operations with fractions, and averaging numbers. These concepts underpin the creation of a box-and-whisker plot, a practical application of both algebra and statistics working in tandem.By using algebraic techniques to calculate quartiles and median, we can then construct a graphical representation of data that helps identify patterns, outliers, and the spread of the dataset. This synergy of algebra and statistics equips students with the tools necessary to analyze and interpret real-world data sets effectively.