Problem 35 - Algebra 2

Question

Make a box-and-whisker plot for each set of values. $$ 25,25,30,35,45,45,50,55,60,60 $$

Step-by-Step Solution

Answer

The box-and-whisker plot for this set of data has minimum=25, first quartile (Q1)=27.5, median=45, third quartile (Q3)=57.5 and maximum=60.

Step 1: Arrange the data in ascending order

Firstly, check if the values are arranged in ascending order or not. If not, arrange so. In this case, the set of values, 25,25,30,35,45,45,50,55,60,60, is already sorted.

Step 2: Determine minimum, maximum and median

The minimum value is 25, the maximum value is 60 and the median (middle value when data is arranged in ascending order) can be found using the formula \(\frac{n+1}{2}\), where \(n\) is the total number of data points. Here the 5.5th data point is the median, which is the average of 5th and 6th data, i.e., \(\frac{45 + 45}{2} = 45\).

Step 3: Determine first quartile and third quartile

To find the first quartile \(Q_1\) (25th percentile) split the data to two halves before the median (for below median data), and do the same for third quartile \(Q_3\) (75th percentile) for data after the median. Applying the median formula again, for \(Q_1\), the 2.5th value is the median of the lower half, which is the average of 2nd and 3rd data, i.e., \(\frac{25 + 30}{2} = 27.5\), and for \(Q_3\), the 2.5th value from the upper half, i.e., the average of 55 and 60, hence \(Q_3 = \frac{55 + 60}{2} = 57.5\).

Step 4: Create the box-and-whisker plot

Finally, create the box-and-whisker plot. Draw a number line and plot the values obtained above. Draw vertical lines at \(Q_1\), the median, and \(Q_3\). Then join the three lines to form a box and draw lines from the box to the minimum and maximum values to create whiskers.

Key Concepts

Understanding Statistics

Statistics involves collecting, analyzing, interpreting, and presenting data. It allows us to make informed decisions based on data. One key function of statistics is to summarize data sets in ways that make them easier to understand. A box-and-whisker plot is one such method. This visual summary illustrates the distribution, central value, and variability of a data set at a glance. In our exercise, the box-and-whisker plot helps depict the given set of numbers and highlights its quartiles and median. Understanding statistical methods like this is crucial because it helps translate numeric data into meaningful insights.

The Role of Quartiles

Quartiles divide a data set into four equal parts. This helps in understanding the spread and center of the data. The key quartiles are:

• First Quartile \(Q_1\): The value below which 25% of the data falls. It marks the lower 25th percentile.
• Median: Also known as the second quartile, it splits the data in half.
• Third Quartile \(Q_3\): The value below which 75% of the data falls. It is the 75th percentile.

Finding the quartiles involves understanding the order and position of data points. In our problem, we calculated \(Q_1\) as 27.5 and \(Q_3\) as 57.5. These quartiles are pivotal as they demonstrate the distribution of the lower and upper halves of the data.

The Power of Data Visualization

Data visualization uses graphic representation to make data easy to understand. Box-and-whisker plots, in particular, offer a clean visual summary of data distributions. They are beneficial because:

• They display the range and central tendency of the data.
• They highlight outliers, quartiles, and spread within data.
• They are useful for directly comparing multiple data sets.

By utilizing the box-and-whisker plot, students and analysts gain insights into where most of the values lie and how they deviate from each other. For our exercise, creating the plot involved marking minimum, maximum, median, and quartile points on a number line.

Understanding the Median

The median is a measure of central tendency that divides a data set into two equal halves. It is the middle number when values are arranged in order, and it is often a better representative of central tendency than the mean in skewed distributions. In our example, the median value was found to be 45, dividing the set into two equal parts. Understanding the median is crucial because it is not influenced by extreme values or outliers. It tells us the central value of the data set, ensuring an unbiased central location. Calculating the median requires ordering the data, and when the number of data points is even, it is the average of the two middle numbers.