Day 52 – Box-and-Whisker

Vocabulary The Median is the middle number of a set of data when the numbers are arranged in numerical order. The Range of a set of data is the difference between the highest and lowest values in the set. The Box-and-Whisker Plot is a graphic way to display the median, quartiles, and extremes of a data set on a number line to show the distribution of the data.

Vocabulary The Upper Quartile is the median of the upper half of a data set. This is located by dividing the data set with the median, and then dividing the upper half that remains with the median again, this median of the upper half being the upper quartile. The Lower Quartile value is the median of the lower half of the data. The upper quartile value is the median of the upper half of the data.

Key Concept Box-and-whisker plots are a method for representing large sets of data so that they can be analyzed and compared. For example, a box- and-whisker plot will allow you to quickly identify key statistical measures (median and range) and major concentrations of data for a set with a large number of data points. One can make inferences about multiple sets of data and compare sets of data over time more efficiently and with greater ease using box-and-whisker plots than when using tables or other graphical representations.

How to make a boxplot A boxplot is a one-dimensional graph of numerical data based on the five-number summary. This summary includes the following statistics: the minimum value, the 25th percentile (known as Q1), the median, the 75th percentile (Q3), and the maximum value. In essence, these five descriptive statistics divide the data set into four parts, where each part contains 25% of the data. To make a boxplot, follow these steps: 1. Find the five-number summary of your data set: The minimum is the smallest value in the dataset, and the maximum is the largest value in the data set. Use the following steps to find the 25th percentile (known as Q1), the 50th percentile (the median), and the 75th percentile (Q3).

Order all the values in the data set from smallest to largest. Multiply k percent times the total number of values in the data, n. The result is known as the index. 3. If the index obtained in Step 2 isn't a whole number, round it up to the nearest whole number and go to Step 4a. If index obtained in Step 2 is a whole number, go to Step 4b. 4. Choose one of the following. a. Count the values in your data set from left to right (from the smallest to the largest value) until you reach the number indicated by Step 3. The corresponding value in your data set is the kth percentile. b. Count the values in your data set from left to right (smallest to largest) until you reach the number indicated by Step 2. The kth percentile is the average of that corresponding value in your data set and the value that directly follows it.

Create a vertical (or horizontal) number line whose scale includes the values in the five-number summary and uses appropriate units of equal distance from each other. Mark the location of each value in the five-number summary just above the number line (for a horizontal boxplot) or just to the right of the number line (for a vertical boxplot). Draw a box around the marks for the 25th percentile and the 75th percentile. Draw a line in the box where the median is located. Determine whether or not outliers are present. To make this determination, calculate the Interquartile Range (IQR), which is found by subtracting Q3 – Q1; then multiply IQR by 1.5. Add this amount to the value of Q3 and subtract this amount from Q1. This gives you a wider boundary around the median than the box does. Any data points that fall outside this boundary are determined to be outliers.

Boxplot of Best Actress ages (1928–2009; n = 83 actresses). Descriptive Statistics for Best Actress ages (1928–2009).

Boxplots can be vertical (straight up and down) with the values on the axis going from bottom (lowest) to top (highest); or they can be horizontal, with the values on the axis going from left (lowest) to the right (highest). REMEMBER: The steps are shown here demonstrate one way of calculating the median and quartiles of the five-number summary and of constructing the boxplot. But there are several other acceptable methods. Do not be too alarmed if your calculator or a friend gives you a boxplot close to but different from what these steps would give.

Example 1. Read, identify and interpret the key components of the box-and whisker plot below. (Use back of page to record any interpretations of the box-and-whisker plot).   2. Identify a set of data containing 12 numbers that would create the box-and-whisker plot in question 1. 3. Create a box-and-whisker plot for the following set of data. 20, 15, 45, 33, 19, 30, 31, 32, 31, 30, 27, 34, 50, 22, 29, 30

Answer key 1. Read, identify and interpret the key components of the box-and whisker plot below.   2. Identify a set of data containing 12 numbers that would create the box-and-whisker plot in question 1. Answers will vary for this question.

Answer key 3. Create a Box-and-Whisker plot for the following set of data.   20, 15, 45, 33, 19, 30, 31, 32, 31, 30, 27, 34, 50, 22, 29, 30

Try it on yourself 1. Draw a box plot for the following set of data. Remember to order the data first, if necessary. 1, 0, 3, 2, 1, 1, 7, 8, 6, 6, 7, 7 2. Draw a box plot for the following set of data. Remember to order the data first, if necessary. 4.7, 3.8, 3.9, 3.9, 4.6, 4.5, 5 3. Draw a box plot for the following set of data. Remember to order the data first, if necessary. 90, 77, 79, 60, 87, 87, 80, 80, 83

Try it on yourself (answers) 1. Draw a box plot for the following set of data. Remember to order the data first, if necessary. 1, 0, 3, 2, 1, 1, 7, 8, 6, 6, 7, 7

Try it on yourself (answers) 2. Draw a box plot for the following set of data. Remember to order the data first, if necessary. 4.7, 3.8, 3.9, 3.9, 4.6, 4.5, 5

Try it on yourself (answers) 3. Draw a box plot for the following set of data. Remember to order the data first, if necessary. 90, 77, 79, 60, 87, 87, 80, 80, 83