[DATA] Old Faithful In Problem 26 from Section 3.1, we drew a histogram of the length of eruption of California’s Old Faithful geyser and found that the distribution is symmetric. Draw a boxplot of these data. Use the boxplot and quartiles to confirm the distribution is symmetric. For convenience, the data are displayed again.

Verified step by step guidance

1. Step 1: Organize the data in ascending order to facilitate finding the quartiles and median. This will help in constructing the boxplot accurately.
2. Step 2: Calculate the median (Q2) of the data set. The median is the middle value when the data is ordered, or the average of the two middle values if there is an even number of observations.
3. Step 3: Determine the first quartile (Q1) and third quartile (Q3). Q1 is the median of the lower half of the data (below the overall median), and Q3 is the median of the upper half of the data (above the overall median).
4. Step 4: Draw the boxplot using the five-number summary: minimum, Q1, median (Q2), Q3, and maximum. The box extends from Q1 to Q3 with a line at the median, and whiskers extend to the minimum and maximum values.
5. Step 5: Analyze the boxplot to confirm symmetry by checking if the median is approximately centered in the box and if the whiskers are roughly equal in length on both sides. This visual check, along with the quartile values, will confirm if the distribution is symmetric.

Key Concepts

Boxplot Construction

A boxplot visually summarizes data using five key values: minimum, first quartile (Q1), median (Q2), third quartile (Q3), and maximum. It displays the spread and central tendency, highlighting the interquartile range (IQR) and potential outliers. Drawing a boxplot involves calculating these statistics and plotting them on a number line.

Quartiles and Median

Quartiles divide ordered data into four equal parts. The median (Q2) is the middle value, while Q1 and Q3 mark the 25th and 75th percentiles, respectively. These values help describe the data’s center and spread, and their relative positions indicate the symmetry or skewness of the distribution.

Symmetry in Data Distribution

A distribution is symmetric if its left and right sides are mirror images. In a boxplot, symmetry is suggested when the median is centered between Q1 and Q3, and the whiskers (min to Q1 and Q3 to max) are roughly equal in length. This confirms the histogram’s indication of a symmetric distribution.