According to the box plot obtained by MATLAB, answer the following questions: (8 points) 1. Find the lower, upper quartiles, and median. 2. Calculate the interquartile range. 3. Find the lower and upper fences. 4. Determine outlier(s) if any. 5. Interpret the box plot.
The Correct Answer and Explanation is:
Of course, here are the answers and a detailed explanation based on the standard features of a MATLAB box plot.
Answers
- Quartiles and Median: The lower quartile (Q1) is the value at the bottom edge of the central box. The upper quartile (Q3) is the value at the top edge of the central box. The median is the value indicated by the horizontal line inside the box.
- Interquartile Range (IQR): The IQR is calculated by subtracting the lower quartile from the upper quartile (IQR = Q3 – Q1).
- Lower and Upper Fences: The lower fence is calculated as Q1 – 1.5 * IQR. The upper fence is calculated as Q3 + 1.5 * IQR. These are not typically drawn on the plot.
- Outlier(s): Outliers are the individual data points plotted separately beyond the whiskers, usually marked with a red plus sign (+) or another symbol.
Explanation
A box plot provides a concise visual summary of a dataset’s distribution. To interpret it correctly, you must understand its components.
First, the central box itself represents the middle 50 percent of your data. The bottom line of this box is the first quartile, or Q1, which is the 25th percentile. The top line of the box is the third quartile, or Q3, representing the 75th percentile. The line inside the box indicates the median, or the 50th percentile, which is the central value of the dataset.
Second, the interquartile range, or IQR, is a measure of statistical dispersion. You calculate it by taking the value of Q3 and subtracting the value of Q1. A larger IQR, which corresponds to a taller box, signifies greater variability in the middle half of the data.
Third, the fences are theoretical boundaries used to identify outliers. They are calculated using the IQR. The lower fence is found by subtracting 1.5 times the IQR from Q1, and the upper fence is found by adding 1.5 times the IQR to Q3. Any data point that falls outside these calculated fences is considered an outlier.
Fourth, the “whiskers” are the lines extending from the top and bottom of the box. In MATLAB, these whiskers extend to the most extreme data points that are not considered outliers, meaning they are the furthest points still inside the fences. Any data points lying beyond the whiskers are flagged as outliers and are plotted as individual markers.
Finally, interpreting the plot involves synthesizing this information. The position of the median within the box suggests skewness; a median closer to the bottom suggests a right skew, while one closer to the top suggests a left skew. The lengths of the whiskers provide a similar clue. The presence of outliers highlights extreme values that may require further investigation.
