Consider the boxplot below. boxplot with five point summary: 43,58,61,67,74 a. What quarter has the smallest spread of data? Fourth First Second Third b. What is that spread? c. What quarter has the largest spread of data? Second Fourth Third
Consider the boxplot below. boxplot with five point summary: 43,58,61,67,74 a. What quarter has the smallest spread of data? Fourth First Second Third b. What is that spread? c. What quarter has the largest spread of data? Second Fourth Third First d. What is that spread? e. Find the Inter Quartile Range (IQR): f. Which interval has the most data in it? 43 – 57 75 – 200 57 – 61 67 – 74 g. What value could represent the 52nd percentile? 45 52 63 70
The Correct Answer and Explanation is:
Given the five-number summary of the boxplot: 43 (min), 58 (Q1), 61 (median/Q2), 67 (Q3), 74 (max), we can answer each question systematically.
a. What quarter has the smallest spread of data?
Answer: Second (Q2 to Q3)
b. What is that spread?
Answer: 67 – 61 = 6
c. What quarter has the largest spread of data?
Answer: Fourth (Q3 to max)
d. What is that spread?
Answer: 74 – 67 = 7
e. Find the Interquartile Range (IQR):
Answer: Q3 – Q1 = 67 – 58 = 9
f. Which interval has the most data in it?
Answer: 57 – 61
g. What value could represent the 52nd percentile?
Answer: 63
🔍 Explanation (300+ words):
A boxplot provides a graphical summary of data using five key numbers: minimum, Q1, median (Q2), Q3, and maximum. These divide the data into four quarters (or quartiles), with each quarter containing 25% of the data. The distance between the quartiles indicates how spread out the data is in each section.
a–b. Smallest Spread of Data (Q2 to Q3)
The second quarter spans from Q2 (61) to Q3 (67), giving a spread of 6. This is the narrowest range among the quarters, meaning values are most tightly packed here. Since each quarter contains 25% of the data regardless of the range, a smaller spread means more clustered data points.
c–d. Largest Spread of Data (Q3 to Max)
The fourth quarter spans from 67 to 74, a spread of 7, which is the largest among the quartiles. This indicates that the upper 25% of values are more dispersed or variable than in other quarters.
e. Interquartile Range (IQR)
The IQR is a measure of statistical dispersion and is calculated as Q3 – Q1 = 67 – 58 = 9. It reflects the range where the middle 50% of the data lies and is important for detecting outliers and variability.
f. Interval with Most Data
Although all quartiles have 25% of data, the interval 57–61 (approximately Q1 to Q2) is the smallest among choices. This suggests data is most concentrated here (narrow interval with 25% of data), making it the interval with most density.
g. 52nd Percentile Estimate
The 52nd percentile lies just above the median (50th percentile). Since Q2 = 61 and Q3 = 67, a value slightly above 61 is appropriate. Among the choices, 63 fits best as a likely value for the 52nd percentile.
Thus, analyzing spread, concentration, and percentiles through the five-number summary and boxplot helps us better understand the dataset’s distribution and central tendencies.