Question

The following boxplot shows the five-number summary for a data set. For these data the minimum is \(35,\) \(Q_{1}\) is \(42,\) the median is \(49, Q_{3}\) is \(56,\) and the maximum is 65 . Is it possible that no observation in the data set equals 42? Explain your answer.

Short answer

It is not guaranteed but unlikely that no observation equals 42 since Q1 is commonly an actual data point, indicating that 25% of the data is less than or equal to 42.

Step-by-step solution

Understanding the Boxplot and Five-Number Summary

To understand if an observation equals 42, interpret the boxplot and the five-number summary. In the boxplot, the minimum, first quartile (Q1), median, third quartile (Q3), and maximum represent the most extreme and the three middle-most values of the data set, respectively.

Analyzing the First Quartile (Q1)

The first quartile (Q1) is the median of the lower half of the dataset (excluding the median if the number of observations is odd). It represents the value below which 25% of the data fall. In this dataset, Q1 is given as 42.

Determining the Possibility of Observations Equaling Q1

Since Q1 is 42, it implies that at least 25% of the values are less than or equal to 42. It does not guarantee that any of the data points are exactly 42, but it is a point where 25% of the data is less or equal to it. Without more specific information about the distribution of the data points, it is not possible to determine if any observation equals 42. Still, it is very common that Q1 is indeed a part of the dataset unless explicitly stated otherwise, so it quite possible that at least one observation equals 42.

Key concepts

Five-Number Summary

The five-number summary is a concise statistical overview that gives a quick insight into the spread and center of a dataset. It consists of five values: the minimum, the first quartile (Q1), the median, the third quartile (Q3), and the maximum. These numbers divide the data into four equal parts, and each part represents a quarter of the dataset.

Understanding the five-number summary can help in comprehending the distribution of the data at a glance. The difference between Q1 and Q3 (interquartile range) is particularly useful in gauging the dataset's variability. Also, by examining the spread between the minimum and maximum, you can get an idea of the range of the data. For instance, in our exercise, the minimum value of 35 and maximum of 65 illustrate that the data points are spread within a range of 30 units.

First Quartile (Q1)

The first quartile (Q1) represents the 25th percentile of the data set, which means 25% of the data values are less than or equal to this number. Q1 is found by dividing the lower half of a data set (below the median) into two equal parts to find the median of this lower half.

Q1 is not just a boundary line; it's a key figure in understanding the dispersion of the lower part of the dataset. In the presented boxplot, Q1 is 42, suggesting that a quarter of the observations are at most 42. While this doesn't confirm the existence of an observation with that exact value, it's often the case that Q1 could be an actual data point, especially in a discrete data set with repeated values.

Data Set Interpretation

Interpreting a dataset involves looking beyond the numbers to understand what they mean in the real-world context. It starts with summary statistics like the five-number summary but requires further analysis to draw accurate conclusions.

An important aspect of data set interpretation is considering what values are likely within the dataset. For example, knowing that the first quartile is 42 tells us about the lower segment of the data distribution, but without additional information or context, assumptions should be made cautiously. In our exercise, it's possible that no observation equals 42, but generally, the first quartile is a common value in the dataset. Understanding the context and the specific details of data collection is crucial for accurate interpretation.

Statistical Distribution

Statistical distribution refers to the way in which data points are spread out or clustered over a range of values. It can be represented graphically, like in a boxplot, and is essential for understanding the overall shape and spread of the data.

When examining the statistical distribution of a data set, key features to observe include symmetry, skewness, and the presence of outliers. For instance, if the distance between Q1 and the median is not the same as the distance between the median and Q3, this could indicate a skew in the data. In the exercise's boxplot, the symmetric distribution of the quartiles might suggest that the data is fairly evenly spread out, but further analysis (like a histogram or a kernel density plot) would be needed to confirm this. Always remember, the visual representation is a simplified overview and might not capture every detail of the underlying distribution.