Question

Each describe a sample. The information given includes the five number summary, the sample size, and the largest and smallest data values in the tails of the distribution. In each case: (a) Clearly identify any outliers, using the IQR method. (b) Draw a boxplot. Five number summary: (42,72,78,80,99)\(;\) \(n=120 .\) Tails: 42, 63, \(65,67,68, \ldots, 88,89,95,96,99\).

Short answer

The outliers identified using the IQR method are: 42, 95, 96, 99. The boxplot would have a box from 72 to 80, with a median line at 78, whiskers extending from 63 to 89, and individual marks for the outliers.

Step-by-step solution

Calculate the Interquartile Range (IQR)

The IQR is essentially the range of the middle 50% of the data. It is calculated as the difference between the third quartile (Q3) and the first quartile (Q1). From the five number summary, Q3 = 80 and Q1 = 72. So, IQR = Q3 - Q1 = 80 - 72 = 8.

Determine Outliers

Using the IQR rule for outliers, any data point is a potential outlier if it is below Q1 - 1.5*IQR or above Q3 + 1.5*IQR. By calculating these numbers we get: 72 - 1.5*8 = 60 and 80 + 1.5*8 = 92. So any data point below the value of 60 or above the value of 92 is an outlier. Looking at our data set, 42 is an outlier because it is below 60. From the higher side, 95, 96, and 99 are outliers because they're above 92.

Construct the Boxplot

The boxplot should have a box extending from Q1 to Q3 (from 72 to 80), with a line in the box marking the median (78). One 'whisker' will extend from the box down to the smallest non-outlier value (63), while the other 'whisker' will extend from the box up to the largest non-outlier value (89). Individual dots or marks will be made to illustrate the outliers which are 42, 95, 96, 99 in this case.

Key concepts

Interquartile Range (IQR)

The Interquartile Range (IQR) is a crucial measure of variability in a dataset, particularly in understanding its central tendency. It defines the range within which the middle 50% of data lies. To find the IQR, one must subtract the first quartile (Q1) from the third quartile (Q3), which means it's the difference between the 75th percentile and the 25th percentile of the data.
For instance, using the provided five-number summary of (42,72,78,80,99), the Q1 value is 72, and the Q3 value is 80. The IQR is then calculated as 80 minus 72, yielding an IQR of 8. This metric becomes particularly useful for detecting outliers and understanding the spread of the central portion of the data, without being influenced by extreme values. An understanding of the IQR is essential for various applications, including summary statistics and data visualization.

Outliers Detection

Outliers are data points that differ significantly from other observations, and their detection is critical as they can distort statistical analyses. The IQR is a fundamental tool for identifying outliers.
In the Boxplot method, outliers are generally considered to be any data points that fall more than 1.5 times the IQR below the first quartile or above the third quartile. To put it in perspective, for the dataset with an IQR of 8, outliers would be any values below (72 - 1.5*8) = 60 or above (80 + 1.5*8) = 92.
In our exercise, 42 is an outlier on the low end, and 95, 96, and 99 are outliers on the high end. Identifying these helps in analyzing the data more accurately, ensuring that the resultant statistics are not skewed by these anomalous values.

Five Number Summary

The five-number summary is a concise statistical snapshot that describes the spread and center of a dataset. It consists of five values: the minimum value, first quartile (Q1), median, third quartile (Q3), and the maximum value.
The summary for the given sample is (42,72,78,80,99), which represents:

• Minimum (smallest value): 42
• Q1 (25th percentile): 72
• Median (50th percentile): 78
• Q3 (75th percentile): 80
• Maximum (largest value): 99

Using the five-number summary, one can quickly get a feel for the distribution of data and understand where most of the values lie. It is fundamental in creating a boxplot, which is a graphical representation of this summary.

Data Visualization

Data visualization is a powerful way to communicate complex information quickly and effectively. A boxplot is a standardized way of displaying the distribution of data based on the five-number summary. It helps to visualize the data's spread, central tendency, and identify outliers at a glance.
The boxplot for our exercise includes a box from Q1 (72) to Q3 (80) with a line at the median (78). It has 'whiskers' that extend to the smallest and largest non-outlier values, and outliers marked as individual points. Data visualization like this makes it easier to understand and interpret complex data, aiding in decision-making and providing insight into the statistical nature of the data presented.