Question

Environmental scientists are increasingly concerned with the accumulation of toxic elements in marine mammals and the transfer of such elements to the animals' offspring. The striped dolphin (Stenella coeruleoalba), considered to be the top predator in the marine food chain, was the subject of one such study. The mercury concentrations (micrograms/gram) in the livers of 28 male striped dolphins were as follows: \begin{array}{rrrr} 1.70 & 183.00 & 221.00 & 286.00 \\ 1.72 & 168.00 & 406.00 & 315.00 \\ 8.80 & 218.00 & 252.00 & 241.00 \\ 5.90 & 180.00 & 329.00 & 397.00 \\ 101.00 & 264.00 & 316.00 & 209.00 \\ 85.40 & 481.00 & 445.00 & 314.00 \\ 118.00 & 485.00 & 278.00 & 318.00 \end{array} a. Calculate the five-number summary for the data. b. Construct a box plot for the data. c. Are there any outliers? d. If you knew that the first four dolphins were all less than 3 years old, while all the others were more than 8 years old, would this information help explain the difference in the magnitude of those four observations? Explain.

Short answer

#Question# Calculate the five-number summary for the data. #Answer# The five-number summary for the data is (1.70, 143.0, 246.5, 315.5, 485.0).

Step-by-step solution

Organize the data

First, we will organize the data in ascending order: \begin{array}{rrrrrrrr} 1.70 & 1.72 & 5.90 & 8.80 & 85.4 & 101.0 & 118.0 & 168.0 \\\ 180.0 & 183.0 & 209.0 & 218.0 & 221.0 & 241.0 & 252.0 & 264.0 \\\ 278.0 & 286.0 & 314.0 & 315.0 & 316.0 & 318.0 & 329.0 & 397.0 \\\ 406.0 & 445.0 & 481.0 & 485.0 \end{array}

Calculate the five-number summary

The five-number summary consists of the minimum, first quartile (Q1), median, third quartile (Q3), and the maximum. Here's how to calculate each: Minimum: 1.70 Maximum: 485.00 Median (Q2): Since we have 28 data points (an even number), the median will be the average of the 14th and 15th data points: (241 + 252) / 2 = 246.5 First quartile (Q1): The average of the 7th and 8th data points: (118.0 + 168.0) / 2 = 143.0 Third quartile (Q3): The average of the 21st and 22nd data points: (315.0 + 316.0) / 2 = 315.5 Five-number summary: (1.70, 143.0, 246.5, 315.5, 485.0)

Construct a box plot for the data

Using the five-number summary obtained in Step 2, draw a horizontal number line and mark the endpoints and quartiles as follows: - Minimum (1.70) - Q1 (143.0) - Median / Q2 (246.5) - Q3 (315.5) - Maximum (485.0) Draw vertical lines at each quartile and connect them with horizontal lines, forming a box. Then, connect the box edges to the minimum and the maximum with whiskers.

Identify outliers

Outliers are data points that are far from the rest of the data. To determine outliers, we will calculate the interquartile range (IQR) and define lower and upper limits: IQR = Q3 - Q1 = 315.5 - 143 = 172.5 Lower limit = Q1 - (1.5 * IQR) = 143 - (1.5 * 172.5) = -115.75 Upper limit = Q3 + (1.5 * IQR) = 315.5 + (1.5 * 172.5) = 574.25 As all data points are within the lower and upper limits, there are no outliers.

Analyze the age difference

We know that the first four dolphins are less than 3 years old, while the others are more than 8 years old. The first four data points (1.70, 1.72, 5.90, and 8.80) are significantly smaller compared to the rest. It can be inferred that the age difference might be one of the factors that explain the difference in mercury concentrations between the younger and older dolphins. Older dolphins may have had more time to accumulate mercury in their bodies than younger dolphins.

Key concepts

Five-number summary

The five-number summary is a concise statistical description that provides insights into the distribution of a data set. It consists of five values: the minimum, the first quartile (Q1), the median (or second quartile, Q2), the third quartile (Q3), and the maximum.

In the context of environmental science, observing the accumulation of toxic elements in organisms, the five-number summary gives a succinct overview of the data. For our marine mammals' study, the five-number summary allows us to see at a glance that the youngest dolphins have much lower mercury levels compared to their elders, with values as low as 1.70 micrograms/gram, indicating potentially less exposure to pollutants due to their age. The upper quartile (316.0 micrograms/gram) and maximum (485.0 micrograms/gram) point towards a significantly higher accumulation of mercury in the older dolphins' livers.

Box plot

A box plot, also known as a box-and-whisker plot, is a visual representation of the five-number summary, and offers a graphical view of the data distribution. The box itself contains the middle 50% of the data, situated between the first quartile (Q1) and the third quartile (Q3). The median cuts through the box, and 'whiskers' extend to the minimum and maximum data points.

For our striped dolphins mercury concentration study, constructing a box plot illuminates the spread and central tendency of the data. It becomes instantly apparent where most of the mercury concentrations lie and how they're spread out, enabling researchers to quickly assess the overall exposure level and identify any potential concerns within the population.

Outliers in data

Outliers are data points that differ significantly from other observations and may indicate a measurement or entry error, or a true deviation from the rest. Identifying outliers is critical in environmental science, as they can reveal anomalies in the ecosystem, such as a dolphin with exceptionally high toxin levels which may warrant further investigation.

In the striped dolphin data, we define outliers by using 1.5 times the interquartile range (IQR) above the third quartile and below the first quartile. Our calculation did not reveal any outliers, meaning all dolphins' mercury concentrations are within the expected range considering the variation in the dataset. However, understanding outliers helps us continuously monitor environmental health and manage potential risks.

Interquartile range (IQR)

The interquartile range (IQR) is the range within which the central 50% of the data lies and is computed as the difference between the third quartile (Q3) and the first quartile (Q1).

In environmental studies like the one on mercury in dolphins, the IQR can help assess the homogeneity of pollutant distribution among the population. With an IQR of 172.5, we observe a relatively wide range of mercury concentrations in dolphin livers, which may be influenced by various factors including age, as younger dolphins have less time to accumulate mercury. This indicates that individual experiences and histories, such as diet and habitat, have a notable impact on the mercury levels found within these animals.