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Chapter 1: Problem 5

The article "Snow Cover and Temperature Relationships in North America and Eurasia" \((J\). Climate and Applied Meteorology, 1983: 460-469) used statistical techniques to relate the amount of snow cover on each continent to average continental temperature. Data presented there included the following ten observations on October snow cover for Eurasia during the years \(1970-1979\) (in million \(\mathrm{km}^{2}\) ): \(\begin{array}{llllllllll}6.5 & 12.0 & 14.9 & 10.0 & 10.7 & 7.9 & 21.9 & 12.5 & 14.5 & 9.2\end{array}\) What would you report as a representative, or typical, value of October snow cover for this period, and what prompted your choice?

Short Answer

Expert verified
The median, 11.35 million km虏, is a typical value, as it is less affected by extreme values.

Step by step solution

01

Arrange the Data

First, organize the data in ascending order to make it easier to identify central tendencies: 6.5, 7.9, 9.2, 10.0, 10.7, 12.0, 12.5, 14.5, 14.9, 21.9.
02

Calculate the Mean

The mean is calculated by adding all the snow cover values together and dividing by the number of observations. Calculate the sum: \(6.5 + 7.9 + 9.2 + 10.0 + 10.7 + 12.0 + 12.5 + 14.5 + 14.9 + 21.9 = 120.1\).Divide by the number of observations, which is 10:\( \text{Mean} = \frac{120.1}{10} = 12.01\text{ million } \mathrm{km}^2\).
03

Find the Median

Since there are 10 observations, the median value is the average of the 5th and 6th values in the ordered list. Identify these values: 10.7 and 12.0.Calculate their average:\( \text{Median} = \frac{10.7 + 12.0}{2} = 11.35\text{ million } \mathrm{km}^2\).
04

Assess the Representative Value

Data typically uses the mean or median as a representative value. Here, the two metrics (mean = 12.01, median = 11.35) suggest that the mean might be slightly skewed by the high value of 21.9. The median likely provides a better central tendency for this dataset.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Mean Calculation
The mean calculation, often known as the average, is a fundamental concept in descriptive statistics. It helps us understand the central tendency of a dataset. To calculate the mean, we sum up all the data points and then divide by the number of points. In this problem, we dealt with ten observations of snow cover values:
  • The sum was calculated as: \(6.5 + 7.9 + 9.2 + 10.0 + 10.7 + 12.0 + 12.5 + 14.5 + 14.9 + 21.9 = 120.1\).
  • The formula for the mean: \(\text{Mean} = \frac{\text{Total Sum}}{\text{Number of Observations}}\).
  • By dividing 120.1 by 10, the mean snow cover for the dataset is \(12.01\text{ million } \mathrm{km}^2\).
The mean provides a quick snapshot but can be influenced by extreme values, as seen with 21.9 in this dataset. This makes it important to consider its reliability in reflecting typical values.
Median Calculation
The median calculation is crucial when you need a more reliable central value, especially in skewed datasets. The median is the middle value that divides a dataset into two equal parts when it is ordered.
  • After arranging the data from smallest to largest (6.5, 7.9, 9.2, 10.0, 10.7, 12.0, 12.5, 14.5, 14.9, 21.9), we had ten points.
  • For even-numbered datasets, like this one, the median is the average of the two central numbers.
  • Thus, the median here is \(\frac{10.7 + 12.0}{2} = 11.35\text{ million } \mathrm{km}^2\).
The median is particularly useful because it is not affected by outliers, like the high value of 21.9. Therefore, it often provides a more accurate measure of the central tendency in such cases.
Data Analysis
Data analysis is the process of inspecting, cleaning, and modeling data to discover useful information. In this scenario, the ten snow cover observations required proper analysis to determine a typical value.
  • First, ordering data helps easily identify means and medians.
  • Calculating both mean and median allowed us to assess which better represented the dataset.
  • While means are useful, they can be influenced by outliers, leading us to prefer medians when such extreme values exist.
Analyzing both the mean and median can provide insight into the distribution shape. Here, although the mean (12.01) and median (11.35) are close, the median offers a more accurate central tendency due to the outlier. Effective data analysis involves using multiple measures to understand data thoroughly.

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Most popular questions from this chapter

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