Math Problem Statement

Use the magnitudes​ (Richter scale) of the 120

earthquakes listed in the accompanying data table. Use technology to find the​ range, variance, and standard deviation. If another​ value, 7.50 ​, is added to those listed in the data​ set, do the measures of variation change​ much?3.32

2.82 2.78 1.94 1.69 2.52

2.44 3.42 3.90 1.59 2.86 1.64 2.59 3.95 2.54 2.45 1.83 2.20 2.43 3.02 2.89 2.33 2.00 3.04 2.78 3.85 2.96 2.08 1.85 2.32 2.41 3.47 3.42 1.56 2.56 1.48 2.20 3.10 2.30 3.25 1.96 1.91 2.39 2.97 2.58 1.50 2.15 2.33 1.90 2.71 2.89 1.82 3.67 2.70 1.44 3.59 3.15 2.57 1.55 1.42 2.83 2.89 2.19 1.66 3.21 1.38 1.75 2.36 1.13 2.34 2.49 1.80 2.01 3.00 1.93 2.43 1.85 2.25 2.35 3.23 4.04 2.10 1.49 2.31 2.34 2.58 2.54 2.18 2.77 2.48 2.71 3.59 2.82 2.79 3.29 1.77 4.66 3.26 2.37 2.01 3.84 2.40 2.88 2.69 2.29 2.84 2.80 2.40 3.41 2.32 1.51 2.40 2.45 2.47 2.68 2.44 2.81 2.65 2.73 2.44

Solution

For the original set of earthquake magnitudes, the following values are calculated:

  • Range: 3.53
  • Variance: 0.4248
  • Standard Deviation: 0.6518

When the value 7.50 is added to the dataset, the new values are:

  • Range: 6.37
  • Variance: 0.6261
  • Standard Deviation: 0.7912

Conclusion: Adding the value 7.50 significantly increases the range (by 2.84) and moderately increases the variance and standard deviation. This indicates that the spread of the data becomes wider with this additional extreme value.

Do you want more details, or have any questions?

Here are some related questions:

  1. What impact does an outlier have on standard deviation?
  2. How does variance differ from standard deviation?
  3. Can adding a value lower than the current minimum impact the measures of variation similarly?
  4. Why is the sample variance (with ddof=1) used instead of population variance?
  5. What is the significance of the range in this context?

Tip: Always check for outliers in a dataset, as they can significantly affect measures of spread such as variance and standard deviation.

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Math Problem Analysis

Mathematical Concepts

Statistics
Measures of Variation
Range
Variance
Standard Deviation

Formulas

Range = Max value - Min value
Variance = (Σ(xi - mean)^2) / N for population or (Σ(xi - mean)^2) / (N - 1) for sample
Standard Deviation = √Variance

Theorems

Properties of Variance and Standard Deviation
Effect of Outliers on Measures of Dispersion

Suitable Grade Level

College-level statistics or high school advanced mathematics

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