Math Problem Statement

Find the outlires

Find the outliers

Solution

To find the outliers from this dataset, we will apply the Interquartile Range (IQR) method. Outliers are typically defined as data points that are below Q11.5×IQRQ1 - 1.5 \times IQR or above Q3+1.5×IQRQ3 + 1.5 \times IQR, where:

  • Q1Q1 is the first quartile (25th percentile),
  • Q3Q3 is the third quartile (75th percentile),
  • IQR=Q3Q1IQR = Q3 - Q1.

Steps for Outlier Detection:

  1. Compute IQR for each country.
  2. Calculate lower and upper bounds for detecting outliers:
    • Lower bound: Q11.5×IQRQ1 - 1.5 \times IQR
    • Upper bound: Q3+1.5×IQRQ3 + 1.5 \times IQR
  3. Check for any values outside these bounds (either lower than the lower bound or higher than the upper bound).

For Canada:

  • Q1=32.4Q1 = 32.4, Q3=33.2Q3 = 33.2
  • IQR=33.232.4=0.8IQR = 33.2 - 32.4 = 0.8
  • Lower bound: 32.41.5×0.8=31.232.4 - 1.5 \times 0.8 = 31.2
  • Upper bound: 33.2+1.5×0.8=34.433.2 + 1.5 \times 0.8 = 34.4

Any values below 31.2 or above 34.4 would be considered outliers. The minimum value for Canada is 30.9, which is below the lower bound of 31.2, indicating an outlier.

For the United States:

  • Q1=36.4Q1 = 36.4, Q3=36.9Q3 = 36.9
  • IQR=36.936.4=0.5IQR = 36.9 - 36.4 = 0.5
  • Lower bound: 36.41.5×0.5=35.6536.4 - 1.5 \times 0.5 = 35.65
  • Upper bound: 36.9+1.5×0.5=37.6536.9 + 1.5 \times 0.5 = 37.65

Since the minimum value for the United States is 35.7, and it is slightly higher than the lower bound of 35.65, there are no outliers in the United States dataset.

Summary of Outliers:

  • Canada: 30.9 (outlier)
  • United States: No outliers

Would you like further details or have any questions?

Further Questions:

  1. How is the IQR method different from other outlier detection methods?
  2. Why is 1.5 used as a multiplier in the IQR method?
  3. Can boxplots visually help in identifying outliers from a dataset?
  4. How would results change if the dataset's standard deviation were higher?
  5. What are some other contexts where outliers might be relevant?

Tip: When detecting outliers, always visualize the data with boxplots or histograms to get a clearer view of its distribution.

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Math Problem Analysis

Mathematical Concepts

Statistics
Outlier Detection
Interquartile Range (IQR)

Formulas

IQR = Q3 - Q1
Lower bound = Q1 - 1.5 * IQR
Upper bound = Q3 + 1.5 * IQR

Theorems

Interquartile Range Method for Outlier Detection

Suitable Grade Level

Grades 9-12