Rev710 V4 Mean, Median, And Mode Page 2 Of 2 081194

Rev710 V4mean Median And Moderev710 V4page 2 Of 2mean Median And

Find the mean, median, mode, range, interquartile range, and standard deviation of the following two examples.

Paper For Above instruction

In this analysis, we explore descriptive statistical measures for two datasets involving income levels, illustrating how each measure provides insights into the data's distribution and variability.

Example 1: Income Data Analysis

The first dataset encompasses six individuals with the following incomes and their corresponding calculations:

  • Mean: The total income summed and divided by the number of individuals yields the mean. Given a total income of 1560, with 6 persons, the mean income is 1560 / 6 = 260.
  • Median: To determine the median, income values are ordered, and the middle value is identified. The median income provided is 265, indicating the middle point in the ordered data.
  • Mode: The most frequently occurring income values are recorded; here, the modes are 200, 250, 180, 300, 350, and 280, suggesting a multimodal distribution.
  • Range: The difference between the maximum and minimum incomes, which are 30 and 6 respectively, gives a range of 30 - 6 = 24.
  • Interquartile Range (IQR): The IQR measures the middle 50% spread of the data, calculated as the difference between the third quartile (XU) and the first quartile (XL). Given XU = 117.5 and XL = 104, the IQR = 117.5 - 104 = 13.5.
  • Standard Deviation: This statistic reflects data variability around the mean. For the income data with a total of 1560 and 6 entries, the variance is computed first, and the square root of variance denotes the standard deviation, which is approximately calculated as follows: Standard Deviation ≈ 45.

Example 2: Income Data Analysis

The second dataset contains 24 individuals with the following attributes:

  • Mean: Total income is 227.5 times the number of persons (24), giving a total income of 24 × 227.5 = 5460. The mean is directly provided as 227.5.
  • Median: Based on cumulative frequencies, the median income is 225, situated within the 25th individual, as the cumulative frequency reaches 54.5, indicating the middle of the distribution.
  • Mode: The most common income categories are 180, 200, 250, and 280, associated with 16, 24, 26, and 30 persons respectively, indicating potential multiple modes.
  • Range: With the maximum income at 100 and the minimum at 14, the range equals 100 - 14 = 86.
  • Interquartile Range (IQR): Derived from the data, with the third quartile (XU) value at 87.5 and the first quartile (XL) at 66.5, thus IQR = 87.5 - 66.5 = 21.
  • Standard Deviation: Given as 45, capturing the dispersion of incomes around the mean in this dataset.

Conclusion

These statistical measures illuminate key aspects of the income datasets. The mean provides an overall average, while the median indicates the central tendency, especially when data may be skewed. The mode reveals the most frequent incomes, and the range along with the interquartile range measures data variability and spread. The standard deviation offers insights into income distribution consistency across individuals. Collectively, these measures are invaluable for economic analysis, policy formulation, and understanding income disparities within populations.

References

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  • Walpole, R. E., Myers, R. H., Myers, S. L., & Ye, K. (2012). Probability & Statistics for Engineering and the Sciences. Pearson.
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  • Gravetter, F. J., & Wallnau, L. B. (2016). Statistics for the Behavioral Sciences. Cengage Learning.

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