Frequencies: 20, 53, 42, 42, 38, 36

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This question requires a factual and analytical approach. The keywords are “frequencies,” indicating a need to analyze a given dataset of numerical values. The approach will involve calculating descriptive statistics and potentially drawing inferences about the underlying distribution.

Points to Remember:

  • Calculate measures of central tendency (mean, median, mode).
  • Calculate measures of dispersion (range, variance, standard deviation).
  • Identify the shape of the distribution (symmetrical, skewed).
  • Draw conclusions based on the statistical analysis.

Introduction:

The provided data represents a frequency distribution of six values: 20, 53, 42, 42, 38, and 36. Frequency distributions are fundamental in statistics, providing a summary of how often different values occur within a dataset. Analyzing this distribution allows us to understand the central tendency, dispersion, and overall shape of the data. This analysis can be useful in various contexts, from understanding customer preferences to analyzing scientific measurements.

Body:

1. Measures of Central Tendency:

  • Mean: The average of the values. (20 + 53 + 42 + 42 + 38 + 36) / 6 = 38.5
  • Median: The middle value when the data is ordered. Ordering the data (20, 36, 38, 42, 42, 53), the median is (38 + 42) / 2 = 40.
  • Mode: The value that appears most frequently. The mode is 42.

2. Measures of Dispersion:

  • Range: The difference between the highest and lowest values. 53 – 20 = 33.
  • Variance: A measure of how spread out the data is. Calculating the variance requires subtracting the mean from each value, squaring the result, summing these squared differences, and dividing by the number of values minus one (for a sample). The variance is approximately 126.3.
  • Standard Deviation: The square root of the variance, providing a measure of dispersion in the same units as the original data. The standard deviation is approximately 11.2.

3. Shape of the Distribution:

The distribution is slightly right-skewed (positively skewed) because the mean (38.5) is less than the median (40). The presence of the outlier (53) pulls the mean towards the higher end, while the median is less affected by this outlier.

Conclusion:

The analysis of the frequency distribution reveals a dataset with a mean of 38.5, a median of 40, and a mode of 42. The range is 33, the variance is approximately 126.3, and the standard deviation is approximately 11.2. The distribution is slightly right-skewed, indicating a few higher values are influencing the overall average. This analysis provides a basic understanding of the central tendency and dispersion of the data. Further analysis, such as examining the context from which this data originated, would be needed to draw more robust conclusions. For instance, knowing the source of these frequencies (e.g., test scores, product sales, etc.) would allow for a more meaningful interpretation of the results and potentially inform future decision-making. A larger sample size would also provide more reliable statistical inferences.

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