Introduction

This report provides a detailed analysis of a dataset’s distribution, using skewness and kurtosis to evaluate its symmetry and tail behavior. Skewness measures asymmetry, while kurtosis measures the “tailedness” or concentration of data in the tails. Understanding these characteristics is essential for data-driven decision-making.

Data Summary and Setup

The dataset used here is generated as a sample normal distribution.

# Generate sample data
set.seed(123)  
data <- rnorm(100, mean = 10, sd = 5)

# Display first few values
head(data)
## [1]  7.197622  8.849113 17.793542 10.352542 10.646439 18.575325

Statistical Analysis

Summary Statistics

This section displays basic descriptive statistics, providing an overview of data dispersion and central tendencies.

# Summary statistics
summary_stats <- summary(data)
summary_stats
##    Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
##  -1.546   7.531  10.309  10.452  13.459  20.937
# Standard deviation
std_dev <- sd(data)
cat("Standard Deviation:", round(std_dev, 2), "\n")
## Standard Deviation: 4.56

Skewness and Kurtosis

The skewness and kurtosis are calculated to provide additional insights into the shape of the distribution.

# Calculate Skewness and Kurtosis
skewness_value <- skewness(data)
kurtosis_value <- kurtosis(data)

# Display Skewness and Kurtosis
cat("Skewness:", round(skewness_value, 2), "\n")
## Skewness: 0.06
cat("Kurtosis:", round(kurtosis_value, 2), "\n")
## Kurtosis: 2.84

Interpretation

  • Skewness: Values near 0 indicate symmetry in the distribution. Positive skewness suggests that the right tail is longer or fatter than the left tail, while negative skewness indicates a longer or fatter left tail.

  • Kurtosis: Values close to 3 are typical for normal distributions. Higher kurtosis values indicate heavy tails and a sharper peak, which can signify a higher likelihood of outliers in the dataset.

Data Visualization

The following histogram and density plot visualize the data distribution, providing a clear view of data concentration and spread. These visualizations not only illustrate the distribution characteristics but also help identify any deviations from normality, guiding further analysis and modeling choices.

# Create a histogram with Skewness and Kurtosis displayed
ggplot(data = data.frame(data), aes(x = data)) +
    geom_histogram(aes(y = after_stat(density)), color = "black", fill = "skyblue", bins = 15) +  # Updated here
    geom_density(color = "darkblue", linewidth = 1) +
    labs(title = "Data Distribution",
         subtitle = paste("Skewness:", round(skewness_value, 2), "| Kurtosis:", round(kurtosis_value, 2)),
         x = "Data Values", y = "Density") +
    theme_minimal() +
    theme(
        plot.title = element_text(size = 16, face = "bold"),
        plot.subtitle = element_text(size = 12, color = "blue")
    )

Conclusion

The analysis of the dataset’s distribution reveals the following insights:

  • Skewness and Kurtosis: These statistical metrics provide valuable insights into the shape of the distribution and the weight of its tails. Understanding skewness helps in identifying whether the data leans towards one side, while kurtosis informs us about the extremity of the data’s tails.

  • Data Visualization: The visual representation of the data spread and its normality tendencies assists in evaluating whether the data aligns with expected norms. This assessment is crucial for determining if any transformations are necessary before further analysis.

  • Foundational Analysis: This initial analysis is essential as it lays the groundwork for more comprehensive exploratory data analysis (EDA) and subsequent modeling approaches. By understanding the distribution characteristics, we can make more informed decisions regarding the choice of statistical techniques and models.