Introduction

This report provides a comprehensive analysis of the distribution of variables in the trees dataset, focusing on skewness and kurtosis to understand the symmetry and tail behavior of the data. These statistical metrics offer valuable insights into the underlying distribution of the variables, which is crucial for assessing normality and guiding further statistical analysis or modeling decisions.

Data Summary and Setup

The trees dataset contains measurements of three physical characteristics of trees: Girth, Height, and Volume. These variables represent the circumference of the tree at breast height (in inches), the height of the tree (in feet), and the volume of the tree (in cubic feet), respectively. The dataset includes data for 31 trees and provides valuable insights into tree growth patterns. The Volume variable, which is particularly of interest, reflects the overall size of the tree and is influenced by both girth and height.

library(moments)
data = trees
head(data)
##   Girth Height Volume
## 1   8.3     70   10.3
## 2   8.6     65   10.3
## 3   8.8     63   10.2
## 4  10.5     72   16.4
## 5  10.7     81   18.8
## 6  10.8     83   19.7

Statistical Analysis

##Summary Statistics

This section presents basic descriptive statistics for the Volume variable from the trees dataset, providing a snapshot of the data’s central tendencies and distribution.

girth_data = na.omit(data$Girth)

#summary
summary_stats = summary(girth_data)
summary_stats
##    Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
##    8.30   11.05   12.90   13.25   15.25   20.60
#calculate standard deviation
std_dev = sd(girth_data)
cat("Standard Deviation:", round(std_dev,2),"\n")
## Standard Deviation: 3.14

##Interpretation of Summary Statistics

Minimum and Maximum values indicate the range of tree volumes in the dataset. Mean and Median provide insights into the central tendency of the data.

##Skewness and Kurtosis Analysis

The skewness and kurtosis of the Volume variable provide deeper insights into its distribution shape and tail behavior.

skewness_value = skewness(girth_data)
kurtosis_value = kurtosis(girth_data)

cat("Skewness:", round(skewness_value, 2), "\n")
## Skewness: 0.53
cat("Kurtosis:", round(kurtosis_value, 2), "\n")
## Kurtosis: 2.44

##Interpretation

Skewness: A skewness value near zero suggests symmetry in the distribution of tree volumes. A positive skew would indicate a longer right tail (larger tree volumes), while negative skewness would suggest a longer left tail (smaller tree volumes).

Kurtosis: Kurtosis values around 3 typically indicate a normal distribution. A higher value (leptokurtic) would suggest heavy tails and more outliers, while a lower value (platykurtic) indicates lighter tails and fewer extreme values than a normal distribution.

#Data Visualization

Visualizations such as histograms and density plots can help us better understand the shape of the distribution for the Volume variable.

#visualize the distribution of Girth

library(ggplot2)
ggplot(data = data.frame(girth_data), aes(x = girth_data)) +
  geom_histogram(aes(y = after_stat(density)), color = "black", fill = "lightblue", bins = 10) +
  geom_density(color = "red", linewidth = 1) +
  labs(title = "Distribution of Tree Girth in Black Cherry Trees",
       subtitle = paste("Skewness:", round(skewness_value, 2), "| Kurtosis:", round(kurtosis_value, 2)),
       x = "Tree Girth (inches)", y = "Density") +
  theme_minimal() +
  theme(
    plot.title = element_text(size = 16, face = "bold"),
    plot.subtitle = element_text(size = 12, color = "red")
)

## Conclusion

The analysis of the Volume variable in the trees dataset provides the following insights:

Skewness and Kurtosis: The skewness and kurtosis values reveal key aspects of the distribution, such as symmetry and the presence of outliers. If the skewness is high, we might consider transforming the data for more accurate modeling.

Data Visualization: The histogram and density plot visually reinforce the insights gained from the statistical measures, allowing us to assess the distribution’s shape.

Foundational Analysis: This initial analysis of the Volume distribution is foundational in deciding on subsequent steps in exploratory data analysis (EDA) or predictive modeling.