Statistical Analysis of Guatemalan Cholesterol Levels: Distributions, Summary Statistics, and Comparative Measures

Guatemalan Cholesterol Levels: A Statistical Case Study

Introduction

This study examines the distribution of serum total cholesterol levels among two groups of Guatemalan adults: low-income rural and high-income urban populations. The data, sourced from a clinical nutrition study, provides an opportunity to apply statistical methods for comparing distributions, identifying outliers, and interpreting measures of central tendency and variability.

Data Description

Cholesterol Level Data

• Rural Group: 49 adult, low-income rural Guatemalans

• Urban Group: 49 adult, high-income urban Guatemalans

• Measurement: Serum total cholesterol levels (mg/dL)

Table 1. Cholesterol Levels (mg/dL)

Group	Cholesterol Levels (mg/dL)
Rural	95, 108, 108, 114, 115, 124, 129, 129, 131, 131, 135, 136, 136, 139, 140, 142, 142, 143, 143, 144, 144, 145, 146, 148, 152, 152, 155, 157, 158, 158, 162, 165, 166, 171, 172, 173, 174, 175, 180, 181, 189, 192, 194, 197, 204, 220, 223, 226, 231
Urban	133, 134, 155, 170, 175, 179, 181, 181, 188, 189, 190, 196, 197, 199, 200, 200, 201, 201, 204, 205, 205, 206, 214, 217, 222, 222, 227, 227, 228, 234, 236, 239, 241, 242, 244, 249, 252, 273, 279, 284, 284, 284, 330

Graphical Summaries

Histograms and Boxplots

• Both groups' cholesterol distributions are mound-shaped (unimodal) with right skewness.

• An outlier is present in the urban group (330 mg/dL), which is much higher than the next largest value.

• Boxplots visually confirm higher cholesterol levels and greater variability in the urban group, with the outlier clearly visible.

Summary Statistics

Five Number Summary and Descriptive Statistics

Group	Lower Quartile (Q1)	Median	Upper Quartile (Q3)	Maximum	Range	Quartile Range (IQR)	Mean	Standard Deviation
Rural	136.00	152.00	174.00	231.00	136.00	38.00	157.02	31.75
Urban	186.00	206.00	228.00	330.00	197.00	42.00	216.87	39.92

Percentiles (Selected)

Group	20th Percentile	50th Percentile (Median)	80th Percentile	95th Percentile
Rural	131.00	152.00	181.00	231.00
Urban	175.00	206.00	243.00	330.00

Measures of Center and Spread

Central Tendency

• Median: Rural group: 152 mg/dL; Urban group: 206 mg/dL

• Mean: Rural group: 157.02 mg/dL; Urban group: 216.87 mg/dL

• The urban group has a median cholesterol level about 54 mg/dL higher than the rural group, and a mean about 59.85 mg/dL higher.

Variability

• Range: Urban group: 197 mg/dL; Rural group: 136 mg/dL

• Interquartile Range (IQR): Urban group: 42 mg/dL; Rural group: 38 mg/dL

• Standard Deviation: Urban group: 39.92 mg/dL; Rural group: 31.75 mg/dL

• Ratios of variability measures (urban/rural): Range ≈ 1.4, IQR ≈ 1.1, Standard deviation ≈ 1.3

Skewness and Distribution Shape

Quantifying Skewness

• Right skewness is present in both groups, stronger in the urban group (especially with the outlier).

• Skewness can be quantified by comparing the range of the right half (max - median) to the left half (median - min): - Rural: $\frac{79}{57} \approx 1.4$ - Urban: $\frac{124}{73} \approx 1.7$ (with outlier); $\frac{78}{73} \approx 1.1$ (without outlier)

• Similar ratios for the tails and center support the conclusion of right skewness.

Interpretation of Skewness

• Right skewness indicates more variability among individuals with higher cholesterol levels than among those with lower levels.

• Outliers can significantly affect measures of spread and skewness.

Comparative Analysis

Summary of Differences

• Urban Guatemalans have higher cholesterol levels and greater variability than rural Guatemalans.

• Distributions are unimodal, with peaks at approximately 140 mg/dL (rural) and 200 mg/dL (urban).

• Variability is about 30-40% greater in the urban group, depending on the measure used.

Possible Explanations (Contextual)

• Rural Guatemalans may have lower cholesterol due to higher physical activity and lower dietary fat intake.

• Greater variability in the urban group may reflect more diverse lifestyles and eating habits.

Key Statistical Concepts Illustrated

• Five Number Summary: Minimum, Q1, Median, Q3, Maximum

• Measures of Center: Mean, Median

• Measures of Spread: Range, Interquartile Range (IQR), Standard Deviation

• Skewness: Quantified by ratios of right/left halves and tails

• Outliers: Identification and impact on summary statistics

• Percentiles: Used for more detailed distribution comparisons

Formulas Used

• Mean: $\bar{x} = \frac{1}{n} \sum_{i=1}^{n} x_i$

• Standard Deviation: $s = \sqrt{\frac{1}{n-1} \sum_{i=1}^{n} (x_i - \bar{x})^2}$

• Interquartile Range (IQR): $IQR = Q_3 - Q_1$

• Skewness Ratio (example): $\text{Skewness Ratio} = \frac{\text{max} - \text{median}}{\text{median} - \text{min}}$

Example Application

• Comparing cholesterol levels between two populations using summary statistics and graphical methods.

• Assessing the impact of outliers on measures of spread and skewness.

• Using percentiles and boxplots for visual and numerical comparison.

Additional info: The study notes infer some context about the data collection and possible explanations for observed differences, as suggested by the original material.