mariposa (version 0.7.3)

w_kurtosis: Measure Population-Representative Kurtosis

Description

w_kurtosis() measures how heavy the tails of your data's distribution are, using survey weights for population-representative results. Kurtosis tells you whether your data has unusually many extreme values (outliers) compared to a normal distribution:

• Positive (excess) kurtosis: Heavier tails than normal (more outliers)

• Negative (excess) kurtosis: Lighter tails than normal (fewer outliers)

• Near zero: Similar tail behavior to a normal distribution

Usage

w_kurtosis(data, ..., weights = NULL, na.rm = TRUE, excess = TRUE)

Value

Population-weighted kurtosis value(s) with sample size information, including the weighted kurtosis, effective sample size (effective N), and the number of valid observations used.

Arguments

data

Your survey data (a data frame or tibble)

...

The numeric variables you want to analyze. You can list multiple variables or use helpers like starts_with("trust")

weights

Survey weights to make results representative of your population. Without weights, you get the simple sample kurtosis.

na.rm

Remove missing values before calculating? (Default: TRUE)

excess

Show excess kurtosis? (Default: TRUE, matching SPSS). Excess kurtosis subtracts 3 so that a normal distribution has kurtosis of 0, making interpretation easier.

Details

Understanding the Results

• Weighted Kurtosis (excess, default):

  • Near 0: Tail behavior similar to a normal distribution

  • Positive (> 0): Heavier tails, more extreme values than normal

  • Negative (< 0): Lighter tails, fewer extreme values than normal

  • Values beyond 2 or below -2 indicate notable departure from normality

• Effective N: How many independent observations your weighted data represents.

• N: The actual number of observations used.

Kurtosis is often checked together with skewness to assess normality. Both should be close to zero for normally distributed data.

When to Use This

Use w_kurtosis() when:

• You need to assess whether a variable follows a normal distribution

• You want to check for unusually many outliers

• You are deciding whether parametric tests are appropriate

• You need SPSS-compatible weighted kurtosis values

Formula

Uses the SPSS Type 2 (sample-corrected) excess kurtosis formula. First, the population excess kurtosis (\(g_2\)) is calculated:

\(g_2 = \frac{m_4}{m_2^2} - 3\)

where \(m_2 = \sum w_i(x_i - \bar{x}_w)^2 / V_1\) and \(m_4 = \sum w_i(x_i - \bar{x}_w)^4 / V_1\) with \(V_1 = \sum w_i\).

Then, the bias-corrected (Type 2) kurtosis is:

\(G_2 = \frac{(n+1) \cdot g_2 + 6}{(n-2)(n-3)} \cdot (n-1)\)

where \(n = V_1\) for weighted data.

References

Joanes, D. N., & Gill, C. A. (1998). Comparing measures of sample skewness and kurtosis. The Statistician, 47(1), 183-189.

IBM Corp. (2023). IBM SPSS Statistics 29 Algorithms. IBM Corporation.

See Also

w_skew for weighted skewness (distribution asymmetry).

describe for comprehensive descriptive statistics including kurtosis.

Other weighted_statistics: w_iqr(), w_mean(), w_median(), w_modus(), w_quantile(), w_range(), w_sd(), w_se(), w_skew(), w_var()

Examples

# Load required packages and data
library(dplyr)
data(survey_data)

# Basic weighted kurtosis (excess kurtosis, default)
survey_data %>% w_kurtosis(age, weights = sampling_weight)

# Multiple variables
survey_data %>% w_kurtosis(age, income, life_satisfaction, weights = sampling_weight)

# Grouped data
survey_data %>% group_by(region) %>% w_kurtosis(age, weights = sampling_weight)

# Raw kurtosis (not excess)
survey_data %>% w_kurtosis(age, weights = sampling_weight, excess = FALSE)

# In summarise context
survey_data %>% summarise(kurt_age = w_kurtosis(age, weights = sampling_weight))

# Unweighted (for comparison)
survey_data %>% w_kurtosis(age)