Measure Population-Representative Kurtosis

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)

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. Give a column name (unquoted or as a string), an expression such as sampling_weight * 2, or a numeric vector with one weight per row.
na.rm: Remove missing values before calculating? (Default: TRUE). With FALSE, the result for a variable that contains missing values is NA (as in base R).
excess: Show excess kurtosis? (Default: TRUE, matching SPSS). Excess kurtosis subtracts 3 so that a normal distribution has kurtosis of 0, making interpretation easier.

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.

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 / Missing: Valid and missing cases. With weights, both are sums of weights (displayed rounded), as SPSS reports them under WEIGHT BY; Kish's effective N is shown by summary().

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.

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)
#> 
#> Weighted Excess Kurtosis Statistics
#> -----------------------------------
#> Weights: sampling_weight
#> 
#>   ---------------------------------
#>   Variable  Kurtosis     N  Missing
#>   ---------------------------------
#>   age         -0.396  2516        0
#>   ---------------------------------

# Multiple variables
survey_data %>% w_kurtosis(age, income, life_satisfaction, weights = sampling_weight)
#> 
#> Weighted Excess Kurtosis Statistics
#> -----------------------------------
#> Weights: sampling_weight
#> 
#>   ------------------------------------------
#>   Variable           Kurtosis     N  Missing
#>   ------------------------------------------
#>   age                  -0.396  2516        0
#>   income                0.388  2201      315
#>   life_satisfaction    -0.598  2437       79
#>   ------------------------------------------

# Grouped data
survey_data %>% group_by(region) %>% w_kurtosis(age, weights = sampling_weight)
#> 
#> Weighted Excess Kurtosis Statistics
#> -----------------------------------
#> Weights: sampling_weight
#> 
#> Group: region = East
#> --------------------
#> 
#>   --------------------------------
#>   Variable  Kurtosis    N  Missing
#>   --------------------------------
#>   age         -0.389  509        0
#>   --------------------------------
#> 
#> Group: region = West
#> --------------------
#> 
#>   ---------------------------------
#>   Variable  Kurtosis     N  Missing
#>   ---------------------------------
#>   age         -0.396  2007        0
#>   ---------------------------------

# Raw kurtosis (not excess)
survey_data %>% w_kurtosis(age, weights = sampling_weight, excess = FALSE)
#> 
#> Weighted Kurtosis Statistics
#> ----------------------------
#> Weights: sampling_weight
#> 
#>   ---------------------------------
#>   Variable  Kurtosis     N  Missing
#>   ---------------------------------
#>   age          2.604  2516        0
#>   ---------------------------------

# In summarise context
survey_data %>% summarise(kurt_age = w_kurtosis(age, weights = sampling_weight))
#> # A tibble: 1 × 1
#>   kurt_age
#>      <dbl>
#> 1   -0.396

# Unweighted (for comparison)
survey_data %>% w_kurtosis(age)
#> 
#> Excess Kurtosis Statistics
#> --------------------------
#> 
#>   ---------------------------------
#>   Variable  Kurtosis     N  Missing
#>   ---------------------------------
#>   age         -0.364  2500        0
#>   ---------------------------------