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
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.
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: 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.
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
#> -----------------------------------
#>
#> --- age ---
#> Variable weighted_kurtosis Effective_N
#> age -0.396 2468.8
#>
# Multiple variables
survey_data %>% w_kurtosis(age, income, life_satisfaction, weights = sampling_weight)
#>
#> Weighted Excess Kurtosis Statistics
#> -----------------------------------
#>
#> --- age ---
#> Variable weighted_kurtosis Effective_N
#> age -0.396 2468.8
#>
#> --- income ---
#> Variable weighted_kurtosis Effective_N
#> income 0.388 2158.9
#>
#> --- life_satisfaction ---
#> Variable weighted_kurtosis Effective_N
#> life_satisfaction -0.598 2390.9
#>
# Grouped data
survey_data %>% group_by(region) %>% w_kurtosis(age, weights = sampling_weight)
#>
#> Weighted Excess Kurtosis Statistics
#> -----------------------------------
#>
#> Group: region = East
#> Warning: Unknown or uninitialised column: `Variable`.
#>
#> Group: region = West
#> Warning: Unknown or uninitialised column: `Variable`.
#>
# Raw kurtosis (not excess)
survey_data %>% w_kurtosis(age, weights = sampling_weight, excess = FALSE)
#>
#> Weighted Kurtosis Statistics
#> ----------------------------
#>
#> --- age ---
#> Variable weighted_kurtosis Effective_N
#> age 2.604 2468.8
#>
# 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
#> --------------------------
#>
#> --- age ---
#> Variable kurtosis N
#> age -0.364 2500
#>