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Measure Population-Representative Skewness

Source: R/w_skew.R
w_skew.Rd

w_skew() measures how asymmetric your data's distribution is, using survey weights for population-representative results. Skewness tells you whether values tend to trail off more to the left or right:

  • Positive skew: A long tail to the right (e.g., income – many low, few very high)

  • Negative skew: A long tail to the left (e.g., exam scores – many high, few very low)

  • Near zero: Roughly symmetric (e.g., height in a population)

Usage

w_skew(data, ..., weights = NULL, na.rm = 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 skewness. 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).

Value

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

Details

Understanding the Results

  • Weighted Skewness: The population-representative skewness value.

    • Near 0: Distribution is roughly symmetric

    • Between -0.5 and 0.5: Approximately symmetric

    • Between -1 and -0.5 or 0.5 and 1: Moderately skewed

    • Beyond -1 or 1: Highly skewed

  • 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().

High skewness suggests you might want to use the median instead of the mean as a measure of center, and non-parametric tests instead of t-tests.

When to Use This

Use w_skew() when:

  • You need to check if a variable is normally distributed

  • You want to decide between parametric and non-parametric tests

  • You need to assess whether the mean is a good summary measure

  • You need SPSS-compatible weighted skewness values

Formula

Uses the SPSS Type 2 (sample-corrected) skewness formula. First, the population skewness (\(g_1\)) is calculated:

\(g_1 = \frac{m_3}{m_2^{3/2}}\)

where \(m_2 = \sum w_i(x_i - \bar{x}_w)^2 / V_1\) and \(m_3 = \sum w_i(x_i - \bar{x}_w)^3 / V_1\) with \(V_1 = \sum w_i\).

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

\(G_1 = g_1 \cdot \frac{\sqrt{n(n-1)}}{n-2}\)

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_kurtosis for weighted kurtosis (tail heaviness).

describe for comprehensive descriptive statistics including skewness.

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

Examples

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

# Basic weighted skewness
survey_data %>% w_skew(age, weights = sampling_weight)
#> 
#> Weighted Skewness Statistics
#> ----------------------------
#> Weights: sampling_weight
#> 
#>   ---------------------------------
#>   Variable  Skewness     N  Missing
#>   ---------------------------------
#>   age          0.159  2516        0
#>   ---------------------------------

# Multiple variables
survey_data %>% w_skew(age, income, life_satisfaction, weights = sampling_weight)
#> 
#> Weighted Skewness Statistics
#> ----------------------------
#> Weights: sampling_weight
#> 
#>   ------------------------------------------
#>   Variable           Skewness     N  Missing
#>   ------------------------------------------
#>   age                   0.159  2516        0
#>   income                0.725  2201      315
#>   life_satisfaction    -0.499  2437       79
#>   ------------------------------------------

# Grouped data
survey_data %>% group_by(region) %>% w_skew(age, weights = sampling_weight)
#> 
#> Weighted Skewness Statistics
#> ----------------------------
#> Weights: sampling_weight
#> 
#> Group: region = East
#> --------------------
#> 
#>   --------------------------------
#>   Variable  Skewness    N  Missing
#>   --------------------------------
#>   age          0.098  509        0
#>   --------------------------------
#> 
#> Group: region = West
#> --------------------
#> 
#>   ---------------------------------
#>   Variable  Skewness     N  Missing
#>   ---------------------------------
#>   age          0.170  2007        0
#>   ---------------------------------

# In summarise context
survey_data %>% summarise(skew_age = w_skew(age, weights = sampling_weight))
#> # A tibble: 1 × 1
#>   skew_age
#>      <dbl>
#> 1    0.159

# Unweighted (for comparison)
survey_data %>% w_skew(age)
#> 
#> Skewness Statistics
#> -------------------
#> 
#>   ---------------------------------
#>   Variable  Skewness     N  Missing
#>   ---------------------------------
#>   age          0.172  2500        0
#>   ---------------------------------