Test Variables for Normality

normality_test() checks whether numeric variables follow a normal distribution, producing the two tests SPSS prints in its EXAMINE "Tests of Normality" table:

Kolmogorov-Smirnov with Lilliefors significance correction (the "Kolmogorov-Smirnov(a)" column in SPSS)
Shapiro-Wilk (computed for n between 3 and 5000, matching the SPSS convention)

Use it before a t-test, ANOVA, or Pearson correlation to check the normality assumption, together with levene_test for the equal-variances assumption.

Usage

normality_test(data, ...)

Arguments

data: Your survey data (a data frame or tibble). If grouped (via dplyr::group_by()), the tests are run separately per group — matching SPSS EXAMINE ... BY factor.
...: Variables to test (unquoted, supports tidyselect). All selected variables must be numeric.

Value

An object of class "normality_test" whose $results tibble holds one row per variable (and group combination) with:

n: Number of valid (non-missing) cases
ks_statistic, ks_df, ks_p: Kolmogorov-Smirnov statistic with Lilliefors-corrected p-value (Dallal-Wilkinson approximation); df equals n as in SPSS
shapiro_w, shapiro_p: Shapiro-Wilk W and p-value (NA when n < 3 or n > 5000)

Details

Understanding the Output

For both tests the null hypothesis is "the variable is normally distributed":

p >= 0.05: No significant deviation from normality detected.
p < 0.05: The variable deviates significantly from a normal distribution.

With large survey samples these tests flag even tiny, practically irrelevant deviations. Combine them with the skewness and kurtosis from describe before deciding against a parametric test.

When to Use This

Before t_test, oneway_anova, or pearson_cor to check the normality assumption
Grouped (via group_by()) to check normality within each comparison group — the form of the assumption that actually matters for group comparisons

If normality is clearly violated, consider the rank-based alternatives: mann_whitney, kruskal_wallis, or spearman_rho.

Technical Details

The Lilliefors-corrected p-value uses the Dallal-Wilkinson (1986) approximation, the same correction SPSS applies in EXAMINE ("Lilliefors Significance Correction"). Shapiro-Wilk uses stats::shapiro.test() and is reported for 3 <= n <= 5000, as in SPSS. Cases with missing values are excluded per variable.

Weights: normality_test() deliberately takes no weights argument. Normality checks are a diagnostic of the sample distribution, and neither Shapiro-Wilk nor the Lilliefors correction has a well-defined fractional-frequency-weight form. Run the test on the unweighted sample.

An SPSS v29 EXAMINE reference run is pending; until it lands the statistics are verified against independent R implementations of the same published formulas (see the SPSS compatibility vignette).

References

Dallal, G. E., & Wilkinson, L. (1986). An analytic approximation to the distribution of Lilliefors's test statistic for normality. The American Statistician, 40(4), 294-296.

Examples

library(dplyr)
data(survey_data)

# Test a single variable
normality_test(survey_data, age)
#> Normality Tests: age
#>   age: KS = 0.028, p < 0.001; Shapiro-Wilk W = 0.990, p < 0.001 (n = 2500)
#> Use summary() for detailed output.

# Several variables at once
normality_test(survey_data, age, income, life_satisfaction)
#> Normality Tests: age, income, life_satisfaction
#>   age: KS = 0.028, p < 0.001; Shapiro-Wilk W = 0.990, p < 0.001 (n = 2500)
#>   income: KS = 0.079, p < 0.001; Shapiro-Wilk W = 0.963, p < 0.001 (n = 2186)
#>   life_satisfaction: KS = 0.203, p < 0.001; Shapiro-Wilk W = 0.884, p < 0.001 (n = 2421)
#> Use summary() for detailed output.

# Within comparison groups (SPSS: EXAMINE ... BY gender)
survey_data %>%
  group_by(gender) %>%
  normality_test(age, income)
#> Normality Tests: age, income
#>   2 group combination(s) x 2 variable(s) [Grouped: gender]
#> Use summary() for detailed output.

# --- Three-layer output ---
result <- normality_test(survey_data, age, income)
result              # compact overview
#> Normality Tests: age, income
#>   age: KS = 0.028, p < 0.001; Shapiro-Wilk W = 0.990, p < 0.001 (n = 2500)
#>   income: KS = 0.079, p < 0.001; Shapiro-Wilk W = 0.963, p < 0.001 (n = 2186)
#> Use summary() for detailed output.
summary(result)     # full SPSS-style table
#> 
#> Normality Tests Results
#> -----------------------
#> - Variables: age, income
#> 
#> Tests of Normality
#>   ------------------------------------------ 
#>   Variable     KS    df   KS p      W    W p 
#>   ------------------------------------------ 
#>   age       0.028  2500  <.001  0.990  <.001 
#>   income    0.079  2186  <.001  0.963  <.001 
#>   ------------------------------------------ 
#> 
#> KS = Kolmogorov-Smirnov statistic with Lilliefors significance correction.
#> W = Shapiro-Wilk statistic (computed for 3 <= n <= 5000, as in SPSS).
#> p < 0.05 indicates a significant deviation from normality.