Shapiro-Wilk Test: p-Value Interpretation, Excel & SPSS

Quick Answer: Shapiro-Wilk Test Result

The verified SPSS output checks normality for five numeric variables: G3, G2, G1, age, and absences. Each variable has N = 649 valid cases. The Shapiro-Wilk Test is significant for every variable, with SPSS displaying Sig. = .000, which should be reported as p < .001. Therefore, the formal normality decision is to reject the null hypothesis of normality for all five variables.

Hypothesis-style decision: The null hypothesis says the variable follows a normal distribution, meaning H₀: the data are normally distributed. The alternative hypothesis says the variable does not follow a normal distribution, meaning H₁: the data are not normally distributed. Since all Shapiro-Wilk p-values are below .05, the formal decision is to reject H₀ for G3, G2, G1, age, and absences.

What Is the Shapiro-Wilk Test?

The Shapiro-Wilk Test is a formal normality test. It checks whether the observed values of a variable are consistent with a normal distribution. The test produces a W statistic and a p-value. The W statistic is usually between 0 and 1. Values closer to 1 indicate that the sample distribution is closer to normal shape, while lower values indicate stronger departure from normality.

The Shapiro-Wilk Test is widely used because it is sensitive to distribution shape. It can detect skewness, heavy tails, outliers, and other departures from normality. In SPSS, it is commonly produced through Analyze > Descriptive Statistics > Explore, together with the Kolmogorov-Smirnov test, histograms, Q-Q plots, and descriptive statistics.

In this guide, the Shapiro-Wilk Test checks normality for G3, G2, G1, age, and absences. The results show that every variable is formally non-normal at p < .001. But the W statistic reveals practical differences: G1 is closest to normal, while absences is the most non-normal.

Shapiro-Wilk Test Formula and W Statistic

The Shapiro-Wilk Test compares ordered sample values with the expected order statistics from a normal distribution. Its W statistic can be written in simplified form as:

In this formula, x_(i) represents the ordered sample values, x̄ is the sample mean, and a_i are constants based on expected normal order statistics. The numerator measures how closely the ordered sample values match the normal pattern. The denominator measures total variability around the mean.

The interpretation is simple even if the full calculation is complex:

Null and Alternative Hypothesis for Shapiro-Wilk Test

The Shapiro-Wilk Test has a clear hypothesis structure. The null hypothesis says the data follow a normal distribution. The alternative hypothesis says the data do not follow a normal distribution.

Dataset and Variables Used

The worked example uses student performance variables. The Shapiro-Wilk Test is applied to five numeric variables: G3, G2, G1, age, and absences. SPSS also checks G3 normality by sex, so the guide includes a group-level normality interpretation.

Before reading normality tests, it is useful to review descriptive statistics, frequency distribution, histogram interpretation, box plot interpretation, and five-number summary.

Verified SPSS Output Interpretation

The SPSS output provides the core evidence for the Shapiro-Wilk Test decision. It includes case processing summary, descriptive statistics, Kolmogorov-Smirnov results, Shapiro-Wilk results, histograms, normal Q-Q plots, detrended Q-Q plots, and group normality output for G3 by sex.

SPSS Case Processing and Descriptive Summary

SPSS Shapiro-Wilk Test Results Across Variables

SPSS Kolmogorov-Smirnov Cross-check

SPSS Group Normality for G3 by Sex

Python Chart-by-Chart Interpretation

The Python charts show the Shapiro-Wilk Test result visually. They include the distribution with normal curve, Q-Q plot, detrended Q-Q plot, W statistic comparison, p-value comparison, skewness and kurtosis context, and group p-value comparison.

Python Chart 1: Distribution with Normal Curve

This chart compares the observed distribution with a normal curve. If the variable were normally distributed, the bars would follow the bell-shaped curve closely. For G3, the distribution does not perfectly match the normal curve. The SPSS output reports Shapiro-Wilk W = .926 and p < .001, so the formal normality test rejects normality.

The chart also helps explain why visual interpretation matters. A large sample may produce a significant Shapiro-Wilk result even when the distribution looks moderately close to normal. Therefore, the histogram should be used to judge the practical size and direction of the departure.

Python Chart 2: Q-Q Plot Normality Check

The Q-Q plot checks whether observed values line up with expected normal quantiles. Points close to the diagonal line support approximate normality. Points bending away from the line show non-normality, tail issues, skewness, or outliers. The SPSS Shapiro-Wilk result for G3 is significant, and the Q-Q pattern supports that the distribution is not perfectly normal.

The Q-Q plot is especially useful because it shows where the normality problem occurs. If departures occur mostly in the tails, the center may still be usable for some robust analyses. If departures occur throughout the plot, the normality concern is more serious.

Python Chart 3: Detrended Q-Q Plot

The detrended Q-Q plot removes the diagonal reference pattern and shows deviations from normality more directly. Values close to zero indicate good alignment with the normal reference. Values far above or below zero show where the observed distribution differs from normal. This makes small departures easier to see than in the ordinary Q-Q plot.

For the Shapiro-Wilk workflow, the detrended Q-Q plot is a strong companion to the p-value. The p-value says whether the departure is statistically significant. The detrended plot shows the location and shape of that departure.

Python Chart 4: Shapiro-Wilk W Across Variables

This chart ranks the variables by Shapiro-Wilk W statistic. A higher W statistic means the variable is closer to normal shape. G1 has the highest W statistic at .986, making it the closest to normal in the output. Absences has the lowest W statistic at .772, making it the strongest non-normal variable.

This chart is important because all variables have p < .001, but they are not equally non-normal. The W statistic helps distinguish mild practical departure from strong practical departure.

Python Chart 5: Shapiro-Wilk p-values Across Variables

This chart compares the Shapiro-Wilk p-values with the alpha level of .05. Since all p-values are below .05, the formal decision is to reject normality for every listed variable. In SPSS, the values appear as .000, which should be reported as p < .001.

The p-value chart is useful for decision reporting. However, it should not be the only chart used. Because the sample size is large, p-values can be very small even for variables that are visually close to normal. That is why the W statistic chart and Q-Q plots are also necessary.

Python Chart 6: Skewness and Kurtosis Context

This chart explains why Shapiro-Wilk rejects normality. Variables with strong skewness or high kurtosis are more likely to have lower W statistics. In the SPSS output, absences has skewness = 2.021 and kurtosis = 5.781, which explains its low W statistic. G1 has skewness = -.003 and kurtosis = .037, which explains why its W statistic is closest to 1.

The chart teaches an important interpretation rule: the Shapiro-Wilk p-value tells whether normality is rejected, while skewness and kurtosis help explain the shape reason for that rejection.

Python Chart 7: Group p-value Comparison

This chart compares G3 normality by sex group. SPSS reports female G3 W = .934, p < .001 and male G3 W = .913, p < .001. Therefore, G3 normality is rejected within both groups. The male group has the lower W statistic, suggesting slightly stronger departure from normality than the female group.

This chart is useful because many statistical assumptions are checked within groups rather than only in the total sample. For example, t-tests and ANOVA often require attention to group-level distributions, variance patterns, sample sizes, and robustness.

R Chart-by-Chart Validation

The R charts validate the same Shapiro-Wilk Test workflow using a separate software environment. The R output confirms the distribution pattern, Q-Q plot interpretation, detrended Q-Q plot behavior, W statistic comparison, p-value comparison, skewness/kurtosis context, and group-level normality interpretation.

R Chart 1: Distribution with Normal Curve

The R distribution chart confirms the Python visual interpretation. The observed distribution does not perfectly match the theoretical normal curve. This agrees with the SPSS Shapiro-Wilk result, which rejects normality for G3 and the other selected variables.

R Chart 2: Q-Q Plot Normality Check

The R Q-Q plot validates the Python Q-Q plot. The points show departures from the normal reference line, especially in parts of the distribution where skewness or tail behavior is visible. This confirms that the Shapiro-Wilk rejection has a visible distribution-shape basis.

R Chart 3: Detrended Q-Q Plot

The R detrended Q-Q plot confirms where the observed values deviate from normal expectation. Deviations away from zero show the same conclusion as SPSS: the distribution is not perfectly normal.

R Chart 4: Shapiro-Wilk W Across Variables

The R W statistic chart confirms the practical ranking. G1 is closest to normal because its W statistic is highest, while absences is most non-normal because its W statistic is lowest. This validates the Python and SPSS interpretations.

R Chart 5: p-values Across Variables

The R p-value chart confirms that all variables reject normality at the selected alpha level. Like the Python p-value chart, it should be read together with W values and visual plots because all variables can be significant while still having different practical severity.

R Chart 6: Skewness and Kurtosis Context

The R skewness-kurtosis chart validates the explanation for the Shapiro-Wilk results. Absences has the strongest shape problem, while G1 is closest to normal shape. This confirms that W statistics and p-values should be interpreted with distribution-shape measures.

R Chart 7: Group p-value Comparison

The R group p-value comparison confirms that both female and male G3 distributions reject normality. It supports the SPSS group interpretation and reinforces that group-level assumption checks can matter in t-tests, ANOVA, and other grouped analyses.

SPSS, R, Python and Excel Workflows for Shapiro-Wilk Test

The same Shapiro-Wilk Test workflow can be reproduced in SPSS, R, Python and Excel. SPSS, R, and Python directly calculate the Shapiro-Wilk W statistic and p-value. Excel does not include a simple built-in Shapiro-Wilk function, so Excel is better used for supporting normality checks such as histograms, Q-Q plot preparation, skewness, kurtosis, and normal probability comparisons.

SPSS Workflow

R Workflow

Python Workflow

Excel Workflow

Code Blocks for Shapiro-Wilk Test

SPSS Syntax for Shapiro-Wilk Test

* Shapiro-Wilk Test in SPSS.
* Variables: G3 G2 G1 age absences.

TITLE "Shapiro-Wilk Test for Normality".

EXAMINE VARIABLES=G3 G2 G1 age absences
  /PLOT BOXPLOT HISTOGRAM NPPLOT
  /COMPARE GROUPS
  /STATISTICS DESCRIPTIVES
  /CINTERVAL 95
  /MISSING LISTWISE
  /NOTOTAL.

* Group normality check for G3 by sex.
EXAMINE VARIABLES=G3 BY sex
  /PLOT BOXPLOT HISTOGRAM NPPLOT
  /COMPARE GROUPS
  /STATISTICS DESCRIPTIVES
  /CINTERVAL 95
  /MISSING LISTWISE
  /NOTOTAL.

OUTPUT EXPORT
  /CONTENTS EXPORT=VISIBLE
  /PDF DOCUMENTFILE="Shapiro-Wilk-Test-SPSS-Output.pdf".

Python Code for Shapiro-Wilk Test

import pandas as pd
from scipy import stats

df = pd.read_csv("dataset.csv")

variables = ["G3", "G2", "G1", "age", "absences"]
alpha = 0.05

rows = []

for var in variables:
    x = pd.to_numeric(df[var], errors="coerce").dropna()
    w_stat, p_value = stats.shapiro(x)

    rows.append({
        "variable": var,
        "n": len(x),
        "shapiro_w": w_stat,
        "p_value": p_value,
        "decision": "Reject normality" if p_value < alpha else "Fail to reject normality"
    })

summary = pd.DataFrame(rows)
print(summary)

# Optional group normality for G3 by sex
group_rows = []
for group, sub in df.groupby("sex"):
    x = pd.to_numeric(sub["G3"], errors="coerce").dropna()
    w_stat, p_value = stats.shapiro(x)

    group_rows.append({
        "group": group,
        "n": len(x),
        "shapiro_w": w_stat,
        "p_value": p_value,
        "decision": "Reject normality" if p_value < alpha else "Fail to reject normality"
    })

group_summary = pd.DataFrame(group_rows)
print(group_summary)

R Code for Shapiro-Wilk Test

# Shapiro-Wilk Test in R

df <- read.csv("dataset.csv")

variables <- c("G3", "G2", "G1", "age", "absences")
alpha <- 0.05

rows <- list()

for(v in variables){
  x <- as.numeric(df[[v]])
  x <- x[!is.na(x)]

  result <- shapiro.test(x)

  rows[[v]] <- data.frame(
    variable = v,
    n = length(x),
    shapiro_w = as.numeric(result$statistic),
    p_value = result$p.value,
    decision = ifelse(result$p.value < alpha, "Reject normality", "Fail to reject normality")
  )
}

summary_table <- do.call(rbind, rows)
print(summary_table)

# Group normality for G3 by sex
group_rows <- list()

for(g in unique(df$sex)){
  x <- as.numeric(df$G3[df$sex == g])
  x <- x[!is.na(x)]

  result <- shapiro.test(x)

  group_rows[[as.character(g)]] <- data.frame(
    group = g,
    n = length(x),
    shapiro_w = as.numeric(result$statistic),
    p_value = result$p.value,
    decision = ifelse(result$p.value < alpha, "Reject normality", "Fail to reject normality")
  )
}

group_summary <- do.call(rbind, group_rows)
print(group_summary)

Excel Formulas for Shapiro-Wilk Supporting Normality Check

Excel does not have a simple built-in Shapiro-Wilk function.

Use Excel to support the normality decision:

Assume values are in A2:A650.

Sample size:
=COUNT(A2:A650)

Mean:
=AVERAGE(A2:A650)

Standard deviation:
=STDEV.S(A2:A650)

Skewness:
=SKEW(A2:A650)

Kurtosis:
=KURT(A2:A650)

Create sorted values:
Sort A2:A650 from smallest to largest.

Rank:
=ROW()-1

Expected probability:
=(rank-0.5)/COUNT($A$2:$A$650)

Expected normal quantile:
=NORM.S.INV(probability_cell)

Create Q-Q plot:
Scatter plot of expected normal quantile against sorted observed value.

Interpretation:
Points close to a straight line support approximate normality.
Strong bends, tail deviations or outliers suggest non-normality.

APA Reporting Wording for Shapiro-Wilk Test

When reporting the Shapiro-Wilk Test, include the W statistic, degrees of freedom or sample size, p-value, and interpretation. Also mention visual checks when the sample size is large.

APA-Style Full-Sample Report

APA-Style Group Normality Report

Student-Friendly Report Example

Common Mistakes in Shapiro-Wilk Test Interpretation

When to Use Shapiro-Wilk Test

Use the Shapiro-Wilk Test when you need a formal normality test for a continuous or approximately continuous variable. It is commonly used before t-tests, ANOVA, correlation, regression diagnostics, and transformation decisions. It is especially useful when paired with Q-Q plots and descriptive shape measures.

For a full normality workflow, combine the Shapiro-Wilk Test with Q-Q plot normality check, P-P plot normality check, Kolmogorov-Smirnov test, Lilliefors test, D’Agostino-Pearson test, and Cramer-von Mises test.

Downloads and Resources for Shapiro-Wilk Test

The resources below include the SPSS output PDF, Python charts, and R validation charts used in this guide.

Related Normality Guides