Quick Answer: P-P Plot Normality Check Result

The verified SPSS output checks normal probability behavior for five numeric variables: G3, G2, G1, age, and absences. Each variable has N = 649 valid cases and 0 missing cases. For the main G3 final grade variable, SPSS reports mean = 11.91, standard deviation = 3.231, skewness = -0.913, and kurtosis = 2.712. This means the P-P plot should show visible probability departures from perfect normality, although the P-P plot usually emphasizes the center of the distribution more than the extreme tails.

The manual P-P calculation for G3 gives pp_n = 649, pp_mean = 11.91, pp_sd = 3.23, maximum absolute P-P difference = .12275, and mean absolute P-P difference = .03317. These values show that the observed cumulative probabilities do not perfectly match the expected normal probabilities. The first sorted cases include repeated G3 scores of 0, which creates lower-tail probability differences.

Hypothesis-style interpretation: A P-P plot is visual, so it does not directly produce a p-value. However, the same SPSS output includes formal normality tests. For G3, the Shapiro-Wilk statistic is .926 and p < .001. Therefore, the formal normality decision is to reject normality. The P-P plot supports this decision by showing that observed cumulative probabilities do not perfectly follow the expected normal probability line.

What Is P-P Plot Normality Check?

P-P Plot Normality Check compares observed cumulative probabilities with expected cumulative probabilities from a theoretical normal distribution. The term P-P means probability-probability. In a normal P-P plot, the diagonal line represents perfect agreement between observed and expected normal cumulative probabilities.

If the data are approximately normal, the plotted points should follow the diagonal line closely. If the points curve away from the line, the distribution may be skewed, heavy-tailed, light-tailed, or affected by outliers. P-P plots are often useful for seeing how well the cumulative distribution fits the normal model, especially around the middle of the distribution.

In this guide, the main example is G3 final grade. The SPSS output shows that G3 has mean = 11.91, SD = 3.231, skewness = -0.913, and kurtosis = 2.712. These values explain why the P-P plot does not perfectly follow the diagonal line. The data have negative skew and heavy-tail behavior, so observed cumulative probabilities differ from expected normal probabilities.

P-P Plot Formula and Probability Calculation

A P-P plot is built by sorting the observed values, calculating the observed cumulative probability, and comparing it with the expected normal cumulative probability for each value.

Here, rank is the ordered position of the value and n is the sample size. The expected normal probability is calculated from the normal distribution after standardizing the observed value:

The P-P difference compares the observed cumulative probability with the expected normal probability:

Null and Alternative Hypothesis for P-P Plot Normality

A P-P plot is visual and does not directly test a hypothesis. However, it supports the same normality hypothesis used in formal tests such as Shapiro-Wilk and Kolmogorov-Smirnov.

Dataset and Variables Used

The worked example uses student performance variables. The full-sample P-P plot normality check includes G3, G2, G1, age, and absences. SPSS also provides a group-level normality context for G3 by sex.

For descriptive context before interpreting P-P plots, review descriptive statistics, frequency distribution, histogram interpretation, box plot interpretation, and five-number summary.

Verified SPSS Output Interpretation

The SPSS output provides normality plots, detrended normality plots, descriptives, tests of normality, manual P-P probability values, and group normality context. The main variable is G3 final grade, but the output also includes G2, G1, age, and absences.

SPSS Case Processing Summary

SPSS Descriptive Statistics for P-P Plot Context

SPSS Tests of Normality

Manual P-P Plot Calculation for G3

First 20 Sorted G3 P-P Cases

SPSS Group P-P Plot Normality Context for G3 by Sex

Python Chart-by-Chart Interpretation

The Python charts show the P-P Plot Normality Check visually. They include the main P-P plot, detrended P-P plot, histogram with normal curve, Q-Q plot comparison, P-P difference comparison across variables, skewness/kurtosis context, and group P-P difference comparison.

Python Chart 1: P-P Plot Normality Check

This chart is the main P-P Plot Normality Check. If G3 followed a perfect normal distribution, the points would stay close to the diagonal reference line. Instead, the points show visible departures, which agrees with the SPSS Shapiro-Wilk result of W = .926 and p < .001.

The P-P plot is useful because it shows cumulative probability agreement. It is often easier to read around the middle of the distribution than the extreme tails. For G3, the plot shows that the observed cumulative probabilities do not perfectly match expected normal probabilities.

Python Chart 2: Detrended P-P Plot

The detrended P-P plot displays the difference between observed and expected probabilities. Values close to zero indicate good normal probability alignment. Values above or below zero show where the observed cumulative probability differs from the normal model.

For G3, the detrended plot confirms that the probability differences are not only random noise. The manual P-P output reports a maximum absolute difference of .12275 and a mean absolute difference of .03317, which helps quantify the visual departure.

Python Chart 3: Distribution with Normal Curve

This chart shows the observed distribution with a normal curve overlay. The histogram helps explain why the P-P plot departs from the line. G3 has a mean of 11.91, standard deviation of 3.231, skewness of -.913, and kurtosis of 2.712. These values indicate non-perfect normal shape.

The histogram and P-P plot should be interpreted together. The histogram shows the overall distribution shape, while the P-P plot shows cumulative probability alignment.

Python Chart 4: Q-Q Plot Comparison

The Q-Q plot comparison helps distinguish two related normality plots. A P-P plot compares cumulative probabilities and often emphasizes the center. A Q-Q plot compares quantiles and is usually more sensitive to tail behavior. This chart shows why both can be useful.

For G3, the Q-Q comparison is helpful because low scores and tail behavior are part of the normality issue. The P-P plot confirms probability departure, while the Q-Q plot helps locate tail-level deviation.

Python Chart 5: P-P Differences Across Variables

This chart compares P-P difference patterns across G3, G2, G1, age, and absences. Variables with larger P-P differences depart more from expected normal cumulative probabilities. Based on SPSS shape values, absences should show the strongest departure because it has skewness = 2.021 and kurtosis = 5.781. G1 should show the closest probability alignment because its skewness and kurtosis are near zero.

The P-P difference comparison is useful because it summarizes normality visually across multiple variables. Instead of looking only at one P-P plot, the chart helps identify which variables need more attention, transformation, or robust methods.

Python Chart 6: Skewness and Kurtosis Context

This chart explains why P-P plot departures happen. Skewness tells whether the distribution is asymmetric, while kurtosis tells whether the distribution has unusual peak or tail behavior. G3 has negative skewness and positive kurtosis, so its cumulative probability pattern differs from the perfect normal line. Absences has strong positive skewness and high kurtosis, so it should show even stronger departure.

This chart is important for teaching interpretation. A P-P plot shows probability behavior, but skewness and kurtosis explain the distribution-shape reason behind that behavior.

Python Chart 7: Group P-P Difference Comparison

This chart compares P-P differences for G3 by sex group. SPSS shows that female students have N = 383, mean = 12.25, SD = 3.124, and Shapiro-Wilk W = .934. Male students have N = 266, mean = 11.41, SD = 3.321, and Shapiro-Wilk W = .913. Both groups reject normality at p < .001.

The group comparison matters because many statistical tests require checking assumptions within groups, not only in the total sample. The male group has a slightly lower W statistic and slightly larger standard deviation, suggesting slightly stronger departure from normality than the female group.

R Chart-by-Chart Validation

The R charts validate the same P-P Plot Normality Check using a separate software workflow. The R results confirm the Python and SPSS interpretation: G3 departs from perfect normality, the detrended P-P plot shows systematic deviations, and group-level probability behavior should be checked when comparing groups.

R Chart 1: P-P Plot Normality Check

The R P-P plot validates the Python P-P plot. The points do not perfectly follow the diagonal line, which supports the SPSS conclusion that G3 is formally non-normal.

R Chart 2: Detrended P-P Plot

The R detrended P-P plot confirms the same probability-deviation pattern as Python and SPSS. Values away from zero show that the observed data do not fully match expected normal cumulative probabilities.

R Chart 3: Distribution with Normal Curve

The R distribution chart confirms that the observed G3 distribution is not a perfect normal bell shape. The histogram supports the P-P plot by showing the overall distribution pattern.

R Chart 4: Q-Q Plot Comparison

The R Q-Q comparison confirms that P-P and Q-Q plots emphasize different parts of the distribution. The P-P plot is useful for cumulative probability fit, while the Q-Q plot is especially useful for tail and quantile behavior.

R Chart 5: P-P Differences Across Variables

The R difference comparison validates the Python difference chart. Variables with stronger skewness and kurtosis show larger P-P departures. Absences is expected to be the strongest case, while G1 is closest to normal.

R Chart 6: Skewness and Kurtosis Context

The R skewness-kurtosis chart validates the explanation for the P-P plot deviations. Shape statistics explain why some variables depart more strongly from the normal probability line than others.

R Chart 7: Group P-P Difference Comparison

The R group comparison confirms the SPSS group normality context. Both female and male groups show non-normality for G3. This supports checking grouped P-P plots when the later analysis compares groups.

SPSS, R, Python and Excel Workflows for P-P Plot Normality Check

The same P-P Plot Normality Check can be created in SPSS, R, Python, and Excel. SPSS creates normality plots directly through Explore. R and Python can calculate observed and expected cumulative probabilities. Excel can build a manual P-P plot using sorted values, plotting probabilities, z scores, and normal cumulative probabilities.

SPSS Workflow

R Workflow

Python Workflow

Excel Workflow

Code Blocks for P-P Plot Normality Check

SPSS Syntax for P-P Plot Normality Check

* P-P Plot Normality Check in SPSS.
* Variables: G3 G2 G1 age absences.

TITLE "P-P Plot Normality Check".

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

* Group normality context 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="P-P-Plot-Normality-Check-SPSS-Output.pdf".

Python Code for P-P Plot Normality Check

import pandas as pd
import numpy as np
from scipy import stats
import matplotlib.pyplot as plt

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

x = pd.to_numeric(df["G3"], errors="coerce").dropna()
n = len(x)

# Formal normality support test
w_stat, p_value = stats.shapiro(x)
print("Shapiro-Wilk W:", w_stat)
print("p-value:", p_value)

# Manual P-P values
observed_values = np.sort(x.to_numpy())
rank = np.arange(1, n + 1)
observed_probability = (rank - 0.5) / n

z_values = (observed_values - x.mean()) / x.std(ddof=1)
expected_probability = stats.norm.cdf(z_values)

pp_difference = observed_probability - expected_probability
abs_difference = np.abs(pp_difference)

pp_summary = {
    "n": n,
    "mean": x.mean(),
    "sd": x.std(ddof=1),
    "max_abs_difference": abs_difference.max(),
    "mean_abs_difference": abs_difference.mean()
}

print(pp_summary)

pp_table = pd.DataFrame({
    "rank": rank,
    "observed_value": observed_values,
    "observed_probability": observed_probability,
    "z_value": z_values,
    "expected_probability": expected_probability,
    "pp_difference": pp_difference,
    "abs_difference": abs_difference
})

print(pp_table.head(20))

# P-P plot
fig, ax = plt.subplots(figsize=(8, 6))
ax.scatter(expected_probability, observed_probability, s=18)
ax.plot([0, 1], [0, 1], linestyle="--")
ax.set_xlabel("Expected normal cumulative probability")
ax.set_ylabel("Observed cumulative probability")
ax.set_title("P-P Plot Normality Check for G3")
plt.tight_layout()
plt.show()

R Code for P-P Plot Normality Check

# P-P Plot Normality Check in R

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

x <- as.numeric(df$G3)
x <- x[!is.na(x)]
n <- length(x)

# Formal normality support test
print(shapiro.test(x))

observed_values <- sort(x)
rank <- seq_along(observed_values)
observed_probability <- (rank - 0.5) / n

z_values <- (observed_values - mean(x)) / sd(x)
expected_probability <- pnorm(z_values)

pp_difference <- observed_probability - expected_probability
abs_difference <- abs(pp_difference)

pp_summary <- data.frame(
  n = n,
  mean = mean(x),
  sd = sd(x),
  max_abs_difference = max(abs_difference),
  mean_abs_difference = mean(abs_difference)
)

print(pp_summary)

pp_table <- data.frame(
  rank = rank,
  observed_value = observed_values,
  observed_probability = observed_probability,
  z_value = z_values,
  expected_probability = expected_probability,
  pp_difference = pp_difference,
  abs_difference = abs_difference
)

print(head(pp_table, 20))

plot(expected_probability, observed_probability,
     main = "P-P Plot Normality Check for G3",
     xlab = "Expected normal cumulative probability",
     ylab = "Observed cumulative probability")
abline(0, 1, lty = 2)

Excel Formulas for Manual P-P Plot

Assume sorted observed values are in A2:A650.

Step 1: Sort the variable from smallest to largest.

Step 2: Create rank in B2:
=ROW()-1

Step 3: Calculate sample size in a fixed cell, for example E1:
=COUNT(A2:A650)

Step 4: Calculate observed cumulative probability in C2:
=(B2-0.5)/$E$1

Step 5: Calculate sample mean in E2:
=AVERAGE($A$2:$A$650)

Step 6: Calculate sample standard deviation in F2:
=STDEV.S($A$2:$A$650)

Step 7: Calculate z value in D2:
=(A2-$E$2)/$F$2

Step 8: Calculate expected normal probability in G2:
=NORM.S.DIST(D2,TRUE)

Step 9: Calculate P-P difference in H2:
=C2-G2

Step 10: Calculate absolute difference in I2:
=ABS(H2)

Step 11: Create P-P plot:
Insert a scatter plot using expected normal probability and observed cumulative probability.

Interpretation:
Points close to the diagonal line support approximate normality.
Systematic curves or large probability differences suggest non-normality.

APA Reporting Wording for P-P Plot Normality Check

When reporting a P-P Plot Normality Check, describe the visual probability pattern and support it with formal tests or shape statistics. Do not report the P-P plot as if it directly gives a p-value. Instead, say what the plot showed and how that agrees or disagrees with Shapiro-Wilk, Kolmogorov-Smirnov, skewness, and kurtosis.

APA-Style P-P Plot Report

APA-Style Group P-P Plot Report

Student-Friendly Report Example

Common Mistakes in P-P Plot Normality Check

When to Use P-P Plot Normality Check

Use P-P Plot Normality Check when you need to visually inspect whether observed cumulative probabilities match expected normal cumulative probabilities. It is especially useful before parametric tests, regression diagnostics, ANOVA, t-tests, correlation, and transformation decisions.

For a complete normality workflow, combine this guide with Q-Q Plot Normality Check, Shapiro-Wilk Test, Kolmogorov-Smirnov Test, Lilliefors Test, D’Agostino-Pearson Test, and Skewness and Kurtosis Normality Check.

Downloads and Resources for P-P Plot Normality Check

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

Related Statistical Guides

Use these related guides to connect P-P Plot Normality Check with normality tests, descriptive statistics, transformations, residual diagnostics, and reporting.