P-P Plot Normality Check: Complete R, Python, SPSS and Excel Improvement Guide 1

What Is a P-P Plot Normality Check?

A P-P plot, or probability-probability plot, compares two cumulative probability distributions. In a P-P Plot Normality Check, the observed cumulative probabilities from the sample are compared with the cumulative probabilities expected from a fitted normal distribution.

The method is visual. If the sample follows the fitted normal distribution closely, the points should lie near the diagonal reference line. If the points form a systematic curve or repeated departure from the diagonal, the distribution is not well described by the fitted normal model.

In this worked example, the outcome variable is G3 final grade. Since G3 is a discrete score with repeated integer values, the P-P plot naturally forms visible vertical clusters. That clustering is not a software error. It is a feature of the data scale and must be explained during interpretation.

If you are building a full normality and assumption-checking workflow, you may also need related guides such as the Kolmogorov-Smirnov Test, DAgostino Pearson Test, Cramer von Mises Test, Brown-Forsythe Test, Goldfeld-Quandt Test, Durbin-Watson Test and Influence Diagnostics.

P-P Plot Normality Check Formula

The P-P Plot Normality Check is built from cumulative probabilities. The sample values are sorted, empirical cumulative probabilities are calculated, and the fitted normal cumulative probability is computed for each sorted value.

Empirical cumulative probability = (i - 0.5) / n

Theoretical normal cumulative probability = Fnormal(xi; mean, standard deviation)

Here, i is the rank of the sorted value, n is the number of valid observations, and Fnormal is the cumulative distribution function of the fitted normal distribution. For this example, the fitted normal distribution used the sample mean and standard deviation of G3:

G3 n    = 649
G3 mean = 11.9060
G3 SD   = 3.2307

The plot then places theoretical normal cumulative probability on one axis and empirical cumulative probability on the other axis. Points close to the diagonal suggest closer agreement with the fitted normal distribution. Curved or systematic departures suggest non-normality.

P-P Plot Normality Check Null Hypothesis and Alternative Hypothesis

The P-P plot itself is a graph, not a formal hypothesis test. However, it is often interpreted beside formal normality tests. For the supporting Shapiro-Wilk and Kolmogorov-Smirnov tests, the hypotheses are:

Because the SPSS Shapiro-Wilk test for G3 was significant and Python also produced extremely small p-values, the formal test decision is to reject the null hypothesis of perfect normality. The visual P-P plot should therefore be described as a helpful diagnostic rather than as proof of normality.

Dataset Used for the P-P Plot Normality Check

This worked example uses the student-por.csv student performance dataset. The key grade variables are G1, G2 and G3. G1 is the first period grade, G2 is the second period grade, and G3 is the final grade. The main normality check in this guide focuses on G3.

External dataset source: UCI Machine Learning Repository: Student Performance dataset.

Verified Results in R, Python and SPSS

The P-P Plot Normality Check was reproduced in R, Python and SPSS. R created the main visual diagnostic workflow. Python validated the same values and produced improved publication-style charts. SPSS confirmed the import, descriptives, P-P plots, Q-Q plot, normality tests and residual checks.

Descriptive Statistics for G1, G2 and G3

P-P Plot Deviation Summary

SPSS Tests of Normality

Python Normality Test Validation

P-P Plot Normality Check Charts and Interpretation

1. Main R P-P Plot for G3

The main P-P plot shows that the G3 distribution partly follows the diagonal reference line, especially in the middle probability region. However, the repeated vertical groups show that the grade variable is discrete. This means the plot should be interpreted with caution and supported by formal normality tests.

2. P-P Plot Comparison for G1, G2 and G3

This comparison makes the main result clearer. G1 and G2 have smaller maximum deviations, while G3 has a maximum deviation of about 0.1227. That is why the article uses G3 as the main normality-check example.

3. Q-Q Plot Supporting the P-P Plot

The Q-Q plot shows stronger tail departures than the P-P plot. This is expected because Q-Q plots are more sensitive to tail behavior, while P-P plots often emphasize cumulative probability agreement around the center of the distribution.

4. Histogram and Normal Curve for G3

The histogram confirms that G3 is not a smooth continuous normal variable. It has repeated integer values, a concentration around the middle grades and a visible low-score tail. This supports the formal test conclusion of non-normality.

5. Empirical CDF versus Fitted Normal CDF

The ECDF chart makes the P-P plot logic easier to understand. The step line is the observed cumulative distribution, while the dashed curve is the fitted normal cumulative distribution. Gaps between these lines correspond to departures visible in the P-P plot.

6. Regression Residual P-P Plot

For regression analysis, normality is usually checked on residuals, not only on the raw dependent variable. The residual P-P plot shows a visible curved pattern, meaning residual normality is also not strongly supported in this model.

7. P-P Plot for G3 by School

The school-wise plot shows that the normality pattern differs between school groups. This does not replace a full subgroup statistical analysis, but it is useful for visual diagnosis and interpretation.

8. P-P Plot for G3 by Sex

The sex-wise plot again shows visible step-like probability clusters. This reinforces the main teaching point: P-P plots are valid diagnostic tools, but their shape must be interpreted in the context of the measurement scale.

9. Maximum P-P Probability Deviation Summary

The deviation summary is a compact way to compare the three grade variables. G3 has the largest maximum deviation, which agrees with the visual plots and formal normality-test results.

Python Validation Charts for P-P Plot Normality Check

The Python workflow produced improved publication-style validation charts. These charts are cleaner for article presentation and confirm that the result is not dependent on R alone.

The Python charts are especially useful for WordPress presentation because the titles, subtitles and footers are clearer. The statistical conclusion remains the same as R and SPSS: G3 does not follow a perfect normal distribution.

How to Run a P-P Plot Normality Check in R, Python, SPSS and Excel

P-P Plot Normality Check in R

In R, the workflow is to read the dataset, sort G3, calculate empirical probabilities, calculate fitted normal probabilities and create a scatter plot with a diagonal reference line.

# P-P Plot Normality Check in R

student <- read.csv("student-por.csv", sep = ",", stringsAsFactors = FALSE)

x <- na.omit(as.numeric(student$G3))
x_sorted <- sort(x)

n <- length(x_sorted)
empirical_probability <- ppoints(n)

mean_g3 <- mean(x_sorted)
sd_g3 <- sd(x_sorted)

normal_probability <- pnorm(x_sorted, mean = mean_g3, sd = sd_g3)

pp_data <- data.frame(
  value = x_sorted,
  empirical_probability = empirical_probability,
  normal_probability = normal_probability
)

plot(
  pp_data$normal_probability,
  pp_data$empirical_probability,
  xlab = "Theoretical normal cumulative probability",
  ylab = "Empirical cumulative probability",
  main = "P-P Plot Normality Check for G3"
)

abline(0, 1, lty = 2)

P-P Plot Normality Check in Python

In Python, SciPy can calculate theoretical normal probabilities and Matplotlib can create the plot.

# P-P Plot Normality Check in Python

import numpy as np
import pandas as pd
import matplotlib.pyplot as plt
from scipy.stats import norm, shapiro, normaltest, jarque_bera

student = pd.read_csv("student-por.csv")

x = student["G3"].dropna().astype(float).sort_values().to_numpy()
n = len(x)

empirical_probability = (np.arange(1, n + 1) - 0.5) / n
mean_g3 = np.mean(x)
sd_g3 = np.std(x, ddof=1)

normal_probability = norm.cdf(x, loc=mean_g3, scale=sd_g3)

plt.scatter(normal_probability, empirical_probability)
plt.plot([0, 1], [0, 1], linestyle="--")
plt.xlabel("Theoretical normal cumulative probability")
plt.ylabel("Empirical cumulative probability")
plt.title("P-P Plot Normality Check for G3")
plt.show()

print("Shapiro-Wilk:", shapiro(x))
print("DAgostino-Pearson:", normaltest(x))
print("Jarque-Bera:", jarque_bera(x))

P-P Plot Normality Check in SPSS

SPSS can create P-P plots with the PPLOT command. The EXAMINE command provides descriptives, normality tests, histograms and normal plots.

* P-P Plot Normality Check in SPSS.

GET DATA
  /TYPE=TXT
  /FILE='D:\low kda score priority basis posts\first post\P P Plot Normality Check\student-por.csv'
  /ENCODING='UTF8'
  /DELCASE=LINE
  /DELIMITERS=","
  /QUALIFIER='"'
  /ARRANGEMENT=DELIMITED
  /FIRSTCASE=2
  /VARIABLES=
    school A2
    sex A1
    age F8.0
    address A1
    famsize A3
    Pstatus A1
    Medu F8.0
    Fedu F8.0
    Mjob A12
    Fjob A12
    reason A12
    guardian A8
    traveltime F8.0
    studytime F8.0
    failures F8.0
    schoolsup A3
    famsup A3
    paid A3
    activities A3
    nursery A3
    higher A3
    internet A3
    romantic A3
    famrel F8.0
    freetime F8.0
    goout F8.0
    Dalc F8.0
    Walc F8.0
    health F8.0
    absences F8.0
    G1 F8.0
    G2 F8.0
    G3 F8.0.

CACHE.
EXECUTE.

PPLOT VARIABLES=G1 G2 G3
  /DISTRIBUTION=NORMAL
  /FRACTION=BLOM
  /TIES=MEAN
  /NOSTANDARDIZE
  /TYPE=P-P
  /PLOT=BOTH
  /NOLOG.

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

P-P Plot Normality Check in Excel

Excel can create a manual P-P plot by calculating empirical cumulative probabilities and fitted normal cumulative probabilities.

After creating the scatter plot, add a diagonal reference line from 0 to 1. Points near the line indicate stronger agreement with the fitted normal distribution.

How to Report the P-P Plot Normality Check Result

A good report should describe the visual pattern, the reason for any visible clustering, the supporting plots and the formal normality tests. Avoid using the P-P plot alone as proof of normality.

Related Data Analysis Guides

After learning the P-P Plot Normality Check, continue with related statistical testing and diagnostic guides on Online Internet Cafe: Kolmogorov-Smirnov Test for distribution comparison, DAgostino Pearson Test for omnibus normality testing, Cramer von Mises Test for goodness-of-fit analysis, Influence Diagnostics for regression outlier checking, Durbin-Watson Test for autocorrelation, Brown-Forsythe Test for robust variance testing and Goldfeld-Quandt Test for heteroscedasticity diagnostics.

When Should You Use a P-P Plot Normality Check?

Use a P-P Plot Normality Check when you want a visual comparison between the empirical cumulative distribution of your data and a fitted theoretical distribution, most commonly the normal distribution. It is useful in education research, psychology, social science, regression diagnostics, thesis data analysis and general statistical assumption checking.

Common Mistakes

1. Treating the P-P plot as a final normality test

A P-P plot is visual evidence. It should be combined with Q-Q plots, histograms, ECDF charts and formal tests.

2. Ignoring the measurement scale

Discrete variables such as grade scores often create step-like patterns. These patterns should be explained instead of being treated as software errors.

3. Reporting p = .000

SPSS often prints very small p-values as .000. In formal writing, report these values as p < .001.

4. Checking only the raw dependent variable in regression

For regression models, normality is usually checked on residuals, not only on the raw dependent variable.

5. Confusing P-P plots with Q-Q plots

P-P plots compare cumulative probabilities. Q-Q plots compare quantiles. They are related, but they are not identical.

Download SPSS Output and Verification Files

The SPSS PDF verifies the clean data import, descriptive statistics, P-P plot output, Q-Q plot output, Explore normality tests, grouped normality checks and regression residual normality check.

External References for P-P Plot Normality Check

This post uses verified R, Python and SPSS outputs together with external software documentation and statistical references. These resources help readers verify the background of normal probability plots, normality testing and distribution diagnostics.

• UCI Machine Learning Repository: Student Performance dataset
• IBM SPSS Statistics documentation
• SciPy statistics documentation
• The R Project for Statistical Computing
• Microsoft Excel support