Q-Q Plot Normality Check: Formula, Interpretation, R, Python, SPSS and Excel Guide

What Is a Q-Q Plot Normality Check?

A Q-Q Plot Normality Check is a visual method used to compare a sample distribution with a theoretical normal distribution. The term Q-Q means quantile-quantile. A quantile is a position-based value in a distribution. In a normal Q-Q plot, the observed sample quantiles are placed against the theoretical quantiles that would be expected if the data were normally distributed.

When a variable is normally distributed, the plotted points fall close to a straight diagonal reference line. When the points curve away from the line, form an S-shape, flatten in the upper tail, drop sharply in the lower tail or show extreme outliers, the variable is not behaving like a normal distribution.

The Q-Q Plot Normality Check is especially useful because it shows where normality fails. A p-value can tell us that a distribution is statistically non-normal, but a Q-Q plot shows whether the problem is in the center, the lower tail, the upper tail, both tails or because of outliers.

In this worked example, the Q-Q plot is applied to the G3 final grade variable from the student-por.csv dataset. The result is not perfectly normal. The center of the distribution looks fairly close to normal, but the lower tail clearly departs because several students have very low grades. This tail departure is also supported by the Shapiro-Wilk p-value below .001.

For a complete assumption-checking workflow, Q-Q Plot Normality Check can be used together with related normality and distribution tests such as the Kolmogorov-Smirnov Test, DAgostino Pearson Test, Cramer von Mises Test and Lilliefors Test. If the analysis is regression-based, residual normality should also be checked along with influence and diagnostic tools such as Influence Diagnostics.

Q-Q Plot Formula and Logic

The Q-Q plot does not test normality by one simple formula in the same way that a t test or z test does. Instead, it compares two ordered sets of quantiles:

Observed sample quantiles  vs  Theoretical normal quantiles

The observed sample values are first sorted from smallest to largest. Each sorted value receives a plotting position. That plotting position is then converted into a theoretical normal quantile. If the sample comes from a normal distribution, the observed values and theoretical normal quantiles should form an approximately straight line.

Step 1: Sort the observed values from smallest to largest
Step 2: Assign each value a cumulative probability position
Step 3: Convert each probability into a theoretical normal quantile
Step 4: Plot observed sample quantiles against theoretical normal quantiles
Step 5: Read the distance from the straight reference line

The main idea is simple: if the data are normal, the observed quantiles should match the normal quantiles. If the observed lower tail is too low, points fall below the line on the left. If the upper tail is too high, points rise above the line on the right. If both tails depart, the plot may form an S-shape.

Normality Hypotheses Used with Q-Q Plot Checking

A Q-Q plot itself is visual, so it does not have a formal null hypothesis unless it is supported by a statistical normality test. In this post, the visual Q-Q Plot Normality Check is supported by the Shapiro-Wilk test and other formal normality tests.

Because the Shapiro-Wilk p-value is below .001, the null hypothesis of normality is rejected. The visual Q-Q plot agrees with this decision because the lower tail does not follow the normal reference line.

Dataset and Variables Used

This worked example uses the student-por.csv student performance dataset. The main variable checked for normality is G3, which represents the final grade. Earlier grade variables G1 and G2 are also used for comparison charts. The school and sex variables are used to check whether the G3 normality pattern changes across groups.

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

Verified Q-Q Plot Normality Check Results

The Q-Q Plot Normality Check was reproduced with Python, R and SPSS-compatible output. The main conclusion is that G3 final grade departs from normality. The central values are not the main problem. The main problem appears in the lower tail, where very low grades create a clear departure from the expected normal quantile pattern.

Descriptive Statistics for G3

Normality Test Summary

The formal tests agree with the visual Q-Q plot. Shapiro-Wilk, Anderson-Darling, Jarque-Bera, Lilliefors, Cramer-von Mises and Kolmogorov-Smirnov all support the conclusion that G3 is not normally distributed. The exact strength differs by test because each test reacts differently to skewness, kurtosis, tail departure, repeated values and sample size.

Chart-by-Chart Interpretation of the Q-Q Plot Normality Check

This section explains each chart used in the Q-Q Plot Normality Check. The purpose is not only to show the graphs, but to interpret what each graph proves about the normality of G3 final grade. The overall result is clear: G3 is partly close to normal in the middle, but the tails, especially the lower tail, do not follow the normal distribution.

Chart 1: Q-Q Plot Normality Check for G3 Final Grade

Chart 2: Histogram and Normal Curve for G3

Chart 3: Deviation from the Q-Q Reference Line

Chart 4: Q-Q Plot Comparison for G1, G2 and G3

Chart 5: Q-Q Plot Normality Check for G3 by School

Chart 6: Q-Q Plot Normality Check for G3 by Sex

Chart 7: Normality Test P-Value Summary

Chart 8: Residual Q-Q Plot Normality Check

Chart 9: Boxplot Support for Normality Checking

Chart 10: Formula and Interpretation Summary Panel

R Code for Q-Q Plot Normality Check

The following R workflow can be used to reproduce the Q-Q Plot Normality Check, histogram, normality statistics and summary output for G3. Update the folder path before running the code.

# Q-Q Plot Normality Check in R
# Update this folder path
folder <- "D:/low kda score priority basis posts/first post/Q Q Plot Normality Check"
data_file <- file.path(folder, "student-por.csv")
out_dir <- file.path(folder, "R_Output")
dir.create(out_dir, showWarnings = FALSE, recursive = TRUE)

# Load data
df <- read.csv(data_file, sep = ";", stringsAsFactors = FALSE)

# Clean G3
g3 <- as.numeric(df$G3)
g3 <- g3[!is.na(g3)]

# Basic statistics
n <- length(g3)
mean_g3 <- mean(g3)
sd_g3 <- sd(g3)
skew_g3 <- mean((g3 - mean_g3)^3) / sd_g3^3
kurt_excess <- mean((g3 - mean_g3)^4) / sd_g3^4 - 3

# Shapiro-Wilk test
shapiro_result <- shapiro.test(g3)

# Q-Q correlation
qq <- qqnorm(g3, plot.it = FALSE)
qq_correlation <- cor(qq$x, qq$y)

# Save summary
summary_table <- data.frame(
  Statistic = c("n", "Mean", "SD", "Skewness", "Excess kurtosis", "Q-Q correlation", "Shapiro-Wilk W", "Shapiro-Wilk p-value"),
  Value = c(n, mean_g3, sd_g3, skew_g3, kurt_excess, qq_correlation, shapiro_result$statistic, shapiro_result$p.value)
)

write.csv(summary_table, file.path(out_dir, "r_qq_plot_normality_summary.csv"), row.names = FALSE)

# Q-Q plot
png(file.path(out_dir, "r_qq_plot_g3.png"), width = 1600, height = 900, res = 150)
qqnorm(g3, main = "R Q-Q Plot Normality Check for G3 Final Grade",
       xlab = "Theoretical normal quantiles",
       ylab = "Sample quantiles of G3")
qqline(g3, lwd = 2)
dev.off()

# Histogram with normal curve
png(file.path(out_dir, "r_histogram_normal_curve_g3.png"), width = 1600, height = 900, res = 150)
hist(g3, probability = TRUE, breaks = 16,
     main = "R Histogram and Normal Curve for G3",
     xlab = "G3 final grade",
     ylab = "Density")
curve(dnorm(x, mean = mean_g3, sd = sd_g3), add = TRUE, lwd = 2, lty = 2)
abline(v = mean_g3, lwd = 2)
dev.off()

R interpretation: If the Q-Q plot points fall close to the line, normality is visually acceptable. In this example, the central values are close to the line, but the lower tail departs strongly. The Shapiro-Wilk p-value below .001 confirms that G3 does not satisfy normality.

Python Code for Q-Q Plot Normality Check

The Python workflow below creates the same Q-Q plot normality check and supporting statistics. It also saves cleaned outputs that can be used in SPSS workflows.

# Q-Q Plot Normality Check in Python
# Update this folder path
import os
import numpy as np
import pandas as pd
import matplotlib.pyplot as plt
from scipy import stats

folder = r"D:\low kda score priority basis posts\first post\Q Q Plot Normality Check"
data_file = os.path.join(folder, "student-por.csv")
out_dir = os.path.join(folder, "Python_Output")
os.makedirs(out_dir, exist_ok=True)

# Load data
df = pd.read_csv(data_file, sep=";")

# Clean G3
df["G3"] = pd.to_numeric(df["G3"], errors="coerce")
g3 = df["G3"].dropna().to_numpy()

# Statistics
n = len(g3)
mean_g3 = np.mean(g3)
sd_g3 = np.std(g3, ddof=1)
skew_g3 = stats.skew(g3, bias=False)
kurt_excess = stats.kurtosis(g3, fisher=True, bias=False)
shapiro_w, shapiro_p = stats.shapiro(g3)

# Q-Q correlation
osm, osr = stats.probplot(g3, dist="norm", fit=False)
qq_corr = np.corrcoef(osm, osr)[0, 1]

summary = pd.DataFrame({
    "Statistic": [
        "n", "Mean", "SD", "Skewness", "Excess kurtosis",
        "Q-Q correlation", "Shapiro-Wilk W", "Shapiro-Wilk p-value"
    ],
    "Value": [
        n, mean_g3, sd_g3, skew_g3, kurt_excess,
        qq_corr, shapiro_w, shapiro_p
    ]
})

summary.to_csv(os.path.join(out_dir, "python_qq_plot_normality_summary.csv"), index=False)

# Q-Q plot
fig, ax = plt.subplots(figsize=(12, 7))
stats.probplot(g3, dist="norm", plot=ax)
ax.set_title("Python Q-Q Plot Normality Check for G3 Final Grade", fontweight="bold")
ax.set_xlabel("Theoretical normal quantiles")
ax.set_ylabel("Sample quantiles of G3")
ax.grid(True, alpha=0.25)
plt.tight_layout()
plt.savefig(os.path.join(out_dir, "python_qq_plot_g3.png"), dpi=200)
plt.close()

# Histogram with normal curve
x = np.linspace(min(g3), max(g3), 300)
normal_curve = stats.norm.pdf(x, mean_g3, sd_g3)

fig, ax = plt.subplots(figsize=(12, 7))
ax.hist(g3, bins=16, density=True, alpha=0.65, label="Observed grade density")
ax.plot(x, normal_curve, linestyle="--", linewidth=2, label="Normal curve")
ax.axvline(mean_g3, linewidth=2, label=f"Observed mean = {mean_g3:.2f}")
ax.set_title("Python Histogram and Normal Curve for G3", fontweight="bold")
ax.set_xlabel("G3 final grade")
ax.set_ylabel("Density")
ax.legend()
ax.grid(True, alpha=0.25)
plt.tight_layout()
plt.savefig(os.path.join(out_dir, "python_histogram_normal_curve_g3.png"), dpi=200)
plt.close()

# Save SPSS-ready clean CSV
clean_file = os.path.join(folder, "student_por_clean_for_spss.csv")
df.to_csv(clean_file, index=False)

print(summary)
print("SPSS-ready file saved as:", clean_file)

Python interpretation: Python confirms the same conclusion: the Q-Q plot is fairly linear in the center, but the lower tail departs from the normal reference line. The Shapiro-Wilk test gives p < .001, so G3 is not normally distributed.

SPSS Syntax for Q-Q Plot Normality Check

SPSS can create Q-Q plots, histograms, boxplots and Shapiro-Wilk normality tests through the EXAMINE procedure. Use the cleaned SPSS-ready file created from Python, or import the original student-por.csv file directly if it opens correctly.

* Q-Q Plot Normality Check in SPSS.
* Update the file path before running.

GET DATA
  /TYPE=TXT
  /FILE='D:\low kda score priority basis posts\first post\Q Q Plot Normality Check\student_por_clean_for_spss.csv'
  /ENCODING='UTF8'
  /DELCASE=LINE
  /DELIMITERS=","
  /QUALIFIER='"'
  /ARRANGEMENT=DELIMITED
  /FIRSTCASE=2
  /VARIABLES=
    school A10
    sex A10
    age F8.2
    G1 F8.2
    G2 F8.2
    G3 F8.2.
EXECUTE.

DATASET NAME QQPlotData.

* Descriptive statistics and normality plots for G3.
EXAMINE VARIABLES=G3
  /PLOT BOXPLOT HISTOGRAM NPPLOT
  /COMPARE GROUPS
  /STATISTICS DESCRIPTIVES
  /CINTERVAL 95
  /MISSING LISTWISE
  /NOTOTAL.

* Q-Q plot and normality check by school.
SORT CASES BY school.
SPLIT FILE LAYERED BY school.

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

SPLIT FILE OFF.

* Q-Q plot and normality check by sex.
SORT CASES BY sex.
SPLIT FILE LAYERED BY sex.

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

SPLIT FILE OFF.

* Optional: Compare G1, G2 and G3 normality.
EXAMINE VARIABLES=G1 G2 G3
  /PLOT BOXPLOT HISTOGRAM NPPLOT
  /COMPARE GROUPS
  /STATISTICS DESCRIPTIVES
  /CINTERVAL 95
  /MISSING LISTWISE
  /NOTOTAL.

How to Read the SPSS Output

Excel Method for Q-Q Plot Normality Check

Excel does not provide a one-click professional Q-Q plot like R, Python or SPSS, but it can still be used to create a basic normal Q-Q plot. The method is based on sorting the data, calculating plotting positions and converting those positions into theoretical normal quantiles.

Excel Steps

Excel interpretation: If the scatter points follow a straight line, the variable is approximately normal. If the points curve away from the line, the variable departs from normality. For G3, Excel will show the same lower-tail departure seen in Python, R and SPSS.

Download Output and Resources

The complete Q-Q Plot Normality Check output PDF is available below. It includes the main charts and interpretation visuals used in this guide.

APA Style Reporting for Q-Q Plot Normality Check

Normality reporting should combine visual interpretation and formal test results. Do not write only “the data are not normal” without explaining why. In this example, the strongest reason is the lower-tail departure visible in the Q-Q plot.

For a shorter report, use the following version:

Visual inspection of the Q-Q plot showed a clear lower-tail departure from normality for G3. The Shapiro-Wilk test was significant, W = 0.9260, p < .001, indicating that G3 was not normally distributed.

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

Use a Q-Q Plot Normality Check whenever your analysis depends on normality or residual normality. This includes t tests, ANOVA, repeated-measures ANOVA, regression diagnostics, z tests for means under normal assumptions and many parametric confidence interval procedures. In practice, the Q-Q plot is often more informative than a p-value because it shows the type and location of non-normality.

References and Related Guides

This guide uses the student-por.csv dataset and focuses on practical Q-Q Plot Normality Check interpretation. For deeper assumption testing and related workflows, read these connected guides: