Quick Answer: Q-Q Plot Normality Check Result

The verified SPSS output checks Q-Q plot normality 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 Q-Q plot should be expected to show visible departures from a perfect normal line, especially in the tails.

The manual Q-Q calculation for G3 gives qq_n = 649, qq_mean = 11.91, qq_sd = 3.23, maximum absolute Q-Q residual = 5.42009, and mean absolute Q-Q residual = .51630. These residuals show that the observed G3 values do not perfectly match expected normal quantiles. The first sorted cases include several observed values of 0, while the expected normal values are positive, creating large negative residuals in the lower tail.

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

What Is Q-Q Plot Normality Check?

Q-Q Plot Normality Check compares the quantiles of observed data with the quantiles expected from a normal distribution. The term Q-Q means quantile-quantile. In a normal Q-Q plot, the horizontal axis usually represents observed values or theoretical values depending on software, while the vertical axis represents the corresponding expected normal values or observed quantiles. The key idea is simple: if the data are normally distributed, the plotted points should form an approximately straight line.

A Q-Q plot is especially useful because it shows where the data depart from normality. A histogram may show the general shape, but a Q-Q plot shows whether the lower tail, upper tail, center, or outliers are causing the problem. If the middle points follow the line but the ends curve away, the main issue is in the tails. If the whole plot bends, the distribution may be skewed. If a few points are far away, outliers may be present.

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 Q-Q plot is not perfectly straight. The data have negative skew and heavy-tail behavior, so the Q-Q plot shows visible departures from the normal reference line.

Q-Q Plot Formula and Quantile Calculation

A Q-Q plot is built by sorting the observed values and comparing them with expected normal quantiles. A common plotting position formula is:

Here, i is the rank of the sorted observation and n is the sample size. After calculating the plotting probability, the expected normal z value is:

For a normal Q-Q plot on the original variable scale, the expected normal value can be calculated as:

The Q-Q residual then compares the observed sorted value with the expected normal value:

Null and Alternative Hypothesis for Q-Q Plot Normality

A Q-Q plot is visual, so it does not directly test a hypothesis. However, it supports the normality decision used in formal tests. The null hypothesis says the variable follows a normal distribution. The alternative hypothesis says the variable does not follow a normal distribution.

Dataset and Variables Used

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

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

Verified SPSS Output Interpretation

The SPSS output provides normal Q-Q plots, detrended Q-Q plots, descriptives, tests of normality, manual Q-Q 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 Q-Q Plot Context

SPSS Tests of Normality

Manual Q-Q Plot Calculation for G3

First 20 Sorted G3 Q-Q Cases

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

Python Chart-by-Chart Interpretation

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

Python Chart 1: Q-Q Plot Normality Check

This chart is the main Q-Q Plot Normality Check. If G3 were perfectly normal, the points would follow the diagonal reference line closely. Instead, the points show visible departures, especially near the lower tail. This agrees with the SPSS result that G3 has Shapiro-Wilk W = .926 and p < .001.

The Q-Q plot is more informative than a p-value alone because it shows where the normality problem occurs. For G3, the lower-tail points are pulled away from the line because there are very low grade values, including zeros. The central points may be closer to the line, but tail departures show that the distribution is not perfectly normal.

Python Chart 2: Detrended Q-Q Plot

The detrended Q-Q plot displays the difference between observed and expected normal values. Values close to zero indicate good normal alignment. Values above or below zero show where the data deviate from normality. This chart makes the departures easier to see than the ordinary Q-Q plot.

For G3, the detrended plot confirms that the deviations are not random noise only. The pattern shows systematic tail departure. This supports the interpretation that G3 has non-normal features, especially because its skewness is -.913 and kurtosis is 2.712.

Python Chart 3: Distribution with Normal Curve

This chart shows the observed distribution with a normal curve overlay. The histogram helps explain why the Q-Q plot departs from the line. G3 has a mean of 11.91, standard deviation of 3.231, and values ranging from 0 to 19. The shape is not a perfect bell curve because the distribution has negative skewness and heavy tails.

The histogram and Q-Q plot should be interpreted together. The histogram shows the overall shape, while the Q-Q plot shows detailed quantile-level departure. Together, they provide stronger evidence than either chart alone.

Python Chart 4: P-P Plot Comparison

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

For normality checking, the Q-Q plot is often preferred when tail behavior matters. In this G3 example, tail departures are important because the lower end includes several very low values. Therefore, the Q-Q plot is more useful than a P-P plot alone.

Python Chart 5: Q-Q Residuals Across Variables

This chart compares Q-Q residual patterns across G3, G2, G1, age, and absences. Variables with larger Q-Q residuals depart more from normal expectation. 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 Q-Q alignment because its skewness and kurtosis are near zero.

The residual comparison chart is useful because it summarizes normality visually across multiple variables. Instead of looking only at one Q-Q 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 Q-Q 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 Q-Q plot shows tail-related departure. Absences has strong positive skewness and high kurtosis, so its Q-Q plot should show even stronger non-normality.

This chart is important for teaching interpretation. A Q-Q plot shows the visual pattern, but skewness and kurtosis explain the shape reason behind that pattern.

Python Chart 7: Group Q-Q Residual Comparison

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

R Chart 1: Q-Q Plot Normality Check

The R Q-Q plot validates the Python Q-Q plot. The points do not form a perfect straight line, which supports the SPSS conclusion that G3 is formally non-normal. This independent validation strengthens the interpretation.

R Chart 2: Detrended Q-Q Plot

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

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 Q-Q plot by showing the overall distribution pattern.

R Chart 4: P-P Plot Comparison

The R P-P comparison confirms that Q-Q and P-P plots emphasize different parts of the distribution. The Q-Q plot is especially useful for tail departure, while the P-P plot gives a cumulative probability perspective.

R Chart 5: Q-Q Residuals Across Variables

The R residual comparison validates the Python residual chart. Variables with stronger skewness and kurtosis show larger Q-Q 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 Q-Q plot deviations. Shape statistics explain why some variables depart more strongly from the normal line than others.

R Chart 7: Group Q-Q Residual 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 Q-Q plots when the later analysis compares groups.

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

The same Q-Q Plot Normality Check can be created in SPSS, R, Python, and Excel. SPSS creates Q-Q plots directly through Explore. R and Python can calculate theoretical quantiles and plot observed values against expected normal values. Excel can build a manual Q-Q plot using sorted values and the NORM.S.INV formula.

SPSS Workflow

R Workflow

Python Workflow

Excel Workflow

Code Blocks for Q-Q Plot Normality Check

SPSS Syntax for Q-Q Plot Normality Check

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

TITLE "Q-Q Plot Normality Check".

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

* Group Q-Q plot 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="Q-Q-Plot-Normality-Check-SPSS-Output.pdf".

Python Code for Q-Q 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)

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

# Q-Q plot
fig, ax = plt.subplots(figsize=(8, 6))
stats.probplot(x, dist="norm", plot=ax)
ax.set_title("Q-Q Plot Normality Check for G3")
ax.set_xlabel("Theoretical normal quantiles")
ax.set_ylabel("Observed ordered values")
plt.tight_layout()
plt.show()

# Manual Q-Q values
observed = np.sort(x.to_numpy())
rank = np.arange(1, n + 1)
probability = (rank - 0.5) / n
theoretical_z = stats.norm.ppf(probability)
expected_value = x.mean() + theoretical_z * x.std(ddof=1)
qq_residual = observed - expected_value

qq_summary = {
    "n": n,
    "mean": x.mean(),
    "sd": x.std(ddof=1),
    "max_abs_residual": np.max(np.abs(qq_residual)),
    "mean_abs_residual": np.mean(np.abs(qq_residual))
}

print(qq_summary)

qq_table = pd.DataFrame({
    "rank": rank,
    "observed": observed,
    "probability": probability,
    "theoretical_z": theoretical_z,
    "expected_value": expected_value,
    "qq_residual": qq_residual,
    "abs_residual": np.abs(qq_residual)
})

print(qq_table.head(20))

R Code for Q-Q Plot Normality Check

# Q-Q Plot Normality Check in R

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

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

# Shapiro-Wilk support test
print(shapiro.test(x))

# Q-Q plot
qqnorm(x, main = "Q-Q Plot Normality Check for G3")
qqline(x, col = "red")

# Manual Q-Q values
observed <- sort(x)
rank <- seq_along(observed)
probability <- (rank - 0.5) / n
theoretical_z <- qnorm(probability)
expected_value <- mean(x) + theoretical_z * sd(x)
qq_residual <- observed - expected_value

qq_summary <- data.frame(
  n = n,
  mean = mean(x),
  sd = sd(x),
  max_abs_residual = max(abs(qq_residual)),
  mean_abs_residual = mean(abs(qq_residual))
)

print(qq_summary)

qq_table <- data.frame(
  rank = rank,
  observed = observed,
  probability = probability,
  theoretical_z = theoretical_z,
  expected_value = expected_value,
  qq_residual = qq_residual,
  abs_residual = abs(qq_residual)
)

print(head(qq_table, 20))

Excel Formulas for Manual Q-Q 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 plotting probability in C2:
=(B2-0.5)/$E$1

Step 5: Calculate theoretical normal z value in D2:
=NORM.S.INV(C2)

Step 6: Calculate sample mean in E2:
=AVERAGE($A$2:$A$650)

Step 7: Calculate sample standard deviation in F2:
=STDEV.S($A$2:$A$650)

Step 8: Calculate expected normal value in G2:
=$E$2 + D2*$F$2

Step 9: Calculate Q-Q residual in H2:
=A2-G2

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

Step 11: Create Q-Q plot:
Insert a scatter plot using expected normal values and observed sorted values.

Interpretation:
Points close to a straight line support approximate normality.
Systematic curves, tail departures or large residuals suggest non-normality.

APA Reporting Wording for Q-Q Plot Normality Check

When reporting a Q-Q Plot Normality Check, describe the visual pattern and support it with formal tests or shape statistics. Do not report the Q-Q 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 Q-Q Plot Report

APA-Style Group Q-Q Plot Report

Student-Friendly Report Example

Common Mistakes in Q-Q Plot Normality Check

When to Use Q-Q Plot Normality Check

Use Q-Q Plot Normality Check when you need to visually inspect whether a variable, residual, or group distribution follows a normal pattern. 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 P-P 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 Q-Q 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 Q-Q Plot Normality Check with normality tests, descriptive statistics, transformations, residual diagnostics, and reporting.