Quick Answer: Skewness and Kurtosis Normality Check Result

The verified SPSS output checks five numeric variables: G3, G2, G1, age, and absences. All variables have N = 649. The most normal-looking variable by skewness and kurtosis is G1, with skewness = -0.003 and kurtosis = 0.037. The strongest non-normal variable is absences, with skewness = 2.021 and kurtosis = 5.781, showing a strong right-skewed and heavy-tailed distribution.

Hypothesis-style interpretation: The normality null hypothesis says the variable follows a normal distribution. The alternative says it does not. SPSS normality tests reject normality for all five variables at p < .001. However, skewness and kurtosis show different practical severity across variables. G1 is very close to symmetric and mesokurtic, while absences is clearly non-normal. Therefore, the practical conclusion is not just “all are non-normal”; the better conclusion is that normality departure is strongest for absences and weakest for G1.

What Is Skewness and Kurtosis Normality Check?

Skewness and Kurtosis Normality Check evaluates distribution shape. Skewness measures whether the distribution is symmetric or pulled toward one tail. Kurtosis measures whether the distribution has normal, heavy, or light tail/peak behavior. Together, they help identify whether a variable is close to a normal distribution or whether it has shape problems that may affect statistical tests.

Skewness is about asymmetry. A skewness value near zero suggests symmetry. A positive skewness value means the right tail is longer or heavier. A negative skewness value means the left tail is longer or heavier. In the verified output, absences has strong positive skewness, while G3 and G2 have negative skewness.

Kurtosis is about tail weight and peak behavior. In SPSS-style excess kurtosis interpretation, a value near zero is similar to a normal distribution. Positive kurtosis suggests heavier tails and a sharper peak, while negative kurtosis suggests lighter tails or a flatter shape. In this output, absences kurtosis = 5.781 and G3 kurtosis = 2.712, so both show heavier-than-normal tail behavior, especially absences.

Skewness, Kurtosis and Z Statistic Formulas

Skewness and kurtosis are usually calculated by software, but their interpretation depends on comparing the statistic to zero and, when needed, dividing by its standard error.

In the verified SPSS output, the skewness standard error is .096 and the kurtosis standard error is .192 for each full-sample variable. A common large-sample rule is that z values beyond approximately ±1.96 indicate statistically noticeable departure at the .05 level. However, with a large sample, small shape differences can become statistically noticeable, so practical magnitude still matters.

Null and Alternative Hypothesis for Normality

The Skewness and Kurtosis Normality Check can be stated as a normality hypothesis. The null hypothesis says the variable is normally distributed. The alternative hypothesis says the variable departs from normality because of skewness, kurtosis, tail behavior, or another shape issue.

Dataset and Variables Used

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

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

Verified SPSS Output Interpretation

The SPSS output provides descriptive statistics, skewness, kurtosis, Kolmogorov-Smirnov normality tests, Shapiro-Wilk normality tests, histograms, Q-Q plots, detrended Q-Q plots, and group-level shape statistics for G3 by sex.

SPSS Descriptive Shape Statistics

SPSS Normality Test Results

SPSS Group Shape Comparison for G3 by Sex

Python Chart-by-Chart Interpretation

The Python charts show the Skewness and Kurtosis Normality Check visually. They include a distribution with normal curve, skewness ranking, kurtosis ranking, shape map, Jarque-Bera p-values, skewness/kurtosis z statistics, and group shape comparison.

Python Chart 1: Distribution with Normal Curve

This chart compares the observed variable distribution with a theoretical normal curve. The normal curve helps show whether the data are symmetric, bell-shaped, and tail-balanced. When the bars follow the curve closely, the variable is closer to normal. When the bars depart from the curve, skewness or kurtosis may be present.

For the student performance variables, G3 shows negative skewness and positive kurtosis. This means the distribution is pulled toward the left tail and has heavier tail behavior than a perfect normal distribution. The distribution chart should be read with the SPSS values: G3 skewness = -.913 and G3 kurtosis = 2.712.

Python Chart 2: Skewness Across Variables

This chart ranks the variables by skewness. Absences has the largest positive skewness at 2.021, meaning most values are low but a smaller number extend far to the right. Age has mild positive skewness at .417. G3 and G2 have negative skewness, meaning their longer tail is toward lower scores. G1 has skewness almost exactly zero at -.003, making it the closest to symmetric by skewness.

The chart is useful because normality is not only about p-values. It shows which variables have practical shape issues. Absences is the clear transformation candidate, while G1 does not need skewness correction based on shape alone.

Python Chart 3: Kurtosis Across Variables

This chart compares tail and peak behavior. Absences has the highest kurtosis at 5.781, indicating heavy-tail behavior and strong departure from normality. G3 also has high positive kurtosis at 2.712, while G2 has 1.662. G1 and age are much closer to zero, meaning they are closer to normal-like tail behavior.

The kurtosis chart is important because a variable can have low skewness but still have non-normal tails. In this output, G1 has both skewness and kurtosis near zero, while absences has both skewness and kurtosis far from zero. That makes the contrast very clear.

Python Chart 4: Skewness-Kurtosis Shape Map

The shape map combines skewness and kurtosis in one chart. Variables near the origin are closer to normal shape. Variables far from the origin have stronger shape problems. G1 appears close to the normal-shape region because its skewness and kurtosis are nearly zero. Absences appears far from the origin because both skewness and kurtosis are large and positive.

This chart is one of the best summary visuals because it prevents one-sided interpretation. A variable may look acceptable on skewness but not on kurtosis, or the reverse. The shape map shows both dimensions together and identifies the strongest normality concern immediately.

Python Chart 5: Jarque-Bera p-values

The Jarque-Bera test checks normality using skewness and kurtosis together. A small p-value indicates that the variable departs from normality in terms of symmetry, tail behavior, or both. The chart helps readers see which variables fail a combined shape-based normality test.

Because the SPSS Kolmogorov-Smirnov and Shapiro-Wilk tests reject normality for all listed variables, the Jarque-Bera chart should be interpreted as another shape-focused confirmation. The practical ranking still matters: absences has the strongest shape problem, while G1 has the mildest skewness-kurtosis departure.

Python Chart 6: Skewness and Kurtosis Z Statistics

This chart converts skewness and kurtosis values into z statistics by dividing each statistic by its standard error. For the full sample, the skewness standard error is .096 and the kurtosis standard error is .192. This gives very large z values for variables such as absences and G3. For example, absences has skewness z around 21.05 and kurtosis z around 30.11. G3 has skewness z around -9.51 and kurtosis z around 14.13.

The z-statistic chart explains why formal tests reject normality. With N = 649, the standard errors are small, so departures from zero can become very clear. G1 has z values close to zero, which supports its practical closeness to normal shape.

Python Chart 7: Group Shape Comparison

This chart compares the shape of G3 separately for female and male groups. SPSS shows that female students have G3 skewness = -.857 and kurtosis = 2.683, while male students have G3 skewness = -.980 and kurtosis = 2.803. Both groups show negative skew and positive kurtosis, and formal normality tests reject normality for both groups.

The group chart is useful because normality assumptions often apply within groups, not only in the total sample. Here, both groups have similar non-normal shape patterns, with the male group slightly more negatively skewed and slightly more kurtotic.

R Chart-by-Chart Validation

The R charts validate the same Skewness and Kurtosis Normality Check using a separate software environment. The R figures confirm the distribution shape, skewness ranking, kurtosis ranking, shape map, Jarque-Bera p-values, z statistics and group shape comparison.

R Chart 1: Distribution with Normal Curve

The R distribution chart confirms the Python visual interpretation. The observed distribution does not perfectly follow the theoretical normal curve, especially where skewness and tail behavior are present. This agrees with the SPSS normality output.

R Chart 2: Skewness Across Variables

The R skewness chart validates the Python ranking. Absences is strongly positively skewed, age is mildly positively skewed, G3 and G2 are negatively skewed, and G1 is nearly symmetric.

R Chart 3: Kurtosis Across Variables

The R kurtosis chart confirms that absences has the strongest heavy-tail behavior, followed by G3 and G2. G1 and age remain close to zero on kurtosis, making them more normal-like in tail behavior.

R Chart 4: Skewness-Kurtosis Shape Map

The R shape map confirms the practical conclusion that G1 is closest to normal shape and absences is farthest away. The chart is especially useful for explaining normality departure in two dimensions rather than one.

R Chart 5: Jarque-Bera p-values

The R Jarque-Bera p-value chart validates the Python shape-based test. Because Jarque-Bera uses skewness and kurtosis together, it is a direct statistical companion to the shape map. Small p-values indicate departure from normality.

R Chart 6: Skewness and Kurtosis Z Statistics

The R z-statistics chart confirms which variables have statistically large departures from normal shape. Absences has the strongest skewness and kurtosis z statistics, while G1 remains closest to zero.

R Chart 7: Group Shape Comparison

The R group shape comparison validates the Python and SPSS group interpretation. Female and male G3 distributions both show negative skew and positive kurtosis. This means normality concerns are visible within groups as well as in the full sample.

SPSS, R, Python and Excel Workflows for Skewness and Kurtosis Normality Check

The same normality workflow can be reproduced in SPSS, R, Python and Excel. The key steps are to calculate skewness, calculate kurtosis, divide by standard errors if z statistics are needed, inspect plots, and then combine all evidence into a practical normality decision.

SPSS Workflow

R Workflow

Python Workflow

Excel Workflow

Code Blocks for Skewness and Kurtosis Normality Check

SPSS Syntax for Skewness and Kurtosis Normality Check

* Skewness and Kurtosis Normality Check in SPSS.
* Variables: G3 G2 G1 age absences.

TITLE "Skewness and Kurtosis Normality Check".

DESCRIPTIVES VARIABLES=G3 G2 G1 age absences
  /STATISTICS=MEAN STDDEV MIN MAX SKEWNESS KURTOSIS.

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

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="Skewness-and-Kurtosis-Normality-Check-SPSS-Output.pdf".

Python Code for Skewness and Kurtosis Normality Check

import pandas as pd
import numpy as np
from scipy import stats

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

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

rows = []
for var in variables:
    x = pd.to_numeric(df[var], errors="coerce").dropna()
    n = len(x)

    skew_value = stats.skew(x, bias=False)
    kurt_value = stats.kurtosis(x, fisher=True, bias=False)

    se_skew = np.sqrt(6 / n)
    se_kurt = np.sqrt(24 / n)

    skew_z = skew_value / se_skew
    kurt_z = kurt_value / se_kurt

    jb_stat, jb_p = stats.jarque_bera(x)
    shapiro_stat, shapiro_p = stats.shapiro(x)

    rows.append({
        "variable": var,
        "n": n,
        "mean": x.mean(),
        "std": x.std(ddof=1),
        "skewness": skew_value,
        "kurtosis": kurt_value,
        "skewness_z": skew_z,
        "kurtosis_z": kurt_z,
        "jarque_bera_stat": jb_stat,
        "jarque_bera_p": jb_p,
        "shapiro_wilk_stat": shapiro_stat,
        "shapiro_wilk_p": shapiro_p
    })

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

# Simple interpretation rule
summary["shape_flag"] = np.where(
    (summary["skewness"].abs() <= 0.5) & (summary["kurtosis"].abs() <= 1),
    "Close to normal shape",
    "Review shape"
)

print(summary[["variable", "skewness", "kurtosis", "shape_flag"]])

R Code for Skewness and Kurtosis Normality Check

# Skewness and Kurtosis Normality Check in R

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

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

# install.packages(c("moments", "tseries"))
library(moments)
library(tseries)

rows <- list()

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

  skew_value <- skewness(x)
  kurt_value <- kurtosis(x) - 3

  se_skew <- sqrt(6 / n)
  se_kurt <- sqrt(24 / n)

  skew_z <- skew_value / se_skew
  kurt_z <- kurt_value / se_kurt

  jb <- jarque.bera.test(x)

  rows[[v]] <- data.frame(
    variable = v,
    n = n,
    mean = mean(x),
    sd = sd(x),
    skewness = skew_value,
    kurtosis = kurt_value,
    skewness_z = skew_z,
    kurtosis_z = kurt_z,
    jarque_bera_p = jb$p.value
  )
}

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

Excel Formulas for Skewness and Kurtosis Normality Check

Assume a variable is 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)

Approximate standard error of skewness:
=SQRT(6/COUNT(A2:A650))

Approximate standard error of kurtosis:
=SQRT(24/COUNT(A2:A650))

Skewness z statistic:
=SKEW(A2:A650)/SQRT(6/COUNT(A2:A650))

Kurtosis z statistic:
=KURT(A2:A650)/SQRT(24/COUNT(A2:A650))

Simple interpretation:
Skewness near 0 = symmetric.
Positive skewness = right-skewed.
Negative skewness = left-skewed.
Kurtosis near 0 = normal-like tails.
Positive kurtosis = heavy tails.
Negative kurtosis = flatter/lighter tails.

APA Reporting Wording for Skewness and Kurtosis Normality Check

When reporting skewness and kurtosis, include the key statistics, normality tests, and practical interpretation. Do not simply say “normal” or “not normal” without explaining which variables have the strongest shape concerns.

APA-Style Full-Sample Report

APA-Style Group Shape Report

Student-Friendly Report Example

Common Mistakes in Skewness and Kurtosis Normality Check

When to Use Skewness and Kurtosis Normality Check

Use Skewness and Kurtosis Normality Check when you need a numeric summary of distribution shape. It is especially useful before parametric tests, regression modeling, correlation analysis, ANOVA, transformation decisions, and normality reporting.

For related normality and transformation topics, use Q-Q plot normality check, P-P plot normality check, standard normal distribution, square root transformation, and reciprocal transformation.

Downloads and Resources for Skewness and Kurtosis 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 Skewness and Kurtosis Normality Check with normality checks, descriptive statistics, transformations, variance checks, confidence intervals and reporting.