Kurtosis in Statistics: Null Hypothesis, Formula, Interpretation, SPSS, R, Python and Excel Guide

What Is Kurtosis?

Kurtosis is a descriptive statistic that helps explain the shape of a distribution, especially the weight of its tails and the concentration of values around the center. In beginner statistics, kurtosis is often described as “peakedness,” but that explanation is incomplete. A better interpretation is that kurtosis describes how much of the distribution is concentrated in the center and how much probability weight appears in the tails compared with a normal distribution.

In this guide, kurtosis is demonstrated with G3 final grade from the student performance dataset. The G3 variable has many scores around the middle grade range, but it also includes very low values such as 0 and 1. These tail values make the distribution depart from a normal reference curve. That is why the histogram, Q-Q plot and boxplot are important. The number alone tells us that kurtosis is positive, but the charts show where that positive kurtosis is coming from.

For a complete shape diagnosis, kurtosis should be read with descriptive statistics, histogram interpretation, Q-Q plot normality checks, P-P plot normality checks and box plot interpretation. If you are deciding whether the distribution is normal enough for a test, also compare kurtosis with formal normality methods such as the Kolmogorov-Smirnov test, Lilliefors test and DAgostino Pearson test.

Kurtosis Formula and Types

Kurtosis is based on the fourth standardized moment. The basic population kurtosis formula is:

Kurtosis = E[(X - μ)^4] / σ^4

Where X is the variable, μ is the mean and σ is the standard deviation. Because deviations are raised to the fourth power, values far from the mean receive much greater weight. That is why kurtosis is sensitive to tail behavior and outliers.

There are two common ways to report kurtosis:

In the current analysis, G3 excess kurtosis = 2.712. Since this is positive and clearly above the common practical threshold of +1, the distribution is interpreted as leptokurtic. This means the tails and peak are stronger than expected under a normal curve. The histogram and boxplot show that very low grade values are especially important in creating this pattern.

Null and Alternative Hypothesis for Kurtosis

Kurtosis is commonly used as a descriptive statistic, but a hypothesis framework makes the interpretation clearer for reports, assignments and SEO-friendly statistical guides. The hypothesis is based on whether the observed excess kurtosis is close to the normal-reference value or significantly different from it.

The approximate test can be written as:

z = excess kurtosis / standard error of kurtosis

For G3, the excess kurtosis is 2.712 and the standard error of kurtosis is 0.192. Therefore, the approximate z value is:

z ≈ 2.712 / 0.192 = 14.104

Dataset and Clean SPSS-Ready Files Used

This kurtosis example uses the student-por.csv student performance dataset. The main outcome variable is G3 final grade. Additional numeric variables were also checked to compare kurtosis across the dataset, including G1, G2, absences, age, failures, studytime, traveltime, famrel, freetime, goout, Dalc, Walc, health, Medu and Fedu.

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

Verified SPSS, R and Python Results

The kurtosis result was checked in SPSS, Python and R. SPSS reports kurtosis as excess kurtosis, where a normal distribution is approximately 0. The main G3 result was positive and large enough to support a leptokurtic interpretation. The Python and R charts visually confirmed the same conclusion: G3 is centered around the middle grades, but its tails, especially the low-score tail, depart from the normal reference pattern.

Hypothesis Result for G3 Kurtosis

SPSS Descriptive Statistics for Main Variables

G3 Detailed SPSS Output

Python Charts and Interpretation

1. Kurtosis Histogram for G3

This histogram compares the observed G3 grade distribution with a normal reference curve. Most G3 scores are concentrated between about 10 and 15, with the dashed mean line at approximately 11.91. Under the null hypothesis, the G3 distribution would have normal-like excess kurtosis, so the bars would follow the normal reference curve more closely in both the center and the tails. Instead, the observed bars show a strong central concentration, a visible group of very low scores near 0, and a right side that extends toward 19. These tail observations are important because kurtosis is highly sensitive to values far from the mean. The main message is not simply that the chart has a high peak; rather, the chart shows a combination of central clustering and tail departures. That is why G3 receives a positive excess kurtosis value of 2.712. The low-score tail also explains why the 5% trimmed mean is slightly higher than the ordinary mean: removing extreme low values raises the average. Therefore, the histogram supports rejecting the null hypothesis of normal-like kurtosis and supports the leptokurtic interpretation, but it should be read together with the Q-Q plot and boxplot before making the final distribution-shape statement.

2. Q-Q Plot Support for Kurtosis: G3

The Q-Q plot gives stronger evidence about tail behavior than the histogram alone. If the null hypothesis were true and G3 had normal-like kurtosis, the plotted points would stay near the diagonal reference line from the lower tail to the upper tail. In this chart, the middle grade values follow the line reasonably well, which means the central part of the distribution is not the main problem. The strongest departures appear in the tails. At the lower end, several observations cluster around 0, creating a flat step pattern far from what a smooth normal distribution would produce. At the upper end, the highest scores also deviate from the diagonal line, although the low-score tail is visually more influential. The staircase pattern happens because G3 is a discrete grade variable with repeated integer values from 0 to 19, not a continuous measurement. This plot supports the kurtosis hypothesis decision because the alternative hypothesis says tail weight differs from normal, and the Q-Q plot shows exactly where those tail deviations occur. It also supports the negative skewness result because the lower tail is more visibly unusual than the upper tail.

3. Boxplot Tail Check for G3

The boxplot explains the tail structure behind the kurtosis value. The middle box is concentrated roughly from 10 to 14, and the median is close to 12. This agrees with the SPSS interquartile range of 4.00, meaning the middle half of students are grouped within about four grade points. The right whisker extends toward the high-grade end, while the lower side includes very low outlying points around 0 and 1. These low outliers are important because kurtosis gives strong weight to values far from the mean. A few tail values can therefore increase excess kurtosis even when the majority of observations are clustered in a normal-looking central band. The null hypothesis expects normal-like tail weight, but the boxplot shows tail observations that make the normal-like assumption weak. This chart also shows why kurtosis and skewness should be interpreted together. The positive kurtosis tells us the tails are heavier than normal, while the negative skewness tells us that the lower tail has stronger influence than the upper tail. In reporting, this means G3 should be described as leptokurtic with low-end tail influence, not simply as “peaked.”

4. Excess Kurtosis Across Numeric Variables

This horizontal bar chart compares excess kurtosis across multiple numeric variables. The vertical reference line at 0 represents the null hypothesis value for normal-like excess kurtosis. The dashed threshold around +1 marks variables with practically heavier tails, while the dashed threshold around -1 marks variables with noticeably lighter or flatter tails. The strongest positive kurtosis appears for failures = 9.82, followed by absences = 5.78, Dalc = 4.35, G3 = 2.71 and G2 = 1.66. These variables have heavier tails because they either contain rare extreme values, many repeated low values, or strong clustering with a few distant observations. For example, absences is a count variable where many students have low absences but some students have very high absence counts. G3 is not as extreme as failures or absences, but it is still clearly above the +1 threshold, so rejecting the null hypothesis for G3 is not only statistically meaningful; it also has a visible practical explanation. Variables such as G1 = 0.04, studytime = 0.04 and age = 0.07 are close to zero, so their excess kurtosis is much closer to the null hypothesis value. On the negative side, Medu = -1.26, health = -1.12 and Fedu = -1.11 are lighter-tailed or flatter, which often happens with bounded ordinal variables. This chart is useful because it shows that kurtosis is not a one-variable issue; different variable types in the same dataset can have very different tail behavior.

5. Kurtosis Result and Interpretation Card

The result card summarizes the values that should be used in a report: N = 649, mean = 11.906, standard deviation = 3.231, Fisher’s excess kurtosis = 2.712, Pearson kurtosis = 5.712, approximate z = 14.104 and an extremely small p-value. The null hypothesis is that excess kurtosis equals 0, meaning normal-like kurtosis. The alternative hypothesis is that excess kurtosis differs from 0. Since the approximate z value is very large and the p-value is below .001, the correct decision is to reject the null hypothesis. The most important interpretation line is that G3 is leptokurtic, meaning heavier tails or a sharper peak than normal. The card is especially useful for students because it separates Fisher’s excess kurtosis from Pearson kurtosis. Many reporting mistakes happen because users forget that SPSS reports excess kurtosis, not Pearson kurtosis. If the report uses SPSS, Python or R excess kurtosis, the normal reference value is 0. If it uses Pearson kurtosis, the normal reference value is 3. The chart displays the p-value as 0.000000, but the written report should use p < .001. The card also correctly reminds the reader that the hypothesis decision should not be reported alone. The histogram, Q-Q plot and boxplot explain the practical reason for rejecting the null hypothesis.

R Validation Charts

The R charts reproduce the same kurtosis story using a separate software workflow. This is useful because it confirms that the result is not only a Python charting artifact or an SPSS output issue. The R histogram, Q-Q plot, boxplot, variable comparison chart and result card all support the same conclusion: reject the null hypothesis of normal-like G3 kurtosis and interpret G3 as leptokurtic.

The R histogram confirms the same central clustering and tail behavior shown in Python. Most scores are concentrated in the middle grade range, while the low-end tail includes unusually small values. Under the null hypothesis, the histogram would look more normal-like in its central shape and tails. Instead, the observed pattern supports a positive excess kurtosis result. This chart validates the conclusion that G3 is leptokurtic rather than normal-like in kurtosis.

The R Q-Q plot again shows that the center of the distribution follows the normal line better than the tails. The repeated grade values create visible steps, and the lowest observations depart strongly from the reference line. This confirms the interpretation that kurtosis is being driven by tail behavior. Since the null hypothesis assumes normal-like tail behavior, the R Q-Q plot supports the decision to reject that null hypothesis.

The R boxplot makes the same practical point as the Python boxplot. The middle 50% of G3 values is compact, but low-end observations lie far from the central box. These values increase the fourth moment used in the kurtosis formula. Therefore, the boxplot supports the decision to reject the null hypothesis of normal-like kurtosis and report G3 as a leptokurtic distribution with low-tail influence.

The R variable-comparison chart confirms that G3 is not the only variable with positive kurtosis. Failures and absences show much stronger tail weight, while G1, age and studytime are close to zero. This comparison prevents overinterpretation of G3. It shows that G3 is meaningfully leptokurtic, but some count-like or bounded variables in the dataset are even more extreme. From a hypothesis perspective, variables close to 0 are closer to the null hypothesis value, while variables far above +1 or below -1 show stronger practical departures from normal-like kurtosis.

The R result card gives the same reportable values: G3 has positive excess kurtosis and should be described as leptokurtic. The agreement between R, Python and SPSS increases confidence in the result. In a final report, the R card can be used as a visual summary, while the SPSS output PDF can be used as the formal verification file. The hypothesis decision remains the same: reject the null hypothesis of normal-like kurtosis and conclude that G3 has a significant positive kurtosis pattern.

How to Calculate Kurtosis in Python, R, SPSS and Excel

Kurtosis in Python

Python can calculate Fisher’s excess kurtosis, Pearson kurtosis, approximate z value and a hypothesis-style p-value. The example below assumes the clean dataset contains the G3 variable.

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

df = pd.read_csv("spss_ready_data.csv")
g3 = pd.to_numeric(df["G3"], errors="coerce").dropna()

n = len(g3)
mean_g3 = g3.mean()
sd_g3 = g3.std(ddof=1)

# Fisher's excess kurtosis: normal reference = 0
excess_kurtosis = stats.kurtosis(g3, fisher=True, bias=False)

# Pearson kurtosis: normal reference = 3
pearson_kurtosis = excess_kurtosis + 3

# Approximate standard error, z value and two-sided p-value
se_kurtosis = np.sqrt(24 / n)
z_kurtosis = excess_kurtosis / se_kurtosis
p_value = 2 * (1 - stats.norm.cdf(abs(z_kurtosis)))

print("Null hypothesis: excess kurtosis = 0")
print("Alternative hypothesis: excess kurtosis != 0")
print("N:", n)
print("Mean:", round(mean_g3, 3))
print("SD:", round(sd_g3, 3))
print("Fisher excess kurtosis:", round(excess_kurtosis, 3))
print("Pearson kurtosis:", round(pearson_kurtosis, 3))
print("Approx z:", round(z_kurtosis, 3))
print("Approx p-value:", p_value)

if p_value < 0.05:
    print("Decision: Reject the null hypothesis.")
else:
    print("Decision: Fail to reject the null hypothesis.")

Kurtosis in R

R can calculate excess kurtosis using packages such as moments or e1071. The code below gives a simple reproducible workflow with null and alternative hypothesis wording.

install.packages("moments")
library(moments)

student <- read.csv("spss_ready_data.csv")
g3 <- na.omit(as.numeric(student$G3))

n <- length(g3)
mean_g3 <- mean(g3)
sd_g3 <- sd(g3)

# moments::kurtosis gives Pearson kurtosis
pearson_kurtosis <- moments::kurtosis(g3)
excess_kurtosis <- pearson_kurtosis - 3

se_kurtosis <- sqrt(24 / n)
z_kurtosis <- excess_kurtosis / se_kurtosis
p_value <- 2 * (1 - pnorm(abs(z_kurtosis)))

cat("Null hypothesis: excess kurtosis = 0\n")
cat("Alternative hypothesis: excess kurtosis != 0\n")
cat("N:", n, "\n")
cat("Mean:", round(mean_g3, 3), "\n")
cat("SD:", round(sd_g3, 3), "\n")
cat("Fisher excess kurtosis:", round(excess_kurtosis, 3), "\n")
cat("Pearson kurtosis:", round(pearson_kurtosis, 3), "\n")
cat("Approx z:", round(z_kurtosis, 3), "\n")
cat("Approx p-value:", p_value, "\n")

if (p_value < 0.05) {
  cat("Decision: Reject the null hypothesis.\n")
} else {
  cat("Decision: Fail to reject the null hypothesis.\n")
}

Kurtosis in SPSS

SPSS reports excess kurtosis in the Frequencies, Explore and Descriptives outputs. In this analysis, SPSS reported G3 kurtosis as 2.712 with standard error 0.192. The hypothesis statement should be written in the report because SPSS does not automatically print “reject H0” for kurtosis.

* Kurtosis Analysis in SPSS.
* Null hypothesis: G3 excess kurtosis = 0.
* Alternative hypothesis: G3 excess kurtosis is not equal to 0.

SET UNICODE=ON.
SET DECIMAL=DOT.
SET PRINTBACK=OFF.
SET MPRINT=OFF.

GET FILE="D:\DATA ANALYSIS\A Basic Descriptive Statistics Guides\Kurtosis\SPSS_Output\sav\kurtosis_spss_ready_data.sav".
DATASET NAME Kurtosis_Data WINDOW=FRONT.

TITLE "Kurtosis Analysis for Student Performance Data".

FREQUENCIES VARIABLES=G3 G2 G1 absences age
 /STATISTICS=MEAN MEDIAN STDDEV VARIANCE SKEWNESS SESKEW KURTOSIS SEKURT MINIMUM MAXIMUM RANGE
 /ORDER=ANALYSIS.

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

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

OUTPUT EXPORT
 /CONTENTS EXPORT=VISIBLE
 /PDF DOCUMENTFILE="D:\DATA ANALYSIS\A Basic Descriptive Statistics Guides\Kurtosis\SPSS_Output\Kurtosis-SPSS-Output.pdf".

Kurtosis in Excel

Excel uses the KURT function to calculate excess kurtosis. This means a normal distribution is interpreted around 0. Excel does not automatically give a hypothesis decision, but you can calculate an approximate z value and p-value manually.

How to Report Kurtosis

A strong kurtosis report should include the variable, sample size, null hypothesis, alternative hypothesis, mean, standard deviation, excess kurtosis value, standard error, approximate z value, p-value decision and visual confirmation. Do not report kurtosis as a single isolated number. Connect it to the histogram, Q-Q plot and boxplot.

If kurtosis is being checked before a hypothesis test or model, combine it with other assumptions. For mean-based testing, see the one-sample z test, one-tailed t test and confidence interval guides. For model diagnostics, also review Ramsey RESET test, Goldfeld-Quandt test, Brown-Forsythe test and Levene test.

Related Data Analysis Guides

Kurtosis is part of descriptive and diagnostic analysis. Continue with Descriptive Statistics, Five Number Summary, Frequency Distribution, Histogram Interpretation, Box Plot Interpretation, Q-Q Plot Normality Check, P-P Plot Normality Check and Effect Size. For distribution transformations, see Reciprocal Transformation. For normal approximation concepts, see the Central Limit Theorem.

Common Mistakes

1. Forgetting to state the null and alternative hypothesis

For SEO and reporting clarity, state the hypothesis when the post uses a decision. For kurtosis, the null hypothesis is usually excess kurtosis = 0, and the alternative hypothesis is that excess kurtosis differs from 0.

2. Saying kurtosis only means peakedness

Kurtosis is often taught as peakedness, but that is incomplete. It is better understood as tail weight and concentration pattern. The G3 result is positive because the distribution has central clustering and important tail departures.

3. Confusing Fisher's excess kurtosis with Pearson kurtosis

SPSS, Excel and many statistical outputs report excess kurtosis, where the normal reference is 0. Pearson kurtosis uses 3 as the normal reference. For G3, excess kurtosis is 2.712 and Pearson kurtosis is 5.712.

4. Reporting p = .000

Software sometimes displays very small p-values as .000 or 0.000000. In final writing, report p < .001.

5. Ignoring charts after rejecting the null hypothesis

Rejecting the null hypothesis tells us that the kurtosis differs from normal-like kurtosis, but it does not explain why. The histogram shows the overall shape, the Q-Q plot shows tail departures and the boxplot shows outlying values. Reporting the decision alone can be misleading.

6. Treating all positive kurtosis as a serious problem

Positive kurtosis means heavier tails than normal, but its importance depends on the research question, sample size, variable type and statistical method. With large samples, some tests are robust, while other procedures may require transformation, robust methods or careful diagnostics.

Download SPSS Output and Verification Files

The SPSS output PDF verifies the descriptive statistics, kurtosis values, normality tests, histogram, Q-Q plot, detrended Q-Q plot and boxplot used in this guide. The hypothesis decision in this post is based on the reported excess kurtosis value and standard error.

External References for Kurtosis

This post uses verified Python, R and SPSS outputs together with standard statistical documentation and dataset references.

• UCI Machine Learning Repository: Student Performance dataset
• IBM SPSS Statistics documentation
• The Comprehensive R Archive Network
• SciPy statistical functions documentation
• Microsoft Excel KURT function documentation