Normal Distribution: Formula, Null Hypothesis, SPSS, R, Python and Excel Guide

What Is Normal Distribution?

A normal distribution is a continuous, bell-shaped distribution where values are symmetrically distributed around the mean. In an ideal normal distribution, the mean, median and mode are equal, the curve is balanced on both sides, and most values are close to the center. The standard normal distribution is a special normal distribution with mean = 0 and standard deviation = 1.

Normal distribution is important because many statistical methods assume that the outcome, residuals or sampling distribution is approximately normal. It affects how we interpret confidence intervals, one-sample tests, regression diagnostics and assumption checks. However, real datasets rarely look perfectly normal. That is why analysts should combine formal normality tests with visual checks such as a Q-Q plot normality check, P-P plot normality check, histogram interpretation and box plot interpretation.

In this worked example, G3 final grade has a mean of 11.91 and a standard deviation of 3.231. The center is clear, but the distribution is not perfectly normal because low scores near 0 create a left-tail problem. This is also visible in the z-score range, where the minimum standardized value is -3.69 and the maximum is only 2.20. A perfectly normal distribution would usually show a more balanced tail pattern.

Normal Distribution Formula

The probability density function of a normal distribution is written as:

f(x) = [1 / (σ√(2π))] × e^[-(x - μ)² / (2σ²)]

The standard normal distribution uses z-scores. A z-score tells how many standard deviations an observed value is from the mean:

z = (x - mean) / standard deviation

For example, a G3 score of 0 is far below the mean. Using the SPSS output values, its approximate z-score is (0 – 11.91) / 3.231 = -3.69. This is a very low standardized value and explains why the left tail creates a normality problem. A G3 score of 19 has a z-score of about 2.20, which is high but not as extreme as the low-score tail.

Null and Alternative Hypothesis for Normal Distribution

A normality test uses a formal hypothesis decision. This is important for SEO and for statistical reporting because readers often search for “normal distribution null hypothesis,” “normality test alternative hypothesis,” or “reject the null hypothesis for normal distribution.”

Dataset and SPSS-Ready Files Used

This guide uses the student-por.csv dataset. The main variable is G3 final grade. Python and R were used to generate visual checks, while SPSS was used to verify descriptive statistics, Kolmogorov-Smirnov normality results, Shapiro-Wilk normality results, z-scores and empirical rule frequencies.

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

Verified SPSS Normal Distribution Results

The SPSS output gives the most important normal distribution statistics for G3 final grade. The distribution has a clear center around 12, but the skewness, kurtosis, formal normality tests and low-end z-scores show that it is not a perfect normal distribution.

Descriptive Statistics for G3

SPSS Tests of Normality

The Kolmogorov-Smirnov test compares the observed cumulative distribution with a theoretical normal distribution. The Shapiro-Wilk test is another formal test of normality. Both tests have p < .001, so the normality null hypothesis is rejected. This result should be reported clearly, but it should not be interpreted without the visual checks because large datasets can make small distribution differences statistically significant.

G3 Frequency Table Pattern

Z-Score and Empirical Rule Results

The empirical rule check is one of the most practical ways to explain the normal distribution result. A normal distribution should have about 68.3% of values within one standard deviation, 95.5% within two standard deviations and 99.7% within three standard deviations. For G3, the one-standard-deviation and two-standard-deviation bands are higher than normal expectation, but the three-standard-deviation band is lower because the low score of 0 has a z-score of about -3.69. This is another reason the final decision is to reject the normality null hypothesis.

Python Charts and Full Interpretation

1. Normal Distribution Histogram Curve for G3

This histogram compares the observed G3 distribution with a fitted normal curve. The blue bars show the observed density of G3 scores, the orange curve shows the normal distribution expected from the sample mean and standard deviation, and the dashed vertical line marks the mean at about 11.91. The central part of the distribution is concentrated between about 10 and 15, so the fitted normal curve captures the broad location of the data. However, the histogram does not follow the curve perfectly. There is a visible bar near 0, showing very low final grades that are much farther left than a clean normal bell curve would expect. The bars around 10 to 11 and 14 to 15 are also uneven, suggesting a discrete grade distribution rather than a smooth continuous normal distribution. This chart supports the formal hypothesis decision: the distribution has a central bell-like region, but the low-score tail and uneven grade clustering mean we reject the null hypothesis of perfect normality.

2. Q-Q Plot for G3

The Q-Q plot checks whether the ordered observed G3 values follow the theoretical quantiles of a normal distribution. If G3 were normally distributed, the points would sit close to the diagonal reference line across the full range. The chart reports a Q-Q line correlation of about r = 0.962, which means the middle of the distribution has a fairly strong linear pattern. However, normality is not judged only by the middle. The left tail shows clear deviation because many low values sit far away from the reference line, especially around score 0. The upper tail also flattens because G3 is bounded at the high end and has only a few scores near 18 and 19. The stepped pattern occurs because G3 is an integer grade variable rather than a continuous measurement. This chart shows partial normal-like behavior in the center but non-normal behavior in the tails. Therefore, it agrees with the SPSS Kolmogorov-Smirnov and Shapiro-Wilk results: reject the normality null hypothesis.

3. P-P Plot for G3

The P-P plot compares observed cumulative probabilities with expected cumulative probabilities under a normal distribution. Points close to the diagonal line suggest agreement between the observed and expected cumulative probabilities. In this chart, many points follow the diagonal reasonably well through the middle range, which means the cumulative pattern of G3 is not completely unrelated to a normal curve. However, the vertical step groups show that the variable has repeated integer scores, and several sections depart from the line. The lower-left region reflects the unusual low-score tail, while the middle probability bands show clustering around common grades such as 10, 11, 12 and 13. The P-P plot is often less sensitive to extreme tails than the Q-Q plot, so it may look more acceptable in the center. Even so, it still shows a distribution that is only approximately normal in the middle and not normal in the tails. For the hypothesis decision, the P-P plot supports the same conclusion: do not accept normality as a clean assumption for G3.

4. Standardized Normal Distribution for G3

This chart converts G3 scores into z-scores and compares the standardized values with the standard normal curve. In a standard normal distribution, the mean is 0 and the standard deviation is 1. SPSS confirms the standardized G3 z-score mean is approximately 0.0000 and the standard deviation is 1.0000. The dashed vertical guide lines show standard deviation positions. Most z-scores fall between about -1 and 1, which is expected for a central grade cluster. However, the left tail extends to about -3.69, while the right tail reaches only about 2.20. This imbalance is a strong visual sign of negative skewness. The very low G3 scores are much farther below the mean than the highest scores are above the mean. That is why the standardized chart does not match the standard normal curve perfectly. It supports rejecting the null hypothesis and helps explain the result in practical units: the problem is not just the center, but the unequal tail distance.

5. Empirical Rule Check

The empirical rule chart compares the observed G3 proportions with the expected proportions for a normal distribution. A normal distribution should place about 0.683 of values within one standard deviation, 0.955 within two standard deviations and 0.997 within three standard deviations. The observed G3 values are 0.773 within one standard deviation, 0.971 within two standard deviations and 0.975 within three standard deviations. The first result means G3 has more values packed near the center than a normal distribution would expect. The second result is also slightly higher than expected. The third result is lower than expected because 16 cases fall outside the three-standard-deviation band, mainly the very low grades. This is a strong practical explanation of the normality rejection: the center is crowded, but the extreme low tail is too far from the mean. Therefore, the empirical rule does not support a clean normal distribution.

R Validation Charts

The R charts validate the same result through a separate workflow. They confirm that the normality decision is not only a Python result. The R histogram, Q-Q plot, P-P plot, z-score chart and empirical rule chart all support the same interpretation: G3 has a central mound but does not fully satisfy normal distribution assumptions.

The R histogram confirms the same central pattern shown by Python. The G3 scores cluster in the middle, but the low-score tail and uneven bar heights prevent the distribution from matching a smooth bell curve. The fitted curve identifies where a normal distribution would place the density, while the observed bars show that the real data are more discrete and tail-heavy on the low side.

The R Q-Q plot again shows that the middle values follow the line better than the tails. The low scores depart strongly from the expected normal quantiles, and the upper scores flatten because G3 is bounded. This visual evidence supports the normality test decision to reject the null hypothesis.

The R P-P plot shows cumulative probability agreement in some middle sections but visible step patterns and deviations across the range. Because G3 is an integer grade variable, repeated values create vertical clusters. The chart supports a moderate central fit but not a full normal distribution.

The R z-score chart confirms that standardizing G3 places the mean around 0 and the standard deviation around 1, but it also shows an unbalanced tail structure. The left tail reaches farther than the right tail, which matches the SPSS skewness value of -0.913. This confirms that the problem is not the scale of measurement; the problem is the shape of the distribution after standardization.

The R empirical rule chart validates the Python and SPSS proportions. The observed proportion within one standard deviation is higher than normal expectation, and the observed proportion within three standard deviations is lower than the 99.7% normal rule. This pattern indicates central clustering plus extreme low-tail influence. It supports the final conclusion that G3 is not normally distributed.

How to Check Normal Distribution in Python, R, SPSS and Excel

Normal Distribution Check in Python

Python can calculate descriptive statistics, z-scores, Q-Q plots, P-P plots, Shapiro-Wilk normality test and empirical rule checks.

import pandas as pd
import numpy as np
from scipy import stats
import matplotlib.pyplot as plt

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)
z = (g3 - mean_g3) / sd_g3

print("H0: G3 is normally distributed.")
print("H1: G3 is not normally distributed.")
print("N:", n)
print("Mean:", round(mean_g3, 2))
print("SD:", round(sd_g3, 3))
print("Skewness:", round(stats.skew(g3, bias=False), 3))
print("Excess kurtosis:", round(stats.kurtosis(g3, fisher=True, bias=False), 3))

shapiro_stat, shapiro_p = stats.shapiro(g3)
print("Shapiro-Wilk:", round(shapiro_stat, 3), "p =", shapiro_p)

# Standard K-S using estimated mean and SD is not identical to SPSS Lilliefors correction.
ks_stat, ks_p = stats.kstest(g3, "norm", args=(mean_g3, sd_g3))
print("K-S:", round(ks_stat, 3), "p =", ks_p)

for k in [1, 2, 3]:
    observed = np.mean(np.abs(z) <= k)
    print(f"Within {k} SD:", round(observed, 3))

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

plt.hist(g3, bins=15, density=True, alpha=0.65, edgecolor="black")
plt.plot(x, pdf, linewidth=3)
plt.axvline(mean_g3, linestyle="--", linewidth=2)
plt.title("Normal Distribution Check for G3")
plt.xlabel("G3")
plt.ylabel("Density")
plt.show()

Normal Distribution Check in R

R can calculate the same statistics and also run the Lilliefors test using the nortest package.

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

n_G3 <- length(g3)
mean_G3 <- mean(g3)
sd_G3 <- sd(g3)
z_G3 <- as.numeric(scale(g3))

cat("H0: G3 is normally distributed.\n")
cat("H1: G3 is not normally distributed.\n")
cat("N:", n_G3, "\n")
cat("Mean:", round(mean_G3, 2), "\n")
cat("SD:", round(sd_G3, 3), "\n")
cat("Minimum z:", round(min(z_G3), 2), "\n")
cat("Maximum z:", round(max(z_G3), 2), "\n")

shapiro.test(g3)

# install.packages("nortest")
library(nortest)
lillie.test(g3)

observed_1sd <- mean(abs(z_G3) <= 1)
observed_2sd <- mean(abs(z_G3) <= 2)
observed_3sd <- mean(abs(z_G3) <= 3)

print(data.frame(
  band = c("mean ± 1 SD", "mean ± 2 SD", "mean ± 3 SD"),
  observed = c(observed_1sd, observed_2sd, observed_3sd),
  normal_expected = c(0.683, 0.955, 0.997)
))

Normal Distribution Check in SPSS

SPSS can produce descriptives, normality tests, histograms, Q-Q plots, detrended Q-Q plots and z-score variables.

* Normal Distribution Analysis for G3 Final Grade.
* H0: G3 is normally distributed.
* H1: G3 is not normally distributed.

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

GET DATA
 /TYPE=TXT
 /FILE="D:\DATA ANALYSIS\B Normality and Assumption Tests\Normal Distribution\SPSS_Output\clean_data\spss_ready_data.csv"
 /ENCODING='UTF8'
 /DELCASE=LINE
 /DELIMITERS=","
 /QUALIFIER='"'
 /ARRANGEMENT=DELIMITED
 /FIRSTCASE=2
 /IMPORTCASE=ALL.

DATASET NAME StudentData WINDOW=FRONT.

TITLE "Normal Distribution Analysis for G3 Final Grade".

DESCRIPTIVES VARIABLES=G3
 /SAVE
 /STATISTICS=MEAN STDDEV MIN MAX.

FREQUENCIES VARIABLES=G3 ZG3
 /STATISTICS=MEAN MEDIAN MODE STDDEV VARIANCE SKEWNESS SESKEW KURTOSIS SEKURT RANGE MINIMUM MAXIMUM
 /ORDER=ANALYSIS.

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

NPAR TESTS
 /K-S(NORMAL)=G3
 /MISSING ANALYSIS.

COMPUTE within_1sd = (ABS(ZG3) <= 1).
COMPUTE within_2sd = (ABS(ZG3) <= 2).
COMPUTE within_3sd = (ABS(ZG3) <= 3).
EXECUTE.

FREQUENCIES VARIABLES=within_1sd within_2sd within_3sd.

OUTPUT EXPORT
 /CONTENTS EXPORT=VISIBLE
 /PDF DOCUMENTFILE="D:\DATA ANALYSIS\B Normality and Assumption Tests\Normal Distribution\SPSS_Output\Normal-Distribution-SPSS-Output.pdf".

Normal Distribution Check in Excel

Excel can calculate a normal distribution curve, z-scores and empirical rule counts. If G3 values are in B2:B650, use these formulas.

How to Report the Normal Distribution Decision

A good normal distribution report should include the variable, sample size, mean, standard deviation, skewness, kurtosis, normality test results, p-value decision and visual interpretation. It should also mention whether the null hypothesis was rejected or not.

When normality is rejected, the next decision depends on the purpose of the analysis. Some methods are robust when sample size is large, especially if group sizes are reasonable. However, for assumption-heavy analysis, check transformations, robust methods, nonparametric alternatives or residual-based diagnostics. For possible transformation workflows, see reciprocal transformation. For variance assumptions, use Levene test, Brown-Forsythe test or Cochran C test. For regression assumption diagnostics, see Ramsey RESET test and Goldfeld-Quandt test.

Related Data Analysis Guides

Normal distribution is connected to many descriptive and inferential topics. Continue with Descriptive Statistics, Frequency Distribution, Histogram Interpretation, Box Plot Interpretation, Central Limit Theorem, Q-Q Plot Normality Check, P-P Plot Normality Check, Kolmogorov-Smirnov Test, Lilliefors Test, DAgostino Pearson Test, Ryan-Joiner Test, Confidence Interval, Effect Size, Five Number Summary, One Sample Z Test and One Tailed T Test.

Common Mistakes

1. Saying p = .000 means the p-value is zero

SPSS displays .000 when the value is very small. In reporting, write p < .001.

2. Accepting normality because the histogram looks roughly bell-shaped

The histogram may look approximately bell-shaped in the middle, but the tails can still violate normality. For G3, the low-score tail is the main issue.

3. Ignoring the Q-Q plot tails

The center of a Q-Q plot can look acceptable while the tails show strong departures. Normality requires the whole distribution to behave reasonably, not only the middle.

4. Confusing normal distribution with standard normal distribution

A normal distribution can have any mean and standard deviation. A standard normal distribution always has mean 0 and standard deviation 1.

5. Treating large-sample normality tests too mechanically

Large samples can make small deviations statistically significant. Use visual evidence, sample size and analysis purpose together.

6. Forgetting that grade data are discrete

G3 is an integer grade variable, so Q-Q and P-P plots show step patterns. This discreteness contributes to departures from a smooth continuous normal curve.

7. Not linking normality to the planned analysis

Normality is not checked for its own sake. It should be connected to the next method, such as a t test, ANOVA, regression model, confidence interval or transformation decision.

Download SPSS Output and Verification Files

The SPSS output PDF verifies the descriptive statistics, G3 frequency table, z-score table, Kolmogorov-Smirnov test, Shapiro-Wilk test, normal curve histogram, Q-Q plot, detrended Q-Q plot and empirical rule counts used in this guide.

External References for Normal Distribution

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
• pandas documentation
• SciPy statistics documentation
• Microsoft Excel support documentation