Quick Answer: Standard Normal Distribution Result

The verified SPSS output standardizes G3 final grade into a z-score variable. The original G3 variable has N = 649, minimum = 0, maximum = 19, mean = 11.91, and standard deviation = 3.231. After z-score transformation, the standardized G3 variable has N = 649, minimum z = -3.685, maximum z = 2.196, mean = .00000, and standard deviation = 1.000000. This confirms that the z-score transformation worked correctly.

Hypothesis-style interpretation: A perfect standard normal variable has mean = 0, standard deviation = 1, skewness = 0, and kurtosis = 0 when excess kurtosis is used. The standardized G3 variable has the expected mean and standard deviation, but its shape is not perfectly normal. SPSS reports skewness = -0.913, kurtosis = 2.712, Kolmogorov-Smirnov D = .124, p < .001, and Shapiro-Wilk W = .926, p < .001. Therefore, the variable is correctly standardized, but it does not perfectly follow a normal distribution.

What Is the Standard Normal Distribution?

The Standard Normal Distribution is a special normal distribution with a mean of 0 and a standard deviation of 1. Values on this distribution are called z-scores. A z-score tells how many standard deviations a value is above or below the mean. A z-score of 0 means the value is exactly at the mean. A z-score of +1 means the value is one standard deviation above the mean. A z-score of -1 means the value is one standard deviation below the mean.

The Standard Normal Distribution is useful because it allows scores measured on different scales to be compared on the same standardized scale. For example, a final grade, an age value, and an absence count are measured differently, but after standardization, each value can be interpreted as a z-score. This makes the Standard Normal Distribution important for probability, confidence intervals, normality checks, regression diagnostics, outlier detection, and hypothesis testing.

In this guide, the main variable is G3 final grade. SPSS first reports the original G3 mean and standard deviation. Then it reports the standardized G3 z-score variable. The standardized variable has mean .00000 and standard deviation 1.000000, confirming a correct z-score transformation. However, the distribution shape is not perfectly normal because skewness, kurtosis, and normality tests show departures from the theoretical normal curve.

Standard Normal Distribution Formula and Z-Score Formula

The Standard Normal Distribution is commonly written as:

This means the standardized variable Z follows a normal distribution with mean 0 and variance 1. The standard deviation is also 1 because the square root of variance 1 is 1.

The z-score formula is:

When working with sample data, the sample version is usually written as:

For the G3 example, the sample mean is 11.91 and the sample standard deviation is 3.231. Therefore, a student with G3 = 15 would have an approximate z-score of:

This means a G3 score of 15 is about 0.96 standard deviations above the mean. A G3 score of 0 has a z-score near -3.685, which means it is far below the mean and contributes strongly to the lower tail of the standardized distribution.

Null and Alternative Hypothesis for Standard Normal Distribution

The Standard Normal Distribution itself is a theoretical reference distribution. In applied data analysis, the hypothesis question is usually whether a standardized variable is consistent with a normal distribution. Because z-standardization always forces mean near 0 and standard deviation near 1, the real diagnostic question is about distribution shape.

Dataset and Standardized Variables Used

The worked example uses the student performance data structure. The main standardized variable is G3 final grade. The SPSS output also lists key numeric measures: G1, G2, G3, age, and absences. The purpose is to show how a raw variable can be converted into a z-score variable and then compared with the Standard Normal Distribution.

Before standardizing a variable, it is useful to review descriptive statistics, frequency distribution, histogram interpretation, box plot interpretation, and five-number summary.

Verified SPSS Output Interpretation

The SPSS output verifies the standardization and normality interpretation. It shows the original G3 descriptive statistics, the standardized G3 z-score statistics, z-score bands, histogram/Q-Q plot output, normality tests, skewness, kurtosis, and descriptive statistics for key numeric variables.

SPSS Descriptive Output for Original G3 and Standardized G3

SPSS z-Score Band Frequencies

SPSS Normality Output for Standardized G3

Python Chart-by-Chart Interpretation

The Python charts show how the standardized G3 variable behaves against the Standard Normal Distribution. The chart set includes the z-score distribution with normal curve, Standard Normal Q-Q plot, empirical CDF versus normal CDF, z-score band percentages, and empirical rule comparison.

Python Chart 1: Z Distribution with Normal Curve

This chart shows the distribution of standardized G3 z-scores against the theoretical Standard Normal curve. The z-score scale places the mean near 0 and the standard deviation near 1. The curve provides the normal reference shape, while the histogram shows the actual standardized G3 data. The important point is that the transformation has centered and scaled the variable correctly, but the bars do not perfectly match the theoretical curve.

The left tail extends farther than the right tail because the minimum z-score is -3.685, while the maximum is 2.196. This asymmetry matches the SPSS skewness value of -0.913. Therefore, the chart supports two conclusions: first, standardization worked; second, the standardized distribution is not perfectly normal in shape.

Python Chart 2: Standard Normal Q-Q Plot

The Q-Q plot checks whether the standardized G3 values follow the theoretical normal line. If the data were perfectly normal, the points would fall close to the diagonal line from the lower tail to the upper tail. In this example, the central values are closer to the reference line, but tail values depart from the line. The strongest departure is expected in the lower tail because some G3 values are far below the mean.

This chart confirms the SPSS normality tests. The z-score variable has the correct mean and standard deviation, but the Q-Q pattern indicates tail departures from normality. For a full normality workflow, this chart should be interpreted alongside the Q-Q plot normality check, Kolmogorov-Smirnov test, and Lilliefors test.

Python Chart 3: Empirical CDF vs Normal CDF

This chart compares the empirical cumulative distribution function with the theoretical normal CDF. The empirical CDF shows the actual cumulative percentage of standardized G3 scores at each z-score level. The normal CDF shows what those cumulative percentages would look like under a perfect Standard Normal Distribution.

Where the two curves overlap, the standardized data behave similarly to the normal reference. Where they separate, the actual distribution differs from normal. This chart is especially helpful because it shows cumulative differences across the entire z-score scale, not just the center or the tails. The observed departures support the SPSS decision that the standardized G3 distribution is not perfectly normal.

Python Chart 4: Z-Score Band Percentages

This chart summarizes how standardized G3 values are distributed across z-score bands. The SPSS frequency output shows that the two central bands labeled -1.00 and 1.00 contain the largest percentages, with 36.4% and 41.0% respectively. Together, these central bands contain most of the observations. This supports the practical idea that many students are near the mean.

The lower-tail bands also matter. The minimum standardized value is -3.685, meaning a small number of observations fall far below the mean. These low values create negative skewness and contribute to the normality-test rejection. Therefore, the z-score band chart is useful for explaining where the distribution differs from the theoretical Standard Normal Distribution.

Python Chart 5: Empirical Rule Comparison

The empirical rule says that, in a perfect normal distribution, about 68% of values fall within ±1 standard deviation, about 95% fall within ±2 standard deviations, and about 99.7% fall within ±3 standard deviations. This chart compares those theoretical expectations with the observed standardized G3 distribution.

The empirical rule comparison helps readers understand whether the standardized variable behaves like a normal variable in practical terms. If the observed percentages are close to 68%, 95%, and 99.7%, the distribution is approximately normal. If the observed percentages differ, the chart helps identify the practical size of the departure. In this example, the normality tests and tail behavior show that the standardized G3 distribution is not perfectly normal, even though it is correctly centered and scaled.

R Chart-by-Chart Validation

The R charts validate the same Standard Normal Distribution workflow using a separate software environment. The R figures confirm the z-score distribution, Q-Q plot pattern, empirical CDF comparison, z-score bands, and empirical rule comparison.

R Chart 1: Z Distribution with Normal Curve

The R distribution chart confirms that the z-score transformation centers the distribution around 0 and scales it to standard deviation 1. It also confirms that the actual standardized G3 distribution does not perfectly match the theoretical normal curve. The same left-tail behavior remains visible.

R Chart 2: Standard Normal Q-Q Plot

The R Q-Q plot validates the Python Q-Q interpretation. Central values generally follow the reference line more closely than tail values. Tail departures show that the standardized G3 scores are not perfectly normal. This agrees with the SPSS Kolmogorov-Smirnov and Shapiro-Wilk tests.

R Chart 3: Empirical CDF vs Normal CDF

The R empirical CDF chart confirms the same cumulative-distribution pattern. The empirical curve follows the normal CDF in some regions but differs in others. This validates the conclusion that the standardized G3 variable is useful for z-score interpretation but not perfectly normal in distribution shape.

R Chart 4: Z-Score Band Percentages

The R z-score band chart confirms the band interpretation from Python and SPSS. Most values are in the central z-score bands, while smaller percentages appear in the extreme bands. The lower-tail bands are important because they explain the negative skewness and the non-normality decision.

R Chart 5: Empirical Rule Comparison

The R empirical rule chart confirms whether the standardized G3 distribution approximates the expected normal percentages within ±1, ±2, and ±3 standard deviations. It validates the Python chart and helps readers see the practical difference between standardization and true normality. The variable is standardized correctly, but its shape still departs from the normal reference.

SPSS, R, Python and Excel Workflows for Standard Normal Distribution

The same Standard Normal Distribution workflow can be reproduced in SPSS, R, Python and Excel. The key steps are to calculate the mean, calculate the standard deviation, compute z-scores, check the z-score mean and standard deviation, and then compare the standardized distribution against the theoretical normal distribution.

SPSS Workflow

R Workflow

Python Workflow

Excel Workflow

Code Blocks for Standard Normal Distribution

SPSS Syntax for Standard Normal Distribution

* Standard Normal Distribution and z-score workflow in SPSS.
* Main variable: G3 final grade.

TITLE "Standard Normal Distribution: G3 z-score transformation".

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

* SPSS creates standardized variable automatically, usually named ZG3.
DESCRIPTIVES VARIABLES=G3 ZG3
  /STATISTICS=MEAN STDDEV MIN MAX.

FREQUENCIES VARIABLES=ZG3
  /FORMAT=NOTABLE
  /STATISTICS=MEAN MEDIAN STDDEV VARIANCE SKEWNESS KURTOSIS MINIMUM MAXIMUM
  /HISTOGRAM NORMAL.

EXAMINE VARIABLES=ZG3
  /PLOT BOXPLOT HISTOGRAM NPPLOT
  /COMPARE GROUPS
  /STATISTICS DESCRIPTIVES
  /CINTERVAL 95
  /MISSING LISTWISE
  /NOTOTAL.

OUTPUT EXPORT
  /CONTENTS EXPORT=VISIBLE
  /PDF DOCUMENTFILE="Standard-Normal-Distribution-SPSS-Output.pdf".

Python Code for Standard Normal Distribution

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

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

x = pd.to_numeric(df["G3"], errors="coerce").dropna()

mean_x = x.mean()
sd_x = x.std(ddof=1)

z = (x - mean_x) / sd_x

print("Original G3 mean:", mean_x)
print("Original G3 SD:", sd_x)
print("Z mean:", z.mean())
print("Z SD:", z.std(ddof=1))
print("Z min:", z.min())
print("Z max:", z.max())

# Normality tests
ks_stat, ks_p = stats.kstest(z, "norm")
shapiro_stat, shapiro_p = stats.shapiro(z)

print("Kolmogorov-Smirnov statistic:", ks_stat)
print("Kolmogorov-Smirnov p:", ks_p)
print("Shapiro-Wilk statistic:", shapiro_stat)
print("Shapiro-Wilk p:", shapiro_p)

# Empirical rule percentages
within_1 = np.mean((z >= -1) & (z <= 1)) * 100
within_2 = np.mean((z >= -2) & (z <= 2)) * 100
within_3 = np.mean((z >= -3) & (z <= 3)) * 100

print("Within ±1 SD:", within_1)
print("Within ±2 SD:", within_2)
print("Within ±3 SD:", within_3)

R Code for Standard Normal Distribution

# Standard Normal Distribution and z-score workflow in R

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

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

z <- as.numeric(scale(x))

cat("Original G3 mean:", mean(x), "\n")
cat("Original G3 SD:", sd(x), "\n")
cat("Z mean:", mean(z), "\n")
cat("Z SD:", sd(z), "\n")
cat("Z min:", min(z), "\n")
cat("Z max:", max(z), "\n")

# Normality checks
shapiro.test(z)

# Q-Q plot
qqnorm(z)
qqline(z)

# Empirical rule
within_1 <- mean(z >= -1 & z <= 1) * 100
within_2 <- mean(z >= -2 & z <= 2) * 100
within_3 <- mean(z >= -3 & z <= 3) * 100

cat("Within ±1 SD:", within_1, "\n")
cat("Within ±2 SD:", within_2, "\n")
cat("Within ±3 SD:", within_3, "\n")

Excel Formulas for Standard Normal Distribution

Assume G3 values are in A2:A650.

Mean:
=AVERAGE(A2:A650)

Sample standard deviation:
=STDEV.S(A2:A650)

Z-score for A2:
=(A2-$B$1)/$B$2

Standard normal cumulative probability:
=NORM.S.DIST(C2,TRUE)

Standard normal density:
=NORM.S.DIST(C2,FALSE)

Observed percentage within ±1 SD:
=COUNTIFS(C2:C650,">=-1",C2:C650,"<=1")/COUNT(C2:C650)

Observed percentage within ±2 SD:
=COUNTIFS(C2:C650,">=-2",C2:C650,"<=2")/COUNT(C2:C650)

Observed percentage within ±3 SD:
=COUNTIFS(C2:C650,">=-3",C2:C650,"<=3")/COUNT(C2:C650)

Interpretation:
A z-score of 0 is at the mean.
A z-score of +1 is one standard deviation above the mean.
A z-score of -1 is one standard deviation below the mean.

APA Reporting Wording for Standard Normal Distribution

When reporting Standard Normal Distribution results, separate the standardization result from the normality result. It is possible for standardization to be correct while normality is rejected.

APA-Style Standardization Report

APA-Style Normality Report

Student-Friendly Report Example

Common Mistakes in Standard Normal Distribution Interpretation

When to Use Standard Normal Distribution

Use the Standard Normal Distribution when you need to convert raw scores into comparable standardized scores, interpret relative standing, calculate normal probabilities, compare empirical distributions with theoretical normal expectations, or prepare variables for normality checks and statistical reporting.

For normality-related follow-up, use Q-Q plot normality check, P-P plot normality check, Kolmogorov-Smirnov test, Lilliefors test, D’Agostino-Pearson test, and Cramer-von Mises test.

Downloads and Resources for Standard Normal Distribution

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 Standard Normal Distribution with normality checks, descriptive statistics, transformations, hypothesis testing, confidence intervals and reporting.