Quick Answer: Square Root Transformation Result

The verified SPSS output shows that the main right-skewed variable in this example was absences. Before square root transformation, absences had N = 649, mean = 3.66, standard deviation = 4.641, minimum = 0, maximum = 32, skewness = 2.021, and kurtosis = 5.781. This means the original absences variable was strongly positively skewed with a long right tail and unusually heavy peak/tail behavior.

After applying sqrt(absences), the transformed variable had mean = 1.4153, standard deviation = 1.2880, minimum = 0, maximum = 5.6569, skewness = .391, and kurtosis = -.711. This is a major improvement in distribution shape. Skewness moved much closer to zero, and kurtosis moved from a highly peaked/heavy-tailed value to a much less extreme value.

Hypothesis-style interpretation: The normality null hypothesis says the transformed variable follows a normal distribution. The alternative hypothesis says it does not. The SPSS normality test for sqrt_absences still reports Kolmogorov-Smirnov = .240, p < .001 and Shapiro-Wilk = .880, p < .001. Therefore, the square root transformation improved skewness and kurtosis, but the transformed variable still does not perfectly pass formal normality tests.

What Is Square Root Transformation?

Square Root Transformation converts each original value into its square root. It is usually applied to non-negative variables, especially count variables. The transformation reduces the relative distance between large values and small values. For example, the difference between 1 and 4 is 3 in the raw scale, but after transformation the values become 1 and 2, so the transformed difference is only 1. This compression is why the square root method can reduce right skewness.

In this guide, the strongest example is the absences variable. The original absences variable ranges from 0 to 32. Many students have low absence counts, while a smaller number have much higher absence counts. This creates a long right tail, high positive skewness, and high kurtosis. After square root transformation, the maximum becomes sqrt(32) = 5.6569, and the distribution becomes less extreme.

Square Root Transformation is not a magic normality tool. It can reduce positive skewness, but it may not fully satisfy formal normality tests, especially in large samples. For broader diagnostic checking, compare this guide with the Q-Q plot normality check, Q-Q plot normality check guide, P-P plot normality check, Kolmogorov-Smirnov test, Lilliefors test, D’Agostino-Pearson test, and Ryan-Joiner test.

Square Root Transformation Formula

The basic Square Root Transformation formula is:

Here, x is the original value and x′ is the transformed value. This formula works when all values are zero or positive. It is especially appropriate for variables such as counts, frequencies, absences, events, and repeated occurrences.

If a variable has negative values, the basic square root formula cannot be applied directly. A common shifted version is:

In this guide, the main variable absences has no negative values, so the direct formula sqrt(absences) is appropriate. Because zero is allowed, sqrt(0) = 0, so no shift is required.

Because the square root scale changes the meaning of the variable, the analyst should explain results carefully. A transformed mean is not directly the same as the raw mean. It is the mean on the square-root scale. For reporting untransformed descriptive results, see the Salar Cafe guide on descriptive statistics and the guide on five-number summary.

Null and Alternative Hypothesis for Square Root Transformation

Square Root Transformation itself is not a hypothesis test. It is a data-preparation and distribution-improvement method. However, we can evaluate whether it improved a variable by comparing skewness, kurtosis, Q-Q plots, histograms, and normality test results before and after transformation.

Dataset and Variables Used

The worked example uses student performance variables. The key transformation variable is absences, because it is a non-negative count variable with strong positive skewness. The output also includes G3, G2, G1, age, and failures before and after transformation to show that the square root transformation does not help every variable equally.

For descriptive context before transformation, use descriptive statistics, frequency distribution, histogram interpretation, box plot interpretation, five-number summary, and coefficient of variation.

Verified SPSS Output Interpretation

The SPSS output verifies the before-and-after transformation results. It reports original variables before square root transformation and transformed variables after square root transformation. The most important result is for absences, because it is the variable that benefits most clearly from square root transformation.

SPSS Before and After Summary for Absences

SPSS Normality Tests Before and After Transformation

SPSS Decision Summary

When checking assumptions for later models, transformation evidence should be combined with residual diagnostics. For regression-based work, the Ramsey RESET test can help assess model specification, and the Goldfeld-Quandt test can help evaluate heteroscedasticity. For grouped comparisons after transformation, guides such as Levene’s test, Brown-Forsythe test, and Cochran C test may also be useful.

Python Chart-by-Chart Interpretation

The Python charts show the Square Root Transformation visually. The main story is that absences is strongly right-skewed before transformation, and the square root version is more balanced, less extreme, and easier to interpret.

Python Chart 1: Original vs Square Root Distribution

This chart compares the original absences distribution with its square-root transformed version. Before transformation, the distribution is highly right-skewed. Many observations are near zero, and a smaller number extend far to the right up to 32 absences. This creates a long right tail and high skewness.

After square root transformation, the right tail is compressed. The maximum value changes from 32 to about 5.657. The transformed histogram becomes more compact and less dominated by extreme absence counts. This visual result matches the SPSS skewness improvement from 2.021 to .391.

Decision/reporting conclusion: Report that the Python distribution chart supports the use of Square Root Transformation for absences because the transformed distribution is visibly less right-skewed and less dominated by extreme high counts.

Python Chart 2: Q-Q Plot Before and After Square Root Transformation

The Q-Q plot shows whether observed values follow the expected normal line. Before transformation, the original absences variable departs strongly from the line, especially in the right tail. This is expected for count data with many low values and a few high values.

After transformation, the Q-Q pattern improves in the central region because extreme high values are compressed. However, the transformed data still do not perfectly follow the normal line. This supports the SPSS normality result: the transformation improves shape, but formal normality is still rejected.

Decision/reporting conclusion: Report that the Q-Q plot shows partial improvement, not perfect normality. For a full guide to interpreting this visual evidence, use the Q-Q plot normality check.

Python Chart 3: Skewness and Kurtosis Before and After

This chart gives the most direct numeric evidence for the transformation benefit. Before transformation, absences has strong positive skewness and high kurtosis. After transformation, skewness moves close to zero and kurtosis becomes much less extreme. The SPSS values confirm this: skewness changes from 2.021 to .391, while kurtosis changes from 5.781 to -.711.

This is the strongest reason to use square root transformation for absences. Even though the normality p-values remain significant, skewness and kurtosis improve substantially. In applied analysis, this can make plots more readable, reduce extreme leverage from high counts, and improve the behavior of some models.

Decision/reporting conclusion: Report that Square Root Transformation was effective for shape improvement because both skewness and kurtosis moved substantially closer to acceptable values.

Python Chart 4: Normality p-values Before and After

This chart explains why the final conclusion must be careful. The square root transformation improves skewness and kurtosis, but the normality p-values remain very small. In SPSS, both the original absences and transformed sqrt_absences have p < .001 for formal normality tests. Therefore, the transformed variable should not be described as normally distributed.

The correct interpretation is that the transformation improves distribution shape but does not fully satisfy normality. This distinction is important for reports. If a teacher, reviewer, or client asks whether the transformation “worked,” the answer is: it worked for reducing skewness and kurtosis, but it did not make the variable perfectly normal under strict formal tests.

Decision/reporting conclusion: Report that formal normality was still rejected after Square Root Transformation, so the transformation should be described as an improvement rather than a full normality correction.

Python Chart 5: Square Root Transformation Curve

This chart explains the mathematics of the transformation. The square root curve rises quickly at small values and more slowly at large values. That means large raw values are compressed more strongly than small raw values. For example, sqrt(4) = 2, sqrt(16) = 4, and sqrt(32) ≈ 5.657. The raw scale from 0 to 32 becomes a transformed scale from 0 to 5.657.

This compression is why square root transformation is useful for right-skewed count variables. It reduces the influence of large counts without changing the order of the observations. A student with more absences still has a higher transformed value, but the difference is less extreme on the transformed scale.

Decision/reporting conclusion: Report that the transformation curve justifies the method mathematically because it compresses high values while preserving rank order.

Python Chart 6: Group Boxplots Before and After

This chart shows the practical effect of transformation across groups. Before transformation, high absence values stretch the boxplots and make group spread harder to compare. After transformation, the scale is compressed, and the group boxes become easier to visually compare. This is useful when the original count variable has a long right tail.

The transformation does not erase group differences or reorder observations. It simply changes the scale so that extreme high values do not dominate the visual display. This makes the transformed boxplots better for comparing distribution shape across groups.

Decision/reporting conclusion: Report that transformed group boxplots are easier to compare visually, but any statistical test should be chosen according to the design, assumptions, and research question. For group spread checks, see Levene’s test and Brown-Forsythe test.

Python Chart 7: Skewness Across Variables

This chart is important because it shows that square root transformation is not equally useful for every variable. Absences improves strongly because it is right-skewed count data. Failures also improves somewhat, but it remains skewed because it has many zero values. Age changes only slightly because it was not highly skewed before transformation. Grade variables such as G3, G2, and G1 may become worse after square root transformation because they are not the best targets for this method.

The final lesson is that transformation must be chosen based on the variable. Do not apply square root transformation automatically to every column. First inspect skewness, histograms, Q-Q plots, measurement scale, and research purpose.

Decision/reporting conclusion: Report that Square Root Transformation should be used selectively. It worked well for absences, partly helped failures, and was not appropriate for every grade variable.

R Chart-by-Chart Validation

The R charts validate the same Square Root Transformation workflow using a separate software environment. The R results confirm the Python and SPSS interpretation: absences improves strongly, while some other variables do not benefit from the transformation.

R Chart 1: Original vs Square Root Distribution

The R distribution chart confirms that the original absences variable is right-skewed and that square root transformation compresses high values. This validates the Python chart and the SPSS descriptives. The transformed distribution becomes more compact, with fewer extreme high-scale values dominating the plot.

Decision/reporting conclusion: Report that the R distribution chart confirms the same improvement shown in Python: the transformed absences variable is less right-skewed and more visually balanced.

R Chart 2: Q-Q Plot Before and After

The R Q-Q plot confirms improved central alignment after transformation but still shows departures from normality. This agrees with the SPSS result that normality tests remain significant after transformation. The transformation helps the center of the distribution but cannot fully remove discreteness and zero-heavy features of the count variable.

Decision/reporting conclusion: Report that the R Q-Q plot supports partial normality improvement, but it does not justify saying the transformed variable is perfectly normal.

R Chart 3: Skewness and Kurtosis Before and After

The R skewness-kurtosis chart confirms the main improvement: skewness and kurtosis move much closer to acceptable values for absences. This supports reporting the transformation as successful for shape improvement. The R chart is especially useful because it independently validates the same conclusion produced by SPSS and Python.

Decision/reporting conclusion: Report that R validation confirms a strong reduction in skewness and kurtosis for absences after Square Root Transformation.

R Chart 4: Normality p-values Before and After

The R normality p-value chart confirms that the transformation does not make the data perfectly normal. It improves descriptive shape, but formal normality tests still reject normality. This is why the report should avoid overstating the transformation result.

Decision/reporting conclusion: Report that the R p-value evidence agrees with SPSS: the transformation improved distribution shape but did not make normality tests non-significant.

R Chart 5: Square Root Transformation Curve

The R transformation curve validates the mathematical explanation. Larger values are compressed more strongly than smaller values. This is the main reason square root transformation reduces right skewness. The curve also shows why the transformation is gentler than a log transformation and often useful for count variables with zeros.

Decision/reporting conclusion: Report that the R transformation curve provides a clear visual explanation of why high absence counts become less extreme on the transformed scale.

R Chart 6: Group Boxplots Before and After

The R group boxplot chart confirms that transformed values are easier to compare visually across groups. Extreme high absence counts have less influence on the plot after transformation. This helps when the analyst wants a clearer descriptive comparison of absence behavior across categories.

Decision/reporting conclusion: Report that R boxplots validate the practical usefulness of transformation for group-level visualization, especially when original count values stretch the vertical scale.

R Chart 7: Skewness Across Variables

The R skewness-across-variables chart confirms that square root transformation should be applied selectively. Absences improves strongly, failures improves partly, age changes slightly, and several grade variables become less suitable after transformation. This validates the main recommendation: inspect the variable before transforming it.

Decision/reporting conclusion: Report that R validation confirms selective use. Square Root Transformation is appropriate for absences in this dataset but should not be mechanically applied to all variables.

SPSS, R, Python and Excel Workflows for Square Root Transformation

The Square Root Transformation workflow is simple across software. The key steps are to inspect the original variable, calculate the square root, compare skewness and kurtosis before and after, inspect histograms and Q-Q plots, and then decide whether the transformed variable should be used in later analysis.

SPSS Workflow

R Workflow

Python Workflow

Excel Workflow

SPSS Syntax, Python Code, R Code and Excel Formulas

SPSS Syntax for Square Root Transformation

* Square Root Transformation in SPSS.
* Main target variable: absences.

SET PRINTBACK=OFF MPRINT=OFF.

TITLE "Square Root Transformation: Before and After Normality Comparison".

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

COMPUTE sqrt_absences = SQRT(absences).
COMPUTE sqrt_G3 = SQRT(G3).
COMPUTE sqrt_G2 = SQRT(G2).
COMPUTE sqrt_G1 = SQRT(G1).
COMPUTE sqrt_age = SQRT(age).
COMPUTE sqrt_failures = SQRT(failures).
EXECUTE.

EXAMINE VARIABLES=sqrt_absences sqrt_G3 sqrt_G2 sqrt_G1 sqrt_age sqrt_failures
  /PLOT BOXPLOT HISTOGRAM NPPLOT
  /COMPARE GROUPS
  /STATISTICS DESCRIPTIVES
  /CINTERVAL 95
  /MISSING LISTWISE
  /NOTOTAL.

OUTPUT SAVE OUTFILE="Square-Root-Transformation-SPSS-Output.spv".

OUTPUT EXPORT
  /CONTENTS EXPORT=VISIBLE
  /PDF DOCUMENTFILE="Square-Root-Transformation-SPSS-Output.pdf".

Python Code for Square Root Transformation

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["absences"], errors="coerce").dropna()
sqrt_x = np.sqrt(x)

summary = pd.DataFrame({
    "measure": ["mean", "std", "min", "max", "skewness", "kurtosis"],
    "before": [
        x.mean(),
        x.std(ddof=1),
        x.min(),
        x.max(),
        stats.skew(x, bias=False),
        stats.kurtosis(x, fisher=True, bias=False)
    ],
    "after_sqrt": [
        sqrt_x.mean(),
        sqrt_x.std(ddof=1),
        sqrt_x.min(),
        sqrt_x.max(),
        stats.skew(sqrt_x, bias=False),
        stats.kurtosis(sqrt_x, fisher=True, bias=False)
    ]
})

print(summary)

before_shapiro = stats.shapiro(x)
after_shapiro = stats.shapiro(sqrt_x)

print("Before Shapiro-Wilk:", before_shapiro)
print("After Shapiro-Wilk:", after_shapiro)

if abs(stats.skew(sqrt_x, bias=False)) < abs(stats.skew(x, bias=False)):
    print("Square root transformation reduced skewness.")
else:
    print("Square root transformation did not reduce skewness.")

plt.figure(figsize=(10, 6))
plt.hist(x, bins=25, alpha=0.6, label="Original absences")
plt.hist(sqrt_x, bins=25, alpha=0.6, label="Square-root absences")
plt.title("Square Root Transformation: Original vs Transformed Distribution")
plt.xlabel("Value")
plt.ylabel("Frequency")
plt.legend()
plt.tight_layout()
plt.savefig("square_root_transformation_distribution.png", dpi=300)
plt.close()

R Code for Square Root Transformation

# Square Root Transformation in R

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

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

sqrt_x <- sqrt(x)

before_summary <- c(
  mean = mean(x),
  sd = sd(x),
  min = min(x),
  max = max(x)
)

after_summary <- c(
  mean = mean(sqrt_x),
  sd = sd(sqrt_x),
  min = min(sqrt_x),
  max = max(sqrt_x)
)

print(before_summary)
print(after_summary)

# Optional package for skewness and kurtosis
# install.packages("moments")
library(moments)

cat("Before skewness:", skewness(x), "\n")
cat("After skewness:", skewness(sqrt_x), "\n")
cat("Before kurtosis:", kurtosis(x) - 3, "\n")
cat("After kurtosis:", kurtosis(sqrt_x) - 3, "\n")

png("square_root_transformation_qq_before.png", width = 1200, height = 900, res = 150)
qqnorm(x, main = "Q-Q Plot Before Square Root Transformation")
qqline(x)
dev.off()

png("square_root_transformation_qq_after.png", width = 1200, height = 900, res = 150)
qqnorm(sqrt_x, main = "Q-Q Plot After Square Root Transformation")
qqline(sqrt_x)
dev.off()

shapiro.test(sqrt_x)

Excel Formulas for Square Root Transformation

Assume original absences values are in A2:A650.

Square root transformation:
=SQRT(A2)

Copy the formula down for all rows.

If negative values exist:
=SQRT(A2-MIN($A$2:$A$650)+1)

Mean before:
=AVERAGE(A2:A650)

Mean after:
=AVERAGE(B2:B650)

Standard deviation before:
=STDEV.S(A2:A650)

Standard deviation after:
=STDEV.S(B2:B650)

Skewness before:
=SKEW(A2:A650)

Skewness after:
=SKEW(B2:B650)

Kurtosis before:
=KURT(A2:A650)

Kurtosis after:
=KURT(B2:B650)

Interpretation:
If skewness moves closer to 0, the transformation improved symmetry.
If kurtosis moves closer to 0, the transformation improved tail/peak behavior.

APA Reporting Wording for Square Root Transformation

When reporting Square Root Transformation, include the reason for transformation, the formula used, the before-and-after skewness/kurtosis values, normality test results, and the final decision about whether the transformed variable was used. If the transformed variable is later used in a group test, regression, or correlation, explain that the analysis was performed on the transformed scale.

APA-Style Transformation Report

Short Report Sentence

Student-Friendly Report Example

When your final analysis reports hypothesis-test results, also include appropriate effect interpretation. Salar Cafe’s effect size guide and confidence interval guide can help connect transformed-variable analysis with practical interpretation.

Common Mistakes in Square Root Transformation Interpretation

For outlier and distribution-shape context, use the guides on box plot interpretation, histogram interpretation, frequency distribution, and five-number summary.

When to Use Square Root Transformation

Use Square Root Transformation when the variable is non-negative, count-based, and positively skewed. It is especially useful when large count values create a long right tail, but the data include zero values that make log transformation less convenient.

For other transformation and assumption topics, compare this guide with reciprocal transformation, Q-Q plot normality check, D’Agostino-Pearson test, Cramer-von Mises test, and Kolmogorov-Smirnov test.

Downloads and Resources for Square Root Transformation

The resources below include the SPSS output PDF, Python charts, and R validation charts used in this guide.

Related Internal Statistical Guides

Use these related Salar Cafe guides to connect Square Root Transformation with normality checks, descriptive statistics, transformations, hypothesis testing, confidence intervals, effect size, grouped data and reporting.