You're on the right track! Assessing skewness and kurtosis is the crucial first step to determine if your variables deviate significantly from a normal distribution and thus require transformation.

Let's break down what these statistics are, why they matter, how to calculate them, and how to interpret their values.

1. Assessing Skewness and Kurtosis Statistics

What are Skewness and Kurtosis?

Both skewness and kurtosis are descriptive statistics that help quantify the shape of a distribution.

• Skewness:


• What it measures: The degree of asymmetry of the distribution around its mean.


• Visual: It tells you if the "tail" of the distribution is longer on one side or the other.


• Types:


• Positive Skew (Right-Skewed): The tail on the right side of the distribution is longer or fatter than the left side. The mean is typically greater than the median. (e.g., income data, reaction times).


• Negative Skew (Left-Skewed): The tail on the left side of the distribution is longer or fatter than the right side. The mean is typically less than the median. (e.g., test scores where most people do well, like a very easy exam).


• Zero Skew: The distribution is perfectly symmetrical (like a normal distribution).


• Kurtosis:


• What it measures: The "tailedness" of the distribution, describing the shape of the tails relative to the center. It indicates the presence of outliers.


• Important Note: Most statistical software (and the formulas used here) compute excess kurtosis, where a perfectly normal distribution has an excess kurtosis of 0. The "raw" or "Pearson" kurtosis (which equals 3 for a normal distribution) is less commonly reported. We'll focus on excess kurtosis.


• Types (Excess Kurtosis):


• Mesokurtic (Kurtosis = 0): The distribution has a similar peakedness and tail thickness to a normal distribution.


• Leptokurtic (Kurtosis > 0): The distribution has fatter tails and a sharper, more defined peak than a normal distribution. This often indicates more outliers than expected in a normal distribution.


• Platykurtic (Kurtosis < 0): The distribution has thinner tails and a flatter, broader peak than a normal distribution. This indicates fewer outliers than expected in a normal distribution.

Why are Skewness and Kurtosis Important for Normality?

Many parametric statistical tests (like t-tests, ANOVA, linear regression) assume that the data, or more specifically, the residuals of the model, are normally distributed.

• Skewness directly violates the symmetry assumption of normality.


• Kurtosis indicates whether the "tails" of your distribution are too heavy or too light compared to a normal distribution, which affects the presence and influence of extreme values.

How to Assess (Practical Steps)

Before jumping to statistics, always start with visual inspection! It provides immediate intuition.

A. Visual Assessment (Highly Recommended First Step)

• Histograms: Look for symmetry, the shape of the peak, and the length of the tails.


• Q-Q Plots (Quantile-Quantile Plots): These are excellent for assessing normality. If the data is normally distributed, the points will fall approximately along a straight diagonal line.


• Skewness: A curve in the Q-Q plot (either above or below the line) indicates skewness.


• Kurtosis: An "S" shape (points starting below, curving above, then ending below the line, or vice-versa) suggests kurtosis (heavy or light tails).


• Box Plots: Check for the median being off-center within the box, and unequal whisker lengths (indicating skewness). Also, look for outliers (which can suggest leptokurtosis).

B. Statistical Assessment (Calculating Skewness and Kurtosis Values)

You'll use statistical software to calculate these.

1. Using R:

# Install if you haven't already
# install.packages("e1071")
# install.packages("psych") # Another good package for descriptive stats
library(e1071) # For skewness and kurtosis functions
library(psych)  # For describe() function which gives many stats
# Example Data (replace with your actual data)
set.seed(123)
data_normal <- rnorm(1000, mean = 50, sd = 10)
data_pos_skew <- rchisq(1000, df = 3) # Chi-squared is positively skewed
data_neg_skew <- max(data_pos_skew) - data_pos_skew # Invert for negative skew
data_leptokurtic <- rt(1000, df = 5) # t-distribution with low df is leptokurtic
# Calculate for a normal-like distribution
skewness(data_normal)
kurtosis(data_normal) # Excess kurtosis
# Calculate for a positively skewed distribution
skewness(data_pos_skew)
kurtosis(data_pos_skew)
# Calculate for a negatively skewed distribution
skewness(data_neg_skew)
kurtosis(data_neg_skew)
# Calculate for a leptokurtic distribution
skewness(data_leptokurtic)
kurtosis(data_leptokurtic)
# Using psych package's describe() for a summary
psych::describe(data_pos_skew)
# Look for 'skew' and 'kurt' columns

2. Using Python (with pandas and scipy):

import pandas as pd
from scipy.stats import skew, kurtosis
import numpy as np
# Example Data (replace with your actual data)
np.random.seed(123)
data_normal = np.random.normal(loc=50, scale=10, size=1000)
data_pos_skew = np.random.chisquare(df=3, size=1000)
data_neg_skew = np.max(data_pos_skew) - data_pos_skew
data_leptokurtic = np.random.standard_t(df=5, size=1000)
# Convert to pandas Series for easy methods
series_normal = pd.Series(data_normal)
series_pos_skew = pd.Series(data_pos_skew)
series_neg_skew = pd.Series(data_neg_skew)
series_leptokurtic = pd.Series(data_leptokurtic)
# Calculate for a normal-like distribution
print(f"Normal Data - Skewness: {skew(data_normal):.3f}, Kurtosis: {kurtosis(data_normal):.3f}") # SciPy default is excess kurtosis
print(f"Normal Series - Skewness: {series_normal.skew():.3f}, Kurtosis: {series_normal.kurt():.3f}") # Pandas default is excess kurtosis
# Calculate for a positively skewed distribution
print(f"Pos Skew Data - Skewness: {skew(data_pos_skew):.3f}, Kurtosis: {kurtosis(data_pos_skew):.3f}")
print(f"Pos Skew Series - Skewness: {series_pos_skew.skew():.3f}, Kurtosis: {series_pos_skew.kurt():.3f}")
# Calculate for a negatively skewed distribution
print(f"Neg Skew Data - Skewness: {skew(data_neg_skew):.3f}, Kurtosis: {kurtosis(data_neg_skew):.3f}")
print(f"Neg Skew Series - Skewness: {series_neg_skew.skew():.3f}, Kurtosis: {series_neg_skew.kurt():.3f}")
# Calculate for a leptokurtic distribution
print(f"Leptokurtic Data - Skewness: {skew(data_leptokurtic):.3f}, Kurtosis: {kurtosis(data_leptokurtic):.3f}")
print(f"Leptokurtic Series - Skewness: {series_leptokurtic.skew():.3f}, Kurtosis: {series_leptokurtic.kurt():.3f}")

3. Using SPSS:

• Go to Analyze > Descriptive Statistics > Frequencies...


• Move your variables of interest to the "Variables" box.


• Click "Statistics..."


• Check Skewness and Kurtosis.


• Click "Continue" and then "OK."


• The output table will show the values for skewness and kurtosis. SPSS reports excess kurtosis.

• For Skewness: =SKEW(range)


• For Kurtosis: =KURT(range) (This function calculates excess kurtosis).

Interpreting the Statistics

There are various rules of thumb for interpreting these values, but remember they are just guidelines. The impact of skewness and kurtosis also depends on your sample size and the specific statistical test you plan to use.

A. Interpreting Skewness

• Value of 0: Perfectly symmetrical.


• Range -0.5 to +0.5: Generally considered "fairly symmetrical." Many sources consider this range acceptable for most analyses without severe issues.


• Range -1.0 to -0.5 or +0.5 to +1.0: Moderately skewed. Transformations might be beneficial.


• Less than -1.0 or Greater than +1.0: Highly skewed. Transformations are very likely necessary.

A more formal way is to compare the skewness (or kurtosis) value to its standard error. Most software (like SPSS) provides the standard error alongside the statistic.

• Calculate the Z-score: Z_skew = Skewness / SE(Skewness)


• If |Zskew| > 1.96 (for $\alpha = 0.05$), the skewness is considered statistically significantly different from zero. Some researchers use |Zskew| > 2 or even |Z_skew| > 3 as a more conservative threshold, especially with large sample sizes, because with very large samples, even trivial skewness can be statistically significant.


• Direction: Positive Z-score indicates positive skew, negative indicates negative skew.

B. Interpreting Kurtosis (Excess Kurtosis)

• Value of 0: Mesokurtic (similar to a normal distribution).


• Range -0.5 to +0.5: Generally considered "acceptable" or sufficiently mesokurtic.


• Greater than +0.5: Leptokurtic (heavy tails, sharper peak, more outliers than normal).


• Less than -0.5: Platykurtic (light tails, flatter peak, fewer outliers than normal).

• Calculate the Z-score: Z_kurt = Kurtosis / SE(Kurtosis)


• If |Zkurt| > 1.96 (for $\alpha = 0.05$), the kurtosis is considered statistically significantly different from zero. Again, |Zkurt| > 2 or |Z_kurt| > 3 might be used as more conservative cutoffs.


• Direction: Positive Z-score indicates leptokurtic, negative indicates platykurtic.

What to Look For and How to Decide

• Visual Inspection First: Does the histogram look lopsided? Do the Q-Q plot points deviate significantly from the line?


• Magnitude of Skewness/Kurtosis: How far are the values from 0? Use the rules of thumb as a guide.


• Statistical Significance (Z-score): Is the deviation from 0 statistically significant, considering your sample size?


• Nature of Your Data & Field: Some fields tolerate more non-normality than others.


• Robustness of Your Test: Some tests are more robust to violations of normality than others (e.g., ANOVA is relatively robust to moderate non-normality, especially with balanced groups and large sample sizes, but not to extreme outliers).

Once you've assessed these, you'll move on to selecting appropriate transformations (which would be the next step in your process!).