How to Conduct Normality Assumptions Tests Using PSPP in Statistics for Social Research

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In PSPP, when conducting normality testing on continuous variables, focus on two key outputs: skewness and kurtosis. Skewness measures the asymmetry of your data distribution. A skewness value close to zero indicates a symmetrical distribution. Positive skewness shows a tail on the right side, while negative skewness indicates a tail on the left.

On the other hand, Kurtosis reflects the ‘tailedness’ of the distribution. A kurtosis value close to zero suggests a distribution similar to the normal distribution in terms of tail thickness. High positive kurtosis indicates heavier tails than a normal distribution, implying more outliers. Negative kurtosis shows thinner tails, suggesting fewer outliers.

To assess normality, compare these values against the standard normal distribution benchmarks. Significant deviations from zero in either skewness or kurtosis can indicate a departure from normality. This assessment helps decide the appropriateness of parametric tests, which assume normal data distribution.

You can use certain numerical thresholds to determine if your data are within the bounds of normality based on skewness and kurtosis values. However, it’s important to note that these thresholds are not strict rules but general guidelines, and interpretations may vary slightly depending on the field of study or the specific statistical approach.

Skewness

1. A skewness value between -1 and +1 is generally acceptable for normal distribution.
2. Values between -1 and -0.5 or between +0.5 and +1 might be moderately skewed.
3. Values less than -1 or greater than +1 are typically regarded as highly skewed.

Kurtosis

1. Kurtosis values are often compared to 3, which is the kurtosis of a normal distribution (sometimes, this is adjusted to 0, depending on whether the software uses excess kurtosis, which subtracts 3 from the raw kurtosis value).
2. A kurtosis value between 2 to 4 (or -1 to +1 for excess kurtosis) is generally acceptable.

Values outside this range might indicate that the data have tails that are too heavy or too light compared to a normal distribution.

These values are more like rough guidelines rather than definitive limits. Statistical decisions should not be based solely on these criteria but should also consider the context of the data, the sample size, and other statistical considerations. Additionally, visual methods such as histograms or Q-Q plots are often used in conjunction with these numerical methods for a more comprehensive assessment of normality.

Step-by-Step Procedures in PSPP

1. Open the dataset in PSPP
2. Click on “Analyze”–>”Descriptive Statistics”–>”Descriptives…”
3. Choose your continuous variables (at the ordinal, interval, or ratio levels) and use the ” >” button to move them to the “Variables” box on the right.
4. Deselect all options under statistics except for “Kurtosis” and “Skewness.”
5. Click “OK.”

In the output window of PSPP, you will see a table similar to this:

╭────────────────────────────┬──┬────────┬─────────┬────────┬─────────╮
│                            │ N│Kurtosis│S.E. Kurt│Skewness│S.E. Skew│
├────────────────────────────┼──┼────────┼─────────┼────────┼─────────┤
│Religiosity_Level           │25│   -1.42│      .90│     .38│      .46│
│Social_Justice_Attitudes_1_5│25│    -.71│      .90│    -.31│      .46│
│Social_Cohesion_Score_1_10  │25│   -1.02│      .90│    -.02│      .46│
│Valid N (listwise)          │25│        │         │        │         │
│Missing N (listwise)        │ 0│        │         │        │         │
╰────────────────────────────┴──┴────────┴─────────┴────────┴─────────╯

To analyze the output for normality based on the kurtosis and skewness values, let’s examine each variable in the provided data:

Religiosity_Level

• Kurtosis: -1.42 (With a standard error of 0.90)
• Skewness: 0.38 (With a standard error of 0.46)

The kurtosis value is slightly below the acceptable range, suggesting a distribution with lighter tails than a normal distribution. The skewness is within the acceptable range (-1 to +1), indicating a fairly symmetrical distribution. However, the standard errors are relatively high, which might affect the reliability of these measures due to the small sample size (N=25).

Social_Justice_Attitudes_1_5

• Kurtosis: -0.71 (With a standard error of 0.90)
• Skewness: -0.31 (With a standard error of 0.46)

Both kurtosis and skewness values are within the acceptable range for a normal distribution. The distribution appears to have slightly lighter tails and is fairly symmetrical.

Social_Cohesion_Score_1_10

• Kurtosis: -1.02 (With a standard error of 0.90)
• Skewness: -0.02 (With a standard error of 0.46)

The kurtosis value is slightly outside the preferred range, indicating lighter tails than a normal distribution. The skewness is almost zero, suggesting a very symmetrical distribution.

In this case, the skewness values for all variables are within the acceptable range, indicating symmetry in the distributions. The kurtosis values for all variables are slightly lower than the normal range, suggesting distributions with lighter tails than a normal distribution. However, the standard errors are quite large compared to the kurtosis and skewness values, which is a common issue in smaller samples and can affect the precision of these estimates.

Therefore, while the data do not show extreme deviations from normality, the reliability of these results might be limited due to the small sample size. Additionally, visual assessments and other statistical tests should be used to confirm these findings.

Example Visualizations

This is what the histograms (distributions) of each variable might look like:

Religiosity Level

• This graph shows a slight positive skew and light tails (platykurtic). The distribution is fairly symmetrical but slightly skewed to the right.

Social Justice Attitudes

• The distribution here is slightly negatively skewed with light tails. It is mostly symmetrical but with a slight skew to the left.

Social Cohesion Score

• This graph presents a distribution with almost no skew and light tails. It’s very symmetrical, closely resembling a normal distribution, but with the tails being lighter than a standard normal distribution.

What Does It Mean?

In this case, we need to look at the Skewness and Kurtosis of each variable to determine if we can use parametric or nonparametric statistical procedures. The results of parametric procedures can be generalized to the population, and nonparametric procedures can only be applied to the sample.

How to proceed is a judgment call by the researcher. In this case, each variable is mostly within the acceptable ranges with slight skews. Using parametric procedures on any statistical tests involving these variables would likely be safe. Still, because the sample sizes are small, it might be necessary to expand the sample size. 

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