Statistical (scientific) determination of normality.

In statistics, normality refers to the distribution of data points in a dataset. A normal distribution, also known as a Gaussian distribution, is a bell-shaped curve that is symmetrical around the mean. Determining whether a dataset follows a normal distribution is important for many statistical analyses, as it allows for the use of certain parametric tests.

There are several methods for determining normality in a dataset:

1. Visual inspection: One common method is to visually inspect the data using a histogram or a Q-Q plot. A histogram can show the shape of the distribution, while a Q-Q plot compares the data to a theoretical normal distribution. If the data points fall along a straight line in the Q-Q plot, it suggests that the data follows a normal distribution.

2. Statistical tests: There are also statistical tests that can be used to determine normality, such as the Shapiro-Wilk test, Kolmogorov-Smirnov test, and Anderson-Darling test. These tests calculate a p-value, which indicates the likelihood that the data comes from a normal distribution. A p-value greater than 0.05 typically suggests that the data is normally distributed.

3. Descriptive statistics: Another way to assess normality is to calculate descriptive statistics such as skewness and kurtosis. Skewness measures the symmetry of the data distribution, while kurtosis measures the peakedness of the distribution. A skewness value close to zero and a kurtosis value close to 3 are indicative of a normal distribution.

It is important to note that normality is not always necessary for statistical analyses, as many tests are robust to deviations from normality. However, if normality is a critical assumption for a particular analysis, it is important to carefully assess whether the data meets this assumption using one or more of the methods described above.

Statistical Tests for Normality

1. Shapiro-Wilk Test

- Non-parametric test that compares the sample distribution to a normal distribution.
- Returns a p-value. If p < 0.05, the sample is considered non-normal.

2. Jarque-Bera Test

- Tests for skewness and kurtosis (peakedness) of the sample distribution.
- Returns a chi-square statistic. If the statistic is significant (p < 0.05), the sample is considered non-normal.

3. Kolmogorov-Smirnov Test

- Compares the empirical distribution function of the sample to a normal distribution function.
- Returns a D statistic (maximum difference between the CDFs). If D > critical value, the sample is considered non-normal.

4. Quantile-Quantile (Q-Q) Plot

- Graphical representation of the data against a normal distribution.
- A linear line indicates normality. Deviations from linearity suggest non-normality.

5. Histogram

- Presents a distribution of the data in bins.
- A bell-shaped histogram suggests normality. Asymmetry or peaks indicate non-normality.

6. Z-Scores

- Standardizes the data by subtracting the mean and dividing by the standard deviation.
- More than 5% of the data having Z-scores > 2 or < -2 indicates non-normality.

7. Skewness and Kurtosis

- Skewness: Measures the asymmetry of the distribution. Values close to 0 indicate symmetry.
- Kurtosis: Measures the peakedness of the distribution. Values close to 3 indicate a normal distribution.

Interpretation

- A sample is considered normal if all or most of the tests (especially Shapiro-Wilk or Jarque-Bera) indicate a p-value > 0.05 and the Q-Q plot shows a linear trend.
- If the sample is non-normal, data transformations (e.g., log transformation) or non-parametric statistical methods may need to be considered.