1. Normal data
Let's review the concept of normal data and how it relates to experimental analysis.
2. The normal distribution
Normal data is drawn from a normal distribution, which has the familiar bell curve shape.
The normal distribution is intrinsically linked to z-scores, which recall, is a standardized measure of how many standard deviations a value is from the population mean.
The most common normal distribution used for z-scores has a mean of zero and a standard deviation of one.
This answers questions such as 'How many standard deviations is this point from the mean?' and 'What is the probability of obtaining this score?'.
3. Normal data and statistical tests
Normal data is an underlying assumption for many statistical tests,
called parametric tests.
There are also nonparametric tests that don't assume normal data.
4. Normal, Z, and alpha
In hypothesis testing, alpha, or the significance level, is often closely linked to the normal distribution.
For normal data, we can visually see the risk of error for a given significance level and compare that result to the p-value, which is related to the z-score.
An alpha of 0.05 on a standard two-tailed test represents a small region in the tails. It means there is a 5% risk of rejecting the null hypothesis when it is actually true - a so-called Type I error.
5. Visualizing normal data
We can visually check data for normality using a kde (or kernel density) plot, available via Seaborn's displot() function.
On this salaries dataset, the data appears approximately normal.
6. QQ plots
A more statistically robust visual tool is a quantile-quantile, or QQ, plot. It plots the quantiles or sections of two distributions against each other.
The qqplot function from statsmodels plots our data. Setting the dist argument to the normal distribution from scipy.stats compares our data against a standard normal distribution.
If the distributions are similar, the dots in the QQ plot hug the line tightly. Our data again seems quite normal.
Here is another example. The dots bow out at the ends, which means that the data is not very normal.
7. Tests for normality
There are also various numerical hypothesis tests for normality.
The Shapiro-Wilk test is known to be good for small datasets.
The D'Agostino K-squared test uses kurtosis and skewness to determine normality. These terms relate to the symmetry and size of a distribution's tails, respectively.
Anderson-Darling is another common test which returns a list of values, rather than just one so we can see normality at different levels of alpha.
Each of these tests has a null hypothesis that the provided dataset is drawn from a normal distribution.
8. A Shapiro-Wilk test
Let's run one of these tests, the Shapiro-Wilk test. We import it from scipy.stats, and set our alpha at 0.05.
The function takes a series of values and returns a test statistic and p-value.
The p-value is greater than alpha, so we have evidence our data that looked quite normal is normal. We fail to reject the null hypothesis and have evidence that the data sample is normal at the alpha level of 0.05.
9. An Anderson-Darling test
To implement an Anderson-Darling test, we provide data and set the dist argument to norm to test for normality.
The result object contains a test statistic and a range of critical values and significance levels.
To interpret, we check the test statistic against each critical value. If the test statistic is higher than the critical value, the null hypothesis is rejected at that particular significance level, and the data is not normal.
0.2748 is less than all the critical values, so we fail to reject the null hypothesis and suspect that the data is normal.
10. Let's practice!
Let's practice visualizing and assessing normal data in experimental design setups.