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## Normality Tests

The distribution plots for the Life Ladder variable does suggest that it is normally distributed but there are some tests that can be used to clarify this. A blogpost on tests for normality at machinelearningmastery.com outlines how it is important when working with a sample of data to know whether to use parametric or nonparametric statistical methods. If methods used assume a Gaussian distribution when it is not the case then findings can be incorrect or misleading. In some cases it is enough to assume the data is normal enough to use parametric methods or to transform the data to be normal enough.

Parametric statistical methods assume that the data has a known and specific distribution, often a Gaussian distribution. If a data sample is not Gaussian, then the assumptions of parametric statistical tests are violated and nonparametric statistical methods must be used.

Normality tests can be used to check if your data sample is from a Gaussian distribution or not.

• Statistical tests calculate statistics on the data to quantify how likely it is that the data was drawn from a normal distribution.
• Graphical methods plot the data to qualitatively evaluate if the data looks Gaussian.

Tests for normality include the Shapiro_Wilk Normality Test, the D’Agostino and Pearson’s Test, the Anderson-Darling Test.

The histograms above show the characteristic bell-shape curve of the normal distribution and indicates that the Life Ladder variable is gaussian or approximately normally distributed.

The Quantile-Quantile Plot (QQ plot) is another plot that can be used to check if the distribution of a data sample. It generates its own sample of the idealised distribution that you want to compare your sample of data to. The idealised samples are divided into groups and each data point in the sample is paired with a similar member from the idealised distribution at the same cumulative distribution.

A scatterplot is drawn with the idealised values on the x-axis and the data sample on the y-axis. The resulting points are plotted as a scatter plot with the idealized value on the x-axis and the data sample on the y-axis. If the result is a straight line of dots on the diagonal from the bottom left to the top right this indicates a perfect match for the distribution whereas if the dots deviate far from the diagonal line.

#### QQ plot of Life Ladder:

Here a QQ plot is created for the Life Ladder sample compared to a Gaussian distribution (the default).

#### Tests for normality:

The QQ plot does seem to indicate to me that the Life Ladder is normally distributed. The scatter plots of points do mostly follow the diagonal pattern for a sample from a Gaussian distribution. I will use some of the normality tests as outlined on the blogpost on tests for normality. These tests assume the sample was drawn from a Gaussian distribution and test this as the null hypothesis. A test-statistic is calculated and a p-value to interpret the test statistic. A threshold level called alpha is used to interpret the test. This is typically 0.05. If the p-value is less than or equal to alpha, then reject the null hypothesis. If the p-value is greater than alpha then fail to reject the null hypothesis. In general a larger p-value indicates that the sample was likely to have been drawn from a Gaussian distribution. A result above 5% doesn’t mean the null hypothesis is true but that it is very likely true given the evidence available.

#### D’Agostino’s K^2 Test

This test calculates the kurtosis and skewness of the data to see if the data distribution is not normal like. The skew is a measure of asymmetry in the data while hurtosis how much of the distribution is in the tails.

#### Anderson-Darling Test:

This is a statistical test that can be used to evaluate whether a data sample comes from one of among many known data samples adn can be used to check whether a data sample is normal. The test is a modified version of the Kolmogorov-Smirnov test which is a nonparametric goodness-of-fit statistical test. The Anderson-Darling test returns a list of critical values rather than a single p-value. The test will check against the Gaussian distribution (dist=’norm’) by default.

The sample of Life Ladder for 2018 does appear to be normally distributed based on the above tests. Therefore I can go ahead and simulate data for this variable using the normal distribution using the sample mean and standard deviation statistics. 