# Testing for Normality, Tests for Normality, Graphical Methods YouTube Lecture Handouts

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## Testing for Normality

Population vs sample

How we test normality

- Difference between theoretical distribution and actual data – we require tests of normality
- When is non-normality a problem?
- Normality can be a problem when the sample size is small (< 50) .
- Highly skewed data create problems.
- Highly leptokurtic data are problematic, but not as much as skewed data.
- Normality becomes a serious concern when there is “activity” in the tails of the data set.
- Outliers are a problem. (Test used are Grubb՚s test and Dixon test)
- “Clumps” of data in the tails are worse.
- Final Words Concerning Normality Testing:
- Since it is a test, state a null and alternate hypothesis.
- If you perform a normality test, do not ignore the results.
- If the data are not normal, use non-parametric tests.
- If the data are normal, use parametric tests.
**If you have groups of data, you MUST test each group for normality**.

### Tests for Normality

- Statistical tests for normality are more precise since actual probabilities are calculated.
- Tests for normality calculate the probability that the sample was drawn from a normal population.

The hypotheses used are:

- Ho: The sample data are not significantly different from a normal population.
- Ha: The sample data are significantly different from a normal population.

When testing for normality:

- Probabilities indicate that the data are normal.
- Probabilities indicate that the data are NOT normal
__SPSS Normality Tests -__Kolmogorov-Smirnov and Shapiro-Wilk.__PAST Normality Tests__- Shapiro-Wilk, Anderson-Darling, Lilliefors, Jarque-Bera.

### Q-Q Plots

Q-Q plots display the observed values against normally distributed data (represented by the line) .

Normally distributed data fall along the line

### Graphical Methods

Graphical methods are typically not very useful when the sample size is small. This is a histogram of the last example. These data do not ‘look’ normal, but they are not statistically different from normal

### W/S Test

### W/S Test for Normality

- A simple test that requires only the sample standard deviation and the data range.
- Should not be confused with the Shapiro-Wilk test.
- Based on the q statistic, which is the ‘studentized’ (meaning t distribution) range, or the range expressed in standard deviation units.
- Where q is the test statistic, w is the range of the data and s is the standard deviation.
- The test statistic q (Kanji 1994, table 14) is often reported as u in the literature.

### Jarque-Bera Test

Normality is one of the assumptions for many statistical tests, like the t test or F test; the Jarque-Bera test is usually run before one of these tests to confirm normality. It is usually used for __large data sets__, because other normality tests are not reliable when n is large

- Where: n is the sample size, S is sample skewness
- K is sample kurtosis
- In general, a large J-B value indicates that errors are
**not**normally distributed. - For sample sizes of 2,000 or larger, this test statistic is compared to a
__chi-squared distribution with 2 degrees of freedom__(normality is rejected if the test statistic is greater than the chi-squared value) . - The chi-square approximation requires large sample sizes to be accurate. For sample sizes less than 2,000, the critical value is determined via simulation.

### Shapiro-Wilks

The test gives you a W value; small values indicate your sample is not normally distributed

Where:

- are the ordered random sample values
- are constants generated from the covariances, variances and means of the sample (size n) from a normally distributed sample.
- The test has limitations, most importantly that the
__test has a bias by sample size__. The larger the sample, the more likely you՚ll get a statistically significant result. - Univariate continuous data
- Numerator is slope of observed data vs expected normal values
- If is true then W should be 1.
- Highly sensitive and use graphical method to assess t-test assumptions

### Kolmogorov-Smirnov (K-S Test)

It compares the __observed versus the expected cumulative relative frequencies__

Kolmogorov-Smirnov test uses the maximal absolute difference between these curves as its test statistic denoted by D.

- It only applies to continuous distributions.
- It tends to be more sensitive near the center of the distribution than at the tails. Determined by stimulation.
- The Kolmogorov-Smirnov (K-S) test is based on the empirical distribution function (ECDF) . Given N
*ordered*data points , the ECDF is defined as

### En

- where
**n (i)**is the number of points less than and the are ordered from smallest to largest value. This is a step function that increases by at the value of each ordered data point. - Calculated value critical value (acceptance criteria) default is 0.565 as critical value

### D՚agostino Test

A very powerful test for departures from normality.

Based on the D statistic, which gives an upper and lower critical value.

Where D is the test statistic, SS is the sum of squares of the data and n is the sample size, and I is the order or rank of observation x. The df for this test is n (sample size) .

First, the data are ordered from smallest to largest or largest to smallest

is middle term of dataset

is observations՚ distance from middle

Notice that as the __sample size increases, the probabilities decrease__. In other words, **it gets harder** to meet the normality assumption as the sample size increases since even small departures from normality are detected.

**W/S or studentized range (q)** :

- Simple, very good for symmetrical distributions and short tails.
- Very bad with asymmetry.

**Shapiro Wilk (W)** :

- Powerful omnibus test. Not good with small samples or discrete data.
- Good power with symmetrical, short, and long tails. Good with asymmetry.

**Jarque-Bera (JB)** :

- Good with symmetric and long-tailed distributions.
- Less powerful with asymmetry, and poor power with bimodal data.

D՚Agostino (D or Y) :

- Good with symmetric and very good with long-tailed distributions.
- Less powerful with asymmetry.

**Anderson-Darling (A)** :

- Similar in power to Shapiro-Wilk but has less power with asymmetry.
- Works well with discrete data.

Distance tests (Kolmogorov-Smirnov, Lillifors, and Chi2) :

- All tend to have lower power. Data have to be very non-normal to reject Ho.
- These tests can outperform other tests when using discrete or grouped data.
- Several goodness-of-fit tests, such as the Anderson-Darling test and the Cramer Von-Mises test, are refinements of the K-S test. As these refined tests are generally considered to be more powerful than the original K-S test, many analysts prefer them. In addition, the advantage for the K-S test of having the critical values be independent of the underlying distribution is not as much of an advantage as first appears.

✍ Manishika