also Gaussian distribution Normal distribution also Gaussian distribution

Probability density Mean value of distribution is  holds - variance of distribution is 2

There is certain inconsistency -  is a symbol for mean in general as well as the specific parameter of normal distribution (which is a mean too) and similar for 2 Key role of normal distribution in statistics rises from central limit theorem. It says, that mean of “very big” random sample is random value with almost normal distribution, even if distribution of population differs from normal one.

From “definition” – variable with normal distribution can take with nonzero probability values from - to + Biological variables haven’t usually normal distribution, but can be ofted “reasonably” approximated by normal distribution.

Skewness and kurtosis i-th moment - mean value Xi i-th central moment, κi – mean value (X- )i So, mean value is the first general moment The first central moment is from definition 0 Variance is the second central moment Skewness is characterized as the third central moment Kurtosis is characterized as the fourth central moment

Skewness Positive skewness – a lot of negative deviances from mean is compensated by lesser number of big positive deviances: 3, 3, 3, 4, 7 μ=4 3=(3-4)3+ (3-4)3+ (3-4)3+ (4-4)3+ (7-4)3 =(-1)+(-1)+(-1)+0+27=24 3 – is in the third powers of measurement units mean median - is dimensionless and set just shape

Skewness Negative skew distribution - a lot of small positive deviations from mean is compensated by lesser number of big negative deviances 5, 5, 5, 1, 4 μ=4 3=(5-4)3+ (5-4)3+ (5-4)3+ (4-4)3+ (1-4)3 =1+1+1+0+(-27)=-24 mean median

Kurtosis – 4th central moment Normal distribution is mesokurtic normal - mesokurtic leptokurtic 2 > 0 platykurtic 2 < 0

standartized normal distribution

“checking“ of normality - graphic Mixture of two types of normal distribution Normal Plot a cumulative histogram of counts on a scale of probabilities Left skewed Right skewed Platykurtic leptokurtic Examples of denzities of probabilities (frequency distributions) together with their cumulative distributions plotted on y-axis in a scale of normal probability

Normality checking– I compute skewness and kurtosis and compare them with expected values for normal distribution. The most of biological data has positive skewed distribution – that’s why computing of skewness give us quite strong test and tell us how much data differ from normality at the same time.

Normality testing - χ2 test I compute mean and variance from data and compare sample data with date with normal distribution with the same mean and variance as data of my own. Then, with help of χ2 test, I compare number of cases in size classes set up from observed data and expected frequencies in normal distribution – classic problem, I must decide the breadth of categories (columns’ width in histogram) – number of degrees of freedom = k-1-2 2 parameters from data

Editors of journals demand such a test, but (almost) no biological data have normal distribution, so if I have a large data, the test is strong and I reject null hypothesis about normality (even if the deviation from normality is small) if I have a few data, the test is desperately weak and even for data with big deviation from normality I cannot reject null hypothesis