1. also Gaussian distribution
Normal distribution
also Gaussian distribution

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

3. 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.

6. 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.

7. 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

8. Skewness
Positive skewness – a lot of negative deviances from mean is compensated by lesser number of big positive deviances:
3, 3, 3, 4, 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

9. 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 = (-27)=-24
mean
median

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

11. standartized normal distribution

12. “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

13. 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.

14. 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

16. 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