Normal distribution

Properties of the normal distribution The center of the curve represents the mean (and median and mode). The curve is symmetrical around the mean. The tails meet the x-axis in infinity. The curve is bell-shaped. The area under the curve is 1 (by definition).

Central Limit Theorem ( Zentraler Grenzwertsatz) Der Mittelwert der Mittelwerte verschiedener Stichproben nähert sich dem wahren Mittelwert an. + Die Mittelwerte verschiedener Stichproben sind normalverteilt, selbst wenn das zu untersuchende Phänomen nicht normalverteilt ist.

Central Limit Theorem ( Zentraler Grenzwertsatz) X1X1 X2X2 X3X3 X4X4 M Sample

Central Limit Theorem ( Zentraler Grenzwertsatz) X1X1 X2X2 X3X3 X4X4 M Sample Sample

Central Limit Theorem ( Zentraler Grenzwertsatz) X1X1 X2X2 X3X3 X4X4 M Sample Sample Sample

Central Limit Theorem ( Zentraler Grenzwertsatz) X1X1 X2X2 X3X3 X4X4 M Sample Sample Sample Sample

Central Limit Theorem ( Zentraler Grenzwertsatz) X1X1 X2X2 X3X3 X4X4 M Sample Sample Sample Sample Sample

Mean of the sample mean = 3.2 5

The sample means are normally distributed (even if the phenomenon in the parent population is not normally distributed).

population

sample

population one sample mean of this sample

population one sample mean of this sample distribution of many sample means

Are your data normally distributed? The distribution in the parent population (normal, slightly skewed, heavily skewed). The number of observations in the individual sample. The total number of individual samples.

Graphical representation of probability

z scores x 1 – x SD

Empirical rule

1.96

Parametric vs. non-parametrical tests If you have ordinal data. If you have interval data that is not normally distributed. If you have interval data but cannot be certain if your data is normally distributed because you don’t have sufficient data (e.g. small number of samples, small individual samples). You use non-parametrical tests:

Confidence intervals Confidence intervals indicate a range within which the mean (or other parameters) of the true population lies given the values of your sample and assuming a certain probability. The standard error is the equivalent of the standard deviation for the sample distribution (i.e. the distribution based on the sample means).

Confidence intervals The mean of the sample means. The SDs of the sample means, i.e. the standard error. The degree of confidence with which you want to state the estimation.

Standard error SamplesMean  8.3 / 5 = 1.66 (mean)

Standard error SamplesMeanIndividual means – Mean of means – – – – – 1.66  8.3 / 5 = 1.66 (mean)

Standard error SamplesMeanIndividual means – Mean of means – – – – – –  8.3 / 5 = 1.66 (mean)

Standard error SamplesMeanIndividual means – Mean of means squared – – – – – –  8.3 / 5 = 1.66 (mean)

Standard error SamplesMeanMean – Mean of means squared – – – – – –  8.3 / 5 = 1.66 (mean)  0.292

Standard error =

Confidence interval [degree of certainty]  [standard error] = x [sample mean] +/–x = confidence interval

Confidence intervals 95% degree of certainty = 1.96 [z-score] Confindence interval of the first sample (mean = 1.5): 1.96  = / = 0.97–2.03 We can be 95% certain that the population mean is located in the range between 0.97 and 2.03.

Standard error (alternative formula) SD N

Exercise SD: (2-7) 2 + (5-7) 2 + (6-7) 2 + (7-7) 2 + (10-7) 2 + (12-7) = 3.58 Standard error:3.58 /  6 = 1.46 Confidence I.:1.46  1.96 = /– 2.86 = 4.14 – 9.86 Mean: 7