Normal distribution

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Normal distribution
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(No Transcript)
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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).

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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.
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Central Limit Theorem(Zentraler Grenzwertsatz)
X1 X2 X3 X4 M
Sample 1 6 2 5 6 4.75
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Central Limit Theorem(Zentraler Grenzwertsatz)
X1 X2 X3 X4 M
Sample 1 6 2 5 6 4.75
Sample 2 2 3 1 6 3
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Central Limit Theorem(Zentraler Grenzwertsatz)
X1 X2 X3 X4 M
Sample 1 6 2 5 6 4.75
Sample 2 2 3 1 6 3
Sample 3 1 1 4 6 3
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Central Limit Theorem(Zentraler Grenzwertsatz)
X1 X2 X3 X4 M
Sample 1 6 2 5 6 4.75
Sample 2 2 3 1 6 3
Sample 3 1 1 4 6 3
Sample 4 6 2 2 1 2.75
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Central Limit Theorem(Zentraler Grenzwertsatz)
X1 X2 X3 X4 M
Sample 1 6 2 5 6 4.75
Sample 2 2 3 1 6 3
Sample 3 1 1 4 6 3
Sample 4 6 2 2 1 2.75
Sample 5 1 5 1 3 2.5
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Mean of the sample mean
4.75 3.0 3.0 2.75 2.5
3.2
5
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The sample means are normally distributed (even
if the phenomenon in the parent population is not
normally distributed).
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population
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population
sample
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population
one sample
mean of this sample
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population
one sample
mean of this sample
distribution of many sample means
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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.

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Graphical representation of probability
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Graphical representation of probability
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z scores
x1 x SD
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Empirical rule
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Empirical rule
1.96
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Parametric vs. non-parametrical tests
You use 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
  dont have sufficient data (e.g. small number of
  samples, small individual samples).

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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).
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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.

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Standard error
Samples Mean
1 2 3 4 5 1.5 1.8 1.3 2.0 1.7
? 8.3 / 5 1.66 (mean)
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Standard error
Samples Mean Individual means Mean of means
1 2 3 4 5 1.5 1.8 1.3 2.0 1.7 1.5 1.66 1.8 1.66 4 1.66 9 1.66 12 1.66
? 8.3 / 5 1.66 (mean)
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Standard error
Samples Mean Individual means Mean of means
1 2 3 4 5 1.5 1.8 1.3 2.0 1.7 1.5 1.66 1.8 1.66 4 1.66 9 1.66 12 1.66 0.16 0.14 0.36 0.36 0.04
? 8.3 / 5 1.66 (mean)
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Standard error
Samples Mean Individual means Mean of means squared
1 2 3 4 5 1.5 1.8 1.3 2.0 1.7 1.5 1.66 1.8 1.66 4 1.66 9 1.66 12 1.66 0.16 0.14 0.36 0.36 0.04 0.0256 0.0196 0.1296 0.1156 0.0016
? 8.3 / 5 1.66 (mean)
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Standard error
Samples Mean Mean Mean of means squared
1 2 3 4 5 1.5 1.8 1.3 2.0 1.7 1.5 1.66 1.8 1.66 4 1.66 9 1.66 12 1.66 0.16 0.14 0.36 0.36 0.04 0.0256 0.0196 0.1296 0.1156 0.0016
? 8.3 / 5 1.66 (mean) ? 0.292
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Standard error
0.292 5 - 1
0.2701
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Confidence interval
degree of certainty ? standard error
x sample mean /x confidence interval
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Confidence intervals
95 degree of certainty 1.96 z-score
Confindence interval of the first sample (mean
1.5) 1.96 ? 0.2701 0.53 1.5 /- 0.53
0.972.03 We can be 95 certain that the
population mean is located in the range between
0.97 and 2.03.
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Standard error (alternative formula)
SD
N
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Exercise
Mean 7
SD (2-7)2 (5-7)2 (6-7)2 (7-7)2 (10-7)2
(12-7)2 6 -1 3.58
Standard error 3.58 / ?6 1.46 Confidence
I. 1.46 ? 1.96 2.86 7 / 2.86 4.14
9.86