Stochastics

Statistics

Mean (arithmetic mean), median, variance and standard deviation are determined for a prime notation. In addition, the distribution is presented as a histogram.

Formulas:

Mean:                x = 1/n · Σ x(i)
 
Variance:           s2 = 1/(n−1)·Σ(x(i) − x)2   bzw.  1/n·Σ(x(i) − x)2
                                 
Standard Dev.:  s = √

The median is that value, which stands in the middle of the sorted list. In case of an even number of values the median ist the mean the two values in the middle of the list.

Example:

Data:
9 6 7 7 3 9 10 1 8 7 9 6 9 8 10 5 10 10 9 11 8

Number of data        n = 21
Maximum            max = 11
Minimum              min = 1
Mean                        x = 7,7142857
Median                     c = 8
Variance                 s² = 6,1142857
Standard deviation  s = 2,4727082
 
               x                       H                 h        
—————————   ———   ——————
         1 ≤ x < 2                 1         0,047619     
         2 ≤ x < 3                 0         0                 
         3 ≤ x < 4                 1         0,047619     
         4 ≤ x < 5                 0         0                 
         5 ≤ x < 6                 1         0,047619     
         6 ≤ x < 7                 2         0,0952381   
         7 ≤ x < 8                 3         0,142857     
         8 ≤ x < 9                 3         0,142857     
         9 ≤ x < 10               5         0,238095     
       10 ≤ x < 11               4         0,190476     

Boxplot:
Q1=6,5;   Q2=8;   Q3=9,5;   IQR=3
Left whisker=3;   Right whisker=11

The box plot is determined by the quartiles
Q1: Median of the lower half of the data (left edge of the box)
Q2: Median of all data (line within the box),
Q3: Median of the upper half of the data (right edge of the box).

IQR = Q3-Q1 (InterQartile Range) is the width of the box.
50% of the data lies in this area.

The two whiskers show the smallest and the largest value, which is no more than 1.5 times the IQR outside the box.
Values that are not within this range (outliers) are indicated by small circles.
The mean value is represented by a plus sign.

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