You entered a number set X of {9,8,7,64,3,2,1}
From the 7 numbers you entered, we want to calculate the mean, variance, standard deviation, standard error of the mean, skewness, average deviation (mean absolute deviation), median, mode, range, Pearsons Skewness Coefficient of that number set, entropy, midrange
1, 2, 3, 7, 8, 9, 64
Rank Ascending
1 is the 1st lowest/smallest number
2 is the 2nd lowest/smallest number
3 is the 3rd lowest/smallest number
7 is the 4th lowest/smallest number
8 is the 5th lowest/smallest number
9 is the 6th lowest/smallest number
64 is the 7th lowest/smallest number
64, 9, 8, 7, 3, 2, 1
Rank Descending
64 is the 1st highest/largest number
9 is the 2nd highest/largest number
8 is the 3rd highest/largest number
7 is the 4th highest/largest number
3 is the 5th highest/largest number
2 is the 6th highest/largest number
1 is the 7th highest/largest number
Sort our number set in ascending order
and assign a ranking to each number:
Number Set Value  1  2  3  7  8  9  64 
Rank  1  2  3  4  5  6  7 
Since we have 7 numbers in our original number set,
we assign ranks from lowest to highest (1 to 7)
Our original number set in unsorted order was 1,2,3,7,8,9,64
Our respective ranked data set is 1,2,3,4,5,6,7
Root Mean Square =  √A 
√N 
where A = x_{1}^{2} + x_{2}^{2} + x_{3}^{2} + x_{4}^{2} + x_{5}^{2} + x_{6}^{2} + x_{7}^{2} and N = 7 number set items
A = 1^{2} + 2^{2} + 3^{2} + 7^{2} + 8^{2} + 9^{2} + 64^{2}
A = 1 + 4 + 9 + 49 + 64 + 81 + 4096
A = 4304
RMS =  √4304 
√7 
RMS =  65.604877867427 
2.6457513110646 
RMS = 24.796313089997
Central tendency contains:
Mean, median, mode, harmonic mean,
geometric mean, midrange, weightedaverage:
μ =  Sum of your number Set 
Total Numbers Entered 
μ =  ΣX_{i} 
n 
μ =  1 + 2 + 3 + 7 + 8 + 9 + 64 
7 
μ =  94 
7 
μ = 13.428571428571
Since our number set contains 7 elements which is an odd number,
our median number is determined as follows:
Number Set = (n_{1},n_{2},n_{3},n_{4},n_{5},n_{6},n_{7})
Median Number = Entry ½(n + 1)
Median Number = Entry ½(8)
Median Number = n_{4}
Our median is entry 4 of our number set highlighted in red:
(1,2,3,7,8,9,64)
Median = 7
The highest frequency of occurence in our number set is 1 times
by the following numbers in green:
()
Since the maximum frequency of any number is 1, no mode exists.
Mode = N/A
Harmonic Mean =  N 
1/x_{1} + 1/x_{2} + 1/x_{3} + 1/x_{4} + 1/x_{5} + 1/x_{6} + 1/x_{7} 
With N = 7 and each x_{i} a member of the number set you entered, we have:
Harmonic Mean =  7 
1/1 + 1/2 + 1/3 + 1/7 + 1/8 + 1/9 + 1/64 
Harmonic Mean =  7 
1 + 0.5 + 0.33333333333333 + 0.14285714285714 + 0.125 + 0.11111111111111 + 0.015625 
Harmonic Mean =  7 
2.2279265873016 
Harmonic Mean = 3.1419347656685
Geometric Mean = (x_{1} * x_{2} * x_{3} * x_{4} * x_{5} * x_{6} * x_{7})^{1/N}
Geometric Mean = (1 * 2 * 3 * 7 * 8 * 9 * 64)^{1/7}
Geometric Mean = 193536^{0.14285714285714}
Geometric Mean = 5.691826851234
MidRange =  Maximum Value in Number Set + Minimum Value in Number Set 
2 
MidRange =  64 + 1 
2 
MidRange =  65 
2 
MidRange = 32.5
Take the first digit of each value in our number set
Use this as our stem value
Use the remaining digits for our leaf portion
{64,9,8,7,3,2,1}
Stem  Leaf 

6  4 
9  
8  
7  
3  
2  
1 
Mean, Variance, Standard Deviation, Median, Mode
μ =  Sum of your number Set 
Total Numbers Entered 
μ =  ΣX_{i} 
n 
μ =  1 + 2 + 3 + 7 + 8 + 9 + 64 
7 
μ =  94 
7 
μ = 13.428571428571
Let's evaluate the square difference from the mean of each term (X_{i}  μ)^{2}:
(X_{1}  μ)^{2} = (1  13.428571428571)^{2} = 12.428571428571^{2} = 154.4693877551
(X_{2}  μ)^{2} = (2  13.428571428571)^{2} = 11.428571428571^{2} = 130.61224489796
(X_{3}  μ)^{2} = (3  13.428571428571)^{2} = 10.428571428571^{2} = 108.75510204082
(X_{4}  μ)^{2} = (7  13.428571428571)^{2} = 6.4285714285714^{2} = 41.326530612245
(X_{5}  μ)^{2} = (8  13.428571428571)^{2} = 5.4285714285714^{2} = 29.469387755102
(X_{6}  μ)^{2} = (9  13.428571428571)^{2} = 4.4285714285714^{2} = 19.612244897959
(X_{7}  μ)^{2} = (64  13.428571428571)^{2} = 50.571428571429^{2} = 2557.4693877551
ΣE(X_{i}  μ)^{2} = 154.4693877551 + 130.61224489796 + 108.75510204082 + 41.326530612245 + 29.469387755102 + 19.612244897959 + 2557.4693877551
ΣE(X_{i}  μ)^{2} = 3041.7142857143
Population  Sample  


 

 
Variance: σ_{p}^{2} = 434.5306122449  Variance: σ_{s}^{2} = 506.95238095238  
Standard Deviation: σ_{p} = √σ_{p}^{2} = √434.5306122449  Standard Deviation: σ_{s} = √σ_{s}^{2} = √506.95238095238  
Standard Deviation: σ_{p} = 20.8454  Standard Deviation: σ_{s} = 22.5156 
Population  Sample  


 

 

 
SEM = 7.8788  SEM = 8.5101 
Skewness =  E(X_{i}  μ)^{3} 
(n  1)σ^{3} 
Let's evaluate the square difference from the mean of each term (X_{i}  μ)^{3}:
(X_{1}  μ)^{3} = (1  13.428571428571)^{3} = 12.428571428571^{3} = 1919.833819242
(X_{2}  μ)^{3} = (2  13.428571428571)^{3} = 11.428571428571^{3} = 1492.7113702624
(X_{3}  μ)^{3} = (3  13.428571428571)^{3} = 10.428571428571^{3} = 1134.1603498542
(X_{4}  μ)^{3} = (7  13.428571428571)^{3} = 6.4285714285714^{3} = 265.67055393586
(X_{5}  μ)^{3} = (8  13.428571428571)^{3} = 5.4285714285714^{3} = 159.97667638484
(X_{6}  μ)^{3} = (9  13.428571428571)^{3} = 4.4285714285714^{3} = 86.854227405248
(X_{7}  μ)^{3} = (64  13.428571428571)^{3} = 50.571428571429^{3} = 129334.88046647
ΣE(X_{i}  μ)^{3} = 1919.833819242 + 1492.7113702624 + 1134.1603498542 + 265.67055393586 + 159.97667638484 + 86.854227405248 + 129334.88046647
ΣE(X_{i}  μ)^{3} = 124275.67346939
Skewness =  E(X_{i}  μ)^{3} 
(n  1)σ^{3} 
Skewness =  124275.67346939 
(7  1)20.8454^{3} 
Skewness =  124275.67346939 
(6)9057.9662779607 
Skewness =  124275.67346939 
54347.797667764 
Skewness = 2.2866735875684
AD =  ΣX_{i}  μ 
n 
Evaluate the absolute value of the difference from the mean
X_{i}  μ:
X_{1}  μ = 1  13.428571428571 = 12.428571428571 = 12.428571428571
X_{2}  μ = 2  13.428571428571 = 11.428571428571 = 11.428571428571
X_{3}  μ = 3  13.428571428571 = 10.428571428571 = 10.428571428571
X_{4}  μ = 7  13.428571428571 = 6.4285714285714 = 6.4285714285714
X_{5}  μ = 8  13.428571428571 = 5.4285714285714 = 5.4285714285714
X_{6}  μ = 9  13.428571428571 = 4.4285714285714 = 4.4285714285714
X_{7}  μ = 64  13.428571428571 = 50.571428571429 = 50.571428571429
ΣX_{i}  μ = 12.428571428571 + 11.428571428571 + 10.428571428571 + 6.4285714285714 + 5.4285714285714 + 4.4285714285714 + 50.571428571429
ΣX_{i}  μ = 101.14285714286
Calculate average deviation (mean absolute deviation)
AD =  ΣX_{i}  μ 
n 
AD =  101.14285714286 
7 
Average Deviation = 14.44898
Since our number set contains 7 elements which is an odd number,
our median number is determined as follows:
Number Set = (n_{1},n_{2},n_{3},n_{4},n_{5},n_{6},n_{7})
Median Number = Entry ½(n + 1)
Median Number = Entry ½(8)
Median Number = n_{4}
Our median is entry 4 of our number set highlighted in red:
(1,2,3,7,8,9,64)
Median = 7
The highest frequency of occurence in our number set is 1 times
by the following numbers in green:
()
Since the maximum frequency of any number is 1, no mode exists.
Mode = N/A
Range = Largest Number in the Number Set  Smallest Number in the Number Set
Range = 64  1
Range = 63
PSC1 =  μ  Mode 
σ 
PSC1 =  3(13.428571428571  N/A) 
20.8454 
Since no mode exists, we do not have a Pearsons Skewness Coefficient 1
PSC2 =  μ  Median 
σ 
PSC1 =  3(13.428571428571  7) 
20.8454 
PSC2 =  3 x 6.4285714285714 
20.8454 
PSC2 =  19.285714285714 
20.8454 
PSC2 = 0.9252
Entropy = Ln(n)
Entropy = Ln(7)
Entropy = 1.9459101490553
MidRange =  Smallest Number in the Set + Largest Number in the Set 
2 
MidRange =  64 + 1 
2 
MidRange =  65 
2 
MidRange = 32.5
We need to sort our number set from lowest to highest shown below:
{1,2,3,7,8,9,64}
V =  y(n + 1) 
100 
V =  75(7 + 1) 
100 
V =  75(8) 
100 
V =  600 
100 
V = 6 ← Rounded down to the nearest integer
Upper quartile (UQ) point = Point # 6 in the dataset which is 9
1,2,3,7,8,9,64V =  y(n + 1) 
100 
V =  25(7 + 1) 
100 
V =  25(8) 
100 
V =  200 
100 
V = 2 ← Rounded up to the nearest integer
Lower quartile (LQ) point = Point # 2 in the dataset which is 2
1,2,3,7,8,9,64
IQR = UQ  LQ
IQR = 9  2
IQR = 7
Lower Inner Fence (LIF) = LQ  1.5 x IQR
Lower Inner Fence (LIF) = 2  1.5 x 7
Lower Inner Fence (LIF) = 2  10.5
Lower Inner Fence (LIF) = 8.5
Upper Inner Fence (UIF) = UQ + 1.5 x IQR
Upper Inner Fence (UIF) = 9 + 1.5 x 7
Upper Inner Fence (UIF) = 9 + 10.5
Upper Inner Fence (UIF) = 19.5
Lower Outer Fence (LOF) = LQ  3 x IQR
Lower Outer Fence (LOF) = 2  3 x 7
Lower Outer Fence (LOF) = 2  21
Lower Outer Fence (LOF) = 19
Upper Outer Fence (UOF) = UQ + 3 x IQR
Upper Outer Fence (UOF) = 9 + 3 x 7
Upper Outer Fence (UOF) = 9 + 21
Upper Outer Fence (UOF) = 30
Suspect Outliers are values between the inner and outer fences
We wish to mark all values in our dataset (v) in red below such that 19 < v < 8.5 and 19.5 < v < 30
1,2,3,7,8,9,64
Highly Suspect Outliers are values outside the outer fences
We wish to mark all values in our dataset (v) in red below such that v < 19 or v > 30
1,2,3,7,8,9,64
Array
Multiply each value by each probability amount
We do this by multiplying each X_{i} x p_{i} to get a weighted score Y
Weighted Average =  
n 
Weighted Average =  
0 
Weighted Average =  
0 
Weighted Average =  0 
0 
Weighted Average = NAN
Show the freqency distribution table for this number set
1, 2, 3, 7, 8, 9, 64
We need to choose the smallest integer k such that 2^{k} ≥ n where n = 0
Therefore, we use 0 intervals
Our maximum value in our number set of 64  1 = 63
Each interval size is the difference of the maximum and minimum value divided by the number of intervals
Interval Size =  63 
0 
Add 1 to this giving us INF + 1 = INF
Class Limits  Class Boundaries  FD  CFD  RFD  CRFD 

0  100% 
Go through our 7 numbers
Determine the ratio of each number to the next one
1:2 → 0.5
2:3 → 0.6667
3:7 → 0.4286
7:8 → 0.875
8:9 → 0.8889
9:64 → 0.1406
Successive Ratio = 1:2,2:3,3:7,7:8,8:9,9:64 or 0.5,0.6667,0.4286,0.875,0.8889,0.1406