You entered a number set X of {27.86,13.29,33.03,44.31,16.58,42.43,39.61,25.51,39.14,16.58,47.13,14.70,57.47,34.44}
From the 14 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
13.29, 14.70, 16.58, 16.58, 25.51, 27.86, 33.03, 34.44, 39.14, 39.61, 42.43, 44.31, 47.13, 57.47
57.47, 47.13, 44.31, 42.43, 39.61, 39.14, 34.44, 33.03, 27.86, 25.51, 16.58, 16.58, 14.70, 13.29
Sort our number set in ascending order
and assign a ranking to each number:
Number Set Value  13.29  14.70  16.58  16.58  25.51  27.86  33.03  34.44  39.14  39.61  42.43  44.31  47.13  57.47 
Rank  1  2  3  4  5  6  7  8  9  10  11  12  13  14 
Since we have 14 numbers in our original number set,
we assign ranks from lowest to highest (1 to 14)
Our original number set in unsorted order was 13.29,14.70,16.58,16.58,25.51,27.86,33.03,34.44,39.14,39.61,42.43,44.31,47.13,57.47
Our respective ranked data set is 1,2,4,4,5,6,7,8,9,10,11,12,13,14
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} + x_{8}^{2} + x_{9}^{2} + x_{10}^{2} + x_{11}^{2} + x_{12}^{2} + x_{13}^{2} + x_{14}^{2} and N = 14 number set items
A = 13.29^{2} + 14.70^{2} + 16.58^{2} + 16.58^{2} + 25.51^{2} + 27.86^{2} + 33.03^{2} + 34.44^{2} + 39.14^{2} + 39.61^{2} + 42.43^{2} + 44.31^{2} + 47.13^{2} + 57.47^{2}
A = 176.6241 + 216.09 + 274.8964 + 274.8964 + 650.7601 + 776.1796 + 1090.9809 + 1186.1136 + 1531.9396 + 1568.9521 + 1800.3049 + 1963.3761 + 2221.2369 + 3302.8009
A = 17035.1516
RMS =  √17035.1516 
√14 
RMS =  130.51877872552 
3.7416573867739 
RMS = 34.882610895074
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 =  13.29 + 14.70 + 16.58 + 16.58 + 25.51 + 27.86 + 33.03 + 34.44 + 39.14 + 39.61 + 42.43 + 44.31 + 47.13 + 57.47 
14 
μ =  452.08 
14 
μ = 32.291428571429
Since our number set contains 14 elements which is an even number,
our median number is determined as follows
Number Set = (n_{1},n_{2},n_{3},n_{4},n_{5},n_{6},n_{7},n_{8},n_{9},n_{10},n_{11},n_{12},n_{13},n_{14})
Median Number 1 = ½(n)
Median Number 1 = ½(14)
Median Number 1 = Number Set Entry 7
Median Number 2 = Median Number 1 + 1
Median Number 2 = Number Set Entry 7 + 1
Median Number 2 = Number Set Entry 8
Median = ½(n_{7} + n_{8})
Our median is the average of entry 7 and entry 8 of our number set highlighted in red:
(13.29,14.70,16.58,16.58,25.51,27.86,33.03,34.44,39.14,39.61,42.43,44.31,47.13,57.47)
Median = ½(33.03 + 34.44)
Median = ½(67.47)
Median = 33.735
The highest frequency of occurence in our number set is 2 times
by the following numbers in green:
()
Our mode is denoted as: 16.58
Mode = 16.58
Harmonic Mean =  N 
1/x_{1} + 1/x_{2} + 1/x_{3} + 1/x_{4} + 1/x_{5} + 1/x_{6} + 1/x_{7} + 1/x_{8} + 1/x_{9} + 1/x_{10} + 1/x_{11} + 1/x_{12} + 1/x_{13} + 1/x_{14} 
With N = 14 and each x_{i} a member of the number set you entered, we have:
Harmonic Mean =  14 
1/13.29 + 1/14.70 + 1/16.58 + 1/16.58 + 1/25.51 + 1/27.86 + 1/33.03 + 1/34.44 + 1/39.14 + 1/39.61 + 1/42.43 + 1/44.31 + 1/47.13 + 1/57.47 
Harmonic Mean =  14 
0.075244544770504 + 0.068027210884354 + 0.060313630880579 + 0.060313630880579 + 0.039200313602509 + 0.035893754486719 + 0.030275507114744 + 0.029036004645761 + 0.025549310168625 + 0.025246149962131 + 0.023568230025925 + 0.022568269013767 + 0.02121790791428 + 0.017400382808422 
Harmonic Mean =  14 
0.5338548471589 
Harmonic Mean = 26.224356816288
Geometric Mean = (x_{1} * x_{2} * x_{3} * x_{4} * x_{5} * x_{6} * x_{7} * x_{8} * x_{9} * x_{10} * x_{11} * x_{12} * x_{13} * x_{14})^{1/N}
Geometric Mean = (13.29 * 14.70 * 16.58 * 16.58 * 25.51 * 27.86 * 33.03 * 34.44 * 39.14 * 39.61 * 42.43 * 44.31 * 47.13 * 57.47)^{1/14}
Geometric Mean = 3.4277862923141E+20^{0.071428571428571}
Geometric Mean = 29.294540856618
MidRange =  Maximum Value in Number Set + Minimum Value in Number Set 
2 
MidRange =  57.47 + 13.29 
2 
MidRange =  70.76 
2 
MidRange = 35.38
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
{57.47,47.13,44.31,42.43,39.61,39.14,34.44,33.03,27.86,25.51,16.58,16.58,14.70,13.29}
Stem  Leaf 

5  7.47 
4  2.43,4.31,7.13 
3  3.03,4.44,9.14,9.61 
2  5.51,7.86 
1  3.29,4.70,6.58,6.58 
Mean, Variance, Standard Deviation, Median, Mode
μ =  Sum of your number Set 
Total Numbers Entered 
μ =  ΣX_{i} 
n 
μ =  13.29 + 14.70 + 16.58 + 16.58 + 25.51 + 27.86 + 33.03 + 34.44 + 39.14 + 39.61 + 42.43 + 44.31 + 47.13 + 57.47 
14 
μ =  452.08 
14 
μ = 32.291428571429
Let's evaluate the square difference from the mean of each term (X_{i}  μ)^{2}:
(X_{1}  μ)^{2} = (13.29  32.291428571429)^{2} = 19.001428571429^{2} = 361.0542877551
(X_{2}  μ)^{2} = (14.70  32.291428571429)^{2} = 17.591428571429^{2} = 309.45835918367
(X_{3}  μ)^{2} = (16.58  32.291428571429)^{2} = 15.711428571429^{2} = 246.8489877551
(X_{4}  μ)^{2} = (16.58  32.291428571429)^{2} = 15.711428571429^{2} = 246.8489877551
(X_{5}  μ)^{2} = (25.51  32.291428571429)^{2} = 6.7814285714286^{2} = 45.987773469388
(X_{6}  μ)^{2} = (27.86  32.291428571429)^{2} = 4.4314285714286^{2} = 19.637559183674
(X_{7}  μ)^{2} = (33.03  32.291428571429)^{2} = 0.73857142857143^{2} = 0.54548775510204
(X_{8}  μ)^{2} = (34.44  32.291428571429)^{2} = 2.1485714285714^{2} = 4.6163591836734
(X_{9}  μ)^{2} = (39.14  32.291428571429)^{2} = 6.8485714285714^{2} = 46.902930612245
(X_{10}  μ)^{2} = (39.61  32.291428571429)^{2} = 7.3185714285714^{2} = 53.561487755102
(X_{11}  μ)^{2} = (42.43  32.291428571429)^{2} = 10.138571428571^{2} = 102.79063061224
(X_{12}  μ)^{2} = (44.31  32.291428571429)^{2} = 12.018571428571^{2} = 144.44605918367
(X_{13}  μ)^{2} = (47.13  32.291428571429)^{2} = 14.838571428571^{2} = 220.18320204082
(X_{14}  μ)^{2} = (57.47  32.291428571429)^{2} = 25.178571428571^{2} = 633.96045918367
ΣE(X_{i}  μ)^{2} = 361.0542877551 + 309.45835918367 + 246.8489877551 + 246.8489877551 + 45.987773469388 + 19.637559183674 + 0.54548775510204 + 4.6163591836734 + 46.902930612245 + 53.561487755102 + 102.79063061224 + 144.44605918367 + 220.18320204082 + 633.96045918367
ΣE(X_{i}  μ)^{2} = 2436.8425714286
Population  Sample  


 

 
Variance: σ_{p}^{2} = 174.06018367347  Variance: σ_{s}^{2} = 187.44942857143  
Standard Deviation: σ_{p} = √σ_{p}^{2} = √174.06018367347  Standard Deviation: σ_{s} = √σ_{s}^{2} = √187.44942857143  
Standard Deviation: σ_{p} = 13.1932  Standard Deviation: σ_{s} = 13.6912 
Population  Sample  


 

 

 
SEM = 3.526  SEM = 3.6591 
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} = (13.29  32.291428571429)^{3} = 19.001428571429^{3} = 6860.5472591866
(X_{2}  μ)^{3} = (14.70  32.291428571429)^{3} = 17.591428571429^{3} = 5443.8146214111
(X_{3}  μ)^{3} = (16.58  32.291428571429)^{3} = 15.711428571429^{3} = 3878.3502390437
(X_{4}  μ)^{3} = (16.58  32.291428571429)^{3} = 15.711428571429^{3} = 3878.3502390437
(X_{5}  μ)^{3} = (25.51  32.291428571429)^{3} = 6.7814285714286^{3} = 311.86280094169
(X_{6}  μ)^{3} = (27.86  32.291428571429)^{3} = 4.4314285714286^{3} = 87.02244083965
(X_{7}  μ)^{3} = (33.03  32.291428571429)^{3} = 0.73857142857143^{3} = 0.40288167055393
(X_{8}  μ)^{3} = (34.44  32.291428571429)^{3} = 2.1485714285714^{3} = 9.9185774460641
(X_{9}  μ)^{3} = (39.14  32.291428571429)^{3} = 6.8485714285714^{3} = 321.21807050729
(X_{10}  μ)^{3} = (39.61  32.291428571429)^{3} = 7.3185714285714^{3} = 391.99357395627
(X_{11}  μ)^{3} = (42.43  32.291428571429)^{3} = 10.138571428571^{3} = 1042.1501506501
(X_{12}  μ)^{3} = (44.31  32.291428571429)^{3} = 12.018571428571^{3} = 1736.0352798746
(X_{13}  μ)^{3} = (47.13  32.291428571429)^{3} = 14.838571428571^{3} = 3267.2041708542
(X_{14}  μ)^{3} = (57.47  32.291428571429)^{3} = 25.178571428571^{3} = 15962.218704446
ΣE(X_{i}  μ)^{3} = 6860.5472591866 + 5443.8146214111 + 3878.3502390437 + 3878.3502390437 + 311.86280094169 + 87.02244083965 + 0.40288167055393 + 9.9185774460641 + 321.21807050729 + 391.99357395627 + 1042.1501506501 + 1736.0352798746 + 3267.2041708542 + 15962.218704446
ΣE(X_{i}  μ)^{3} = 2271.1938089388
Skewness =  E(X_{i}  μ)^{3} 
(n  1)σ^{3} 
Skewness =  2271.1938089388 
(14  1)13.1932^{3} 
Skewness =  2271.1938089388 
(13)2296.4153347896 
Skewness =  2271.1938089388 
29853.399352264 
Skewness = 0.076078230895554
AD =  ΣX_{i}  μ 
n 
Evaluate the absolute value of the difference from the mean
X_{i}  μ:
X_{1}  μ = 13.29  32.291428571429 = 19.001428571429 = 19.001428571429
X_{2}  μ = 14.70  32.291428571429 = 17.591428571429 = 17.591428571429
X_{3}  μ = 16.58  32.291428571429 = 15.711428571429 = 15.711428571429
X_{4}  μ = 16.58  32.291428571429 = 15.711428571429 = 15.711428571429
X_{5}  μ = 25.51  32.291428571429 = 6.7814285714286 = 6.7814285714286
X_{6}  μ = 27.86  32.291428571429 = 4.4314285714286 = 4.4314285714286
X_{7}  μ = 33.03  32.291428571429 = 0.73857142857143 = 0.73857142857143
X_{8}  μ = 34.44  32.291428571429 = 2.1485714285714 = 2.1485714285714
X_{9}  μ = 39.14  32.291428571429 = 6.8485714285714 = 6.8485714285714
X_{10}  μ = 39.61  32.291428571429 = 7.3185714285714 = 7.3185714285714
X_{11}  μ = 42.43  32.291428571429 = 10.138571428571 = 10.138571428571
X_{12}  μ = 44.31  32.291428571429 = 12.018571428571 = 12.018571428571
X_{13}  μ = 47.13  32.291428571429 = 14.838571428571 = 14.838571428571
X_{14}  μ = 57.47  32.291428571429 = 25.178571428571 = 25.178571428571
ΣX_{i}  μ = 19.001428571429 + 17.591428571429 + 15.711428571429 + 15.711428571429 + 6.7814285714286 + 4.4314285714286 + 0.73857142857143 + 2.1485714285714 + 6.8485714285714 + 7.3185714285714 + 10.138571428571 + 12.018571428571 + 14.838571428571 + 25.178571428571
ΣX_{i}  μ = 158.45714285714
Calculate average deviation (mean absolute deviation)
AD =  ΣX_{i}  μ 
n 
AD =  158.45714285714 
14 
Average Deviation = 11.31837
Since our number set contains 14 elements which is an even number,
our median number is determined as follows
Number Set = (n_{1},n_{2},n_{3},n_{4},n_{5},n_{6},n_{7},n_{8},n_{9},n_{10},n_{11},n_{12},n_{13},n_{14})
Median Number 1 = ½(n)
Median Number 1 = ½(14)
Median Number 1 = Number Set Entry 7
Median Number 2 = Median Number 1 + 1
Median Number 2 = Number Set Entry 7 + 1
Median Number 2 = Number Set Entry 8
Median = ½(n_{7} + n_{8})
Our median is the average of entry 7 and entry 8 of our number set highlighted in red:
(13.29,14.70,16.58,16.58,25.51,27.86,33.03,34.44,39.14,39.61,42.43,44.31,47.13,57.47)
Median = ½(33.03 + 34.44)
Median = ½(67.47)
Median = 33.735
The highest frequency of occurence in our number set is 2 times
by the following numbers in green:
()
Our mode is denoted as: 16.58
Mode = 16.58
Range = Largest Number in the Number Set  Smallest Number in the Number Set
Range = 57.47  13.29
Range = 44.18
PSC1 =  μ  Mode 
σ 
PSC1 =  3(32.291428571429  16.58) 
13.1932 
PSC1 =  3 x 15.711428571429 
13.1932 
PSC1 =  47.134285714286 
13.1932 
PSC1 = 3.5726
PSC2 =  μ  Median 
σ 
PSC1 =  3(32.291428571429  33.735) 
13.1932 
PSC2 =  3 x 1.4435714285714 
13.1932 
PSC2 =  4.3307142857143 
13.1932 
PSC2 = 0.3283
Entropy = Ln(n)
Entropy = Ln(14)
Entropy = 2.6390573296153
MidRange =  Smallest Number in the Set + Largest Number in the Set 
2 
MidRange =  57.47 + 13.29 
2 
MidRange =  70.76 
2 
MidRange = 35.38
We need to sort our number set from lowest to highest shown below:
{13.29,14.70,16.58,16.58,25.51,27.86,33.03,34.44,39.14,39.61,42.43,44.31,47.13,57.47}
V =  y(n + 1) 
100 
V =  75(14 + 1) 
100 
V =  75(15) 
100 
V =  1125 
100 
V = 11 ← Rounded down to the nearest integer
Upper quartile (UQ) point = Point # 11 in the dataset which is 42.43
13.29,14.70,16.58,16.58,25.51,27.86,33.03,34.44,39.14,39.61,42.43,44.31,47.13,57.47V =  y(n + 1) 
100 
V =  25(14 + 1) 
100 
V =  25(15) 
100 
V =  375 
100 
V = 4 ← Rounded up to the nearest integer
Lower quartile (LQ) point = Point # 4 in the dataset which is 16.58
13.29,14.70,16.58,16.58,25.51,27.86,33.03,34.44,39.14,39.61,42.43,44.31,47.13,57.47
IQR = UQ  LQ
IQR = 42.43  16.58
IQR = 25.85
Lower Inner Fence (LIF) = LQ  1.5 x IQR
Lower Inner Fence (LIF) = 16.58  1.5 x 25.85
Lower Inner Fence (LIF) = 16.58  38.775
Lower Inner Fence (LIF) = 22.195
Upper Inner Fence (UIF) = UQ + 1.5 x IQR
Upper Inner Fence (UIF) = 42.43 + 1.5 x 25.85
Upper Inner Fence (UIF) = 42.43 + 38.775
Upper Inner Fence (UIF) = 81.205
Lower Outer Fence (LOF) = LQ  3 x IQR
Lower Outer Fence (LOF) = 16.58  3 x 25.85
Lower Outer Fence (LOF) = 16.58  77.55
Lower Outer Fence (LOF) = 60.97
Upper Outer Fence (UOF) = UQ + 3 x IQR
Upper Outer Fence (UOF) = 42.43 + 3 x 25.85
Upper Outer Fence (UOF) = 42.43 + 77.55
Upper Outer Fence (UOF) = 119.98
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 60.97 < v < 22.195 and 81.205 < v < 119.98
13.29,14.70,16.58,16.58,25.51,27.86,33.03,34.44,39.14,39.61,42.43,44.31,47.13,57.47
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 < 60.97 or v > 119.98
13.29,14.70,16.58,16.58,25.51,27.86,33.03,34.44,39.14,39.61,42.43,44.31,47.13,57.47
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
13.29, 14.70, 16.58, 16.58, 25.51, 27.86, 33.03, 34.44, 39.14, 39.61, 42.43, 44.31, 47.13, 57.47
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 57.47  13.29 = 44.18
Each interval size is the difference of the maximum and minimum value divided by the number of intervals
Interval Size =  44.18 
0 
Add 1 to this giving us INF + 1 = INF
Class Limits  Class Boundaries  FD  CFD  RFD  CRFD 

0  100% 
Go through our 14 numbers
Determine the ratio of each number to the next one
13.29:14.70 → 0.9041
14.70:16.58 → 0.8866
16.58:16.58 → 1
16.58:25.51 → 0.6499
25.51:27.86 → 0.9156
27.86:33.03 → 0.8435
33.03:34.44 → 0.9591
34.44:39.14 → 0.8799
39.14:39.61 → 0.9881
39.61:42.43 → 0.9335
42.43:44.31 → 0.9576
44.31:47.13 → 0.9402
47.13:57.47 → 0.8201
Successive Ratio = 13.29:14.70,14.70:16.58,16.58:16.58,16.58:25.51,25.51:27.86,27.86:33.03,33.03:34.44,34.44:39.14,39.14:39.61,39.61:42.43,42.43:44.31,44.31:47.13,47.13:57.47 or 0.9041,0.8866,1,0.6499,0.9156,0.8435,0.9591,0.8799,0.9881,0.9335,0.9576,0.9402,0.8201