You entered a number set X of {11.92,34.86,26.72,24.50,38.93,8.59,29.31,23.39,24.13,30.05,21.54,35.97,7.48,35.97}
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, mid-range
7.48, 8.59, 11.92, 21.54, 23.39, 24.13, 24.50, 26.72, 29.31, 30.05, 34.86, 35.97, 35.97, 38.93
38.93, 35.97, 35.97, 34.86, 30.05, 29.31, 26.72, 24.50, 24.13, 23.39, 21.54, 11.92, 8.59, 7.48
Sort our number set in ascending order
and assign a ranking to each number:
Number Set Value | 7.48 | 8.59 | 11.92 | 21.54 | 23.39 | 24.13 | 24.50 | 26.72 | 29.31 | 30.05 | 34.86 | 35.97 | 35.97 | 38.93 |
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 7.48,8.59,11.92,21.54,23.39,24.13,24.50,26.72,29.31,30.05,34.86,35.97,35.97,38.93
Our respective ranked data set is 1,2,3,4,5,6,7,8,9,10,11,13,13,14
Root Mean Square = | √A |
√N |
where A = x12 + x22 + x32 + x42 + x52 + x62 + x72 + x82 + x92 + x102 + x112 + x122 + x132 + x142 and N = 14 number set items
A = 7.482 + 8.592 + 11.922 + 21.542 + 23.392 + 24.132 + 24.502 + 26.722 + 29.312 + 30.052 + 34.862 + 35.972 + 35.972 + 38.932
A = 55.9504 + 73.7881 + 142.0864 + 463.9716 + 547.0921 + 582.2569 + 600.25 + 713.9584 + 859.0761 + 903.0025 + 1215.2196 + 1293.8409 + 1293.8409 + 1515.5449
A = 10259.8788
RMS = | √10259.8788 |
√14 |
RMS = | 101.29105982267 |
3.7416573867739 |
RMS = 27.071174442833
Central tendency contains:
Mean, median, mode, harmonic mean,
geometric mean, mid-range, weighted-average:
μ = | Sum of your number Set |
Total Numbers Entered |
μ = | ΣXi |
n |
μ = | 7.48 + 8.59 + 11.92 + 21.54 + 23.39 + 24.13 + 24.50 + 26.72 + 29.31 + 30.05 + 34.86 + 35.97 + 35.97 + 38.93 |
14 |
μ = | 353.36 |
14 |
μ = 25.24
Since our number set contains 14 elements which is an even number,
our median number is determined as follows
Number Set = (n1,n2,n3,n4,n5,n6,n7,n8,n9,n10,n11,n12,n13,n14)
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(n7 + n8)
Our median is the average of entry 7 and entry 8 of our number set highlighted in red:
(7.48,8.59,11.92,21.54,23.39,24.13,24.50,26.72,29.31,30.05,34.86,35.97,35.97,38.93)
Median = ½(24.50 + 26.72)
Median = ½(51.22)
Median = 25.61
The highest frequency of occurence in our number set is 2 times
by the following numbers in green:
()
Our mode is denoted as: 35.97
Mode = 35.97
Harmonic Mean = | N |
1/x1 + 1/x2 + 1/x3 + 1/x4 + 1/x5 + 1/x6 + 1/x7 + 1/x8 + 1/x9 + 1/x10 + 1/x11 + 1/x12 + 1/x13 + 1/x14 |
With N = 14 and each xi a member of the number set you entered, we have:
Harmonic Mean = | 14 |
1/7.48 + 1/8.59 + 1/11.92 + 1/21.54 + 1/23.39 + 1/24.13 + 1/24.50 + 1/26.72 + 1/29.31 + 1/30.05 + 1/34.86 + 1/35.97 + 1/35.97 + 1/38.93 |
Harmonic Mean = | 14 |
0.13368983957219 + 0.11641443538999 + 0.083892617449664 + 0.046425255338904 + 0.042753313381787 + 0.041442188147534 + 0.040816326530612 + 0.037425149700599 + 0.034118048447629 + 0.033277870216306 + 0.028686173264487 + 0.027800945232138 + 0.027800945232138 + 0.025687130747496 |
Harmonic Mean = | 14 |
0.72023023865147 |
Harmonic Mean = 19.438228567316
Geometric Mean = (x1 * x2 * x3 * x4 * x5 * x6 * x7 * x8 * x9 * x10 * x11 * x12 * x13 * x14)1/N
Geometric Mean = (7.48 * 8.59 * 11.92 * 21.54 * 23.39 * 24.13 * 24.50 * 26.72 * 29.31 * 30.05 * 34.86 * 35.97 * 35.97 * 38.93)1/14
Geometric Mean = 9.4267014924235E+180.071428571428571
Geometric Mean = 22.662687541866
Mid-Range = | Maximum Value in Number Set + Minimum Value in Number Set |
2 |
Mid-Range = | 38.93 + 7.48 |
2 |
Mid-Range = | 46.41 |
2 |
Mid-Range = 23.205
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
{38.93,35.97,35.97,34.86,30.05,29.31,26.72,24.50,24.13,23.39,21.54,11.92,8.59,7.48}
Stem | Leaf |
---|---|
3 | 0.05,4.86,5.97,5.97,8.93 |
2 | 1.54,3.39,4.13,4.50,6.72,9.31 |
1 | 1.92 |
8 | .59 |
7 | .48 |
Mean, Variance, Standard Deviation, Median, Mode
μ = | Sum of your number Set |
Total Numbers Entered |
μ = | ΣXi |
n |
μ = | 7.48 + 8.59 + 11.92 + 21.54 + 23.39 + 24.13 + 24.50 + 26.72 + 29.31 + 30.05 + 34.86 + 35.97 + 35.97 + 38.93 |
14 |
μ = | 353.36 |
14 |
μ = 25.24
Let's evaluate the square difference from the mean of each term (Xi - μ)2:
(X1 - μ)2 = (7.48 - 25.24)2 = -17.762 = 315.4176
(X2 - μ)2 = (8.59 - 25.24)2 = -16.652 = 277.2225
(X3 - μ)2 = (11.92 - 25.24)2 = -13.322 = 177.4224
(X4 - μ)2 = (21.54 - 25.24)2 = -3.72 = 13.69
(X5 - μ)2 = (23.39 - 25.24)2 = -1.852 = 3.4225
(X6 - μ)2 = (24.13 - 25.24)2 = -1.112 = 1.2321
(X7 - μ)2 = (24.50 - 25.24)2 = -0.740000000000012 = 0.54760000000001
(X8 - μ)2 = (26.72 - 25.24)2 = 1.482 = 2.1904
(X9 - μ)2 = (29.31 - 25.24)2 = 4.072 = 16.5649
(X10 - μ)2 = (30.05 - 25.24)2 = 4.812 = 23.1361
(X11 - μ)2 = (34.86 - 25.24)2 = 9.622 = 92.5444
(X12 - μ)2 = (35.97 - 25.24)2 = 10.732 = 115.1329
(X13 - μ)2 = (35.97 - 25.24)2 = 10.732 = 115.1329
(X14 - μ)2 = (38.93 - 25.24)2 = 13.692 = 187.4161
ΣE(Xi - μ)2 = 315.4176 + 277.2225 + 177.4224 + 13.69 + 3.4225 + 1.2321 + 0.54760000000001 + 2.1904 + 16.5649 + 23.1361 + 92.5444 + 115.1329 + 115.1329 + 187.4161
ΣE(Xi - μ)2 = 1341.0724
Population | Sample | ||||||||
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Variance: σp2 = 95.790885714286 | Variance: σs2 = 103.15941538462 | ||||||||
Standard Deviation: σp = √σp2 = √95.790885714286 | Standard Deviation: σs = √σs2 = √103.15941538462 | ||||||||
Standard Deviation: σp = 9.7873 | Standard Deviation: σs = 10.1567 |
Population | Sample | ||||||||
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SEM = 2.6158 | SEM = 2.7145 |
Skewness = | E(Xi - μ)3 |
(n - 1)σ3 |
Let's evaluate the square difference from the mean of each term (Xi - μ)3:
(X1 - μ)3 = (7.48 - 25.24)3 = -17.763 = -5601.816576
(X2 - μ)3 = (8.59 - 25.24)3 = -16.653 = -4615.754625
(X3 - μ)3 = (11.92 - 25.24)3 = -13.323 = -2363.266368
(X4 - μ)3 = (21.54 - 25.24)3 = -3.73 = -50.653
(X5 - μ)3 = (23.39 - 25.24)3 = -1.853 = -6.3316250000001
(X6 - μ)3 = (24.13 - 25.24)3 = -1.113 = -1.367631
(X7 - μ)3 = (24.50 - 25.24)3 = -0.740000000000013 = -0.40522400000001
(X8 - μ)3 = (26.72 - 25.24)3 = 1.483 = 3.241792
(X9 - μ)3 = (29.31 - 25.24)3 = 4.073 = 67.419143
(X10 - μ)3 = (30.05 - 25.24)3 = 4.813 = 111.284641
(X11 - μ)3 = (34.86 - 25.24)3 = 9.623 = 890.277128
(X12 - μ)3 = (35.97 - 25.24)3 = 10.733 = 1235.376017
(X13 - μ)3 = (35.97 - 25.24)3 = 10.733 = 1235.376017
(X14 - μ)3 = (38.93 - 25.24)3 = 13.693 = 2565.726409
ΣE(Xi - μ)3 = -5601.816576 + -4615.754625 + -2363.266368 + -50.653 + -6.3316250000001 + -1.367631 + -0.40522400000001 + 3.241792 + 67.419143 + 111.284641 + 890.277128 + 1235.376017 + 1235.376017 + 2565.726409
ΣE(Xi - μ)3 = -6530.893902
Skewness = | E(Xi - μ)3 |
(n - 1)σ3 |
Skewness = | -6530.893902 |
(14 - 1)9.78733 |
Skewness = | -6530.893902 |
(13)937.53761587762 |
Skewness = | -6530.893902 |
12187.989006409 |
Skewness = -0.53584671749915
AD = | Σ|Xi - μ| |
n |
Evaluate the absolute value of the difference from the mean
|Xi - μ|:
|X1 - μ| = |7.48 - 25.24| = |-17.76| = 17.76
|X2 - μ| = |8.59 - 25.24| = |-16.65| = 16.65
|X3 - μ| = |11.92 - 25.24| = |-13.32| = 13.32
|X4 - μ| = |21.54 - 25.24| = |-3.7| = 3.7
|X5 - μ| = |23.39 - 25.24| = |-1.85| = 1.85
|X6 - μ| = |24.13 - 25.24| = |-1.11| = 1.11
|X7 - μ| = |24.50 - 25.24| = |-0.74000000000001| = 0.74000000000001
|X8 - μ| = |26.72 - 25.24| = |1.48| = 1.48
|X9 - μ| = |29.31 - 25.24| = |4.07| = 4.07
|X10 - μ| = |30.05 - 25.24| = |4.81| = 4.81
|X11 - μ| = |34.86 - 25.24| = |9.62| = 9.62
|X12 - μ| = |35.97 - 25.24| = |10.73| = 10.73
|X13 - μ| = |35.97 - 25.24| = |10.73| = 10.73
|X14 - μ| = |38.93 - 25.24| = |13.69| = 13.69
Σ|Xi - μ| = 17.76 + 16.65 + 13.32 + 3.7 + 1.85 + 1.11 + 0.74000000000001 + 1.48 + 4.07 + 4.81 + 9.62 + 10.73 + 10.73 + 13.69
Σ|Xi - μ| = 110.26
Calculate average deviation (mean absolute deviation)
AD = | Σ|Xi - μ| |
n |
AD = | 110.26 |
14 |
Average Deviation = 7.87571
Since our number set contains 14 elements which is an even number,
our median number is determined as follows
Number Set = (n1,n2,n3,n4,n5,n6,n7,n8,n9,n10,n11,n12,n13,n14)
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(n7 + n8)
Our median is the average of entry 7 and entry 8 of our number set highlighted in red:
(7.48,8.59,11.92,21.54,23.39,24.13,24.50,26.72,29.31,30.05,34.86,35.97,35.97,38.93)
Median = ½(24.50 + 26.72)
Median = ½(51.22)
Median = 25.61
The highest frequency of occurence in our number set is 2 times
by the following numbers in green:
()
Our mode is denoted as: 35.97
Mode = 35.97
Range = Largest Number in the Number Set - Smallest Number in the Number Set
Range = 38.93 - 7.48
Range = 31.45
PSC1 = | μ - Mode |
σ |
PSC1 = | 3(25.24 - 35.97) |
9.7873 |
PSC1 = | 3 x -10.73 |
9.7873 |
PSC1 = | -32.19 |
9.7873 |
PSC1 = -3.289
PSC2 = | μ - Median |
σ |
PSC1 = | 3(25.24 - 25.61) |
9.7873 |
PSC2 = | 3 x -0.36999999999999 |
9.7873 |
PSC2 = | -1.11 |
9.7873 |
PSC2 = -0.1134
Entropy = Ln(n)
Entropy = Ln(14)
Entropy = 2.6390573296153
Mid-Range = | Smallest Number in the Set + Largest Number in the Set |
2 |
Mid-Range = | 38.93 + 7.48 |
2 |
Mid-Range = | 46.41 |
2 |
Mid-Range = 23.205
We need to sort our number set from lowest to highest shown below:
{7.48,8.59,11.92,21.54,23.39,24.13,24.50,26.72,29.31,30.05,34.86,35.97,35.97,38.93}
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 34.86
7.48,8.59,11.92,21.54,23.39,24.13,24.50,26.72,29.31,30.05,34.86,35.97,35.97,38.93V = | 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 21.54
7.48,8.59,11.92,21.54,23.39,24.13,24.50,26.72,29.31,30.05,34.86,35.97,35.97,38.93
IQR = UQ - LQ
IQR = 34.86 - 21.54
IQR = 13.32
Lower Inner Fence (LIF) = LQ - 1.5 x IQR
Lower Inner Fence (LIF) = 21.54 - 1.5 x 13.32
Lower Inner Fence (LIF) = 21.54 - 19.98
Lower Inner Fence (LIF) = 1.56
Upper Inner Fence (UIF) = UQ + 1.5 x IQR
Upper Inner Fence (UIF) = 34.86 + 1.5 x 13.32
Upper Inner Fence (UIF) = 34.86 + 19.98
Upper Inner Fence (UIF) = 54.84
Lower Outer Fence (LOF) = LQ - 3 x IQR
Lower Outer Fence (LOF) = 21.54 - 3 x 13.32
Lower Outer Fence (LOF) = 21.54 - 39.96
Lower Outer Fence (LOF) = -18.42
Upper Outer Fence (UOF) = UQ + 3 x IQR
Upper Outer Fence (UOF) = 34.86 + 3 x 13.32
Upper Outer Fence (UOF) = 34.86 + 39.96
Upper Outer Fence (UOF) = 74.82
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 -18.42 < v < 1.56 and 54.84 < v < 74.82
7.48,8.59,11.92,21.54,23.39,24.13,24.50,26.72,29.31,30.05,34.86,35.97,35.97,38.93
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 < -18.42 or v > 74.82
7.48,8.59,11.92,21.54,23.39,24.13,24.50,26.72,29.31,30.05,34.86,35.97,35.97,38.93
Array
Multiply each value by each probability amount
We do this by multiplying each Xi x pi 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
7.48, 8.59, 11.92, 21.54, 23.39, 24.13, 24.50, 26.72, 29.31, 30.05, 34.86, 35.97, 35.97, 38.93
We need to choose the smallest integer k such that 2k ≥ n where n = 0
Therefore, we use 0 intervals
Our maximum value in our number set of 38.93 - 7.48 = 31.45
Each interval size is the difference of the maximum and minimum value divided by the number of intervals
Interval Size = | 31.45 |
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
7.48:8.59 → 0.8708
8.59:11.92 → 0.7206
11.92:21.54 → 0.5534
21.54:23.39 → 0.9209
23.39:24.13 → 0.9693
24.13:24.50 → 0.9849
24.50:26.72 → 0.9169
26.72:29.31 → 0.9116
29.31:30.05 → 0.9754
30.05:34.86 → 0.862
34.86:35.97 → 0.9691
35.97:35.97 → 1
35.97:38.93 → 0.924
Successive Ratio = 7.48:8.59,8.59:11.92,11.92:21.54,21.54:23.39,23.39:24.13,24.13:24.50,24.50:26.72,26.72:29.31,29.31:30.05,30.05:34.86,34.86:35.97,35.97:35.97,35.97:38.93 or 0.8708,0.7206,0.5534,0.9209,0.9693,0.9849,0.9169,0.9116,0.9754,0.862,0.9691,1,0.924