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You entered a number set X of {2,15,15,18,30}
From the 5 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
2, 15, 15, 18, 30
Rank Ascending
2 is the 1st lowest/smallest number
15 is the 2nd lowest/smallest number
15 is the 3rd lowest/smallest number
18 is the 4th lowest/smallest number
30 is the 5th lowest/smallest number
30, 18, 15, 15, 2
Rank Descending
30 is the 1st highest/largest number
18 is the 2nd highest/largest number
15 is the 3rd highest/largest number
15 is the 4th highest/largest number
2 is the 5th highest/largest number
Sort our number set in ascending order
and assign a ranking to each number:
Number Set Value | 2 | 15 | 15 | 18 | 30 |
Rank | 1 | 2 | 3 | 4 | 5 |
Since we have 5 numbers in our original number set,
we assign ranks from lowest to highest (1 to 5)
Our original number set in unsorted order was 2,15,15,18,30
Our respective ranked data set is 1,3,3,4,5
Root Mean Square = | √A |
√N |
where A = x12 + x22 + x32 + x42 + x52 and N = 5 number set items
A = 22 + 152 + 152 + 182 + 302
A = 4 + 225 + 225 + 324 + 900
A = 1678
RMS = | √1678 |
√5 |
RMS = | 40.963398296528 |
2.2360679774998 |
RMS = 18.319388636087
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 = | 2 + 15 + 15 + 18 + 30 |
5 |
μ = | 80 |
5 |
μ = 16
Since our number set contains 5 elements which is an odd number,
our median number is determined as follows:
Number Set = (n1,n2,n3,n4,n5)
Median Number = Entry ½(n + 1)
Median Number = Entry ½(6)
Median Number = n3
Our median is entry 3 of our number set highlighted in red:
(2,15,15,18,30)
Median = 15
()
Our mode is denoted as: 15
Mode = 15
Harmonic Mean = | N |
1/x1 + 1/x2 + 1/x3 + 1/x4 + 1/x5 |
With N = 5 and each xi a member of the number set you entered, we have:
Harmonic Mean = | 5 |
1/2 + 1/15 + 1/15 + 1/18 + 1/30 |
Harmonic Mean = | 5 |
0.5 + 0.066666666666667 + 0.066666666666667 + 0.055555555555556 + 0.033333333333333 |
Harmonic Mean = | 5 |
0.72222222222222 |
Harmonic Mean = 6.9230769230769
Geometric Mean = (x1 * x2 * x3 * x4 * x5)1/N
Geometric Mean = (2 * 15 * 15 * 18 * 30)1/5
Geometric Mean = 2430000.2
Geometric Mean = 11.943215116605
Mid-Range = | Maximum Value in Number Set + Minimum Value in Number Set |
2 |
Mid-Range = | 30 + 2 |
2 |
Mid-Range = | 32 |
2 |
Mid-Range = 16
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
Stem | Leaf |
---|---|
3 | 0 |
1 | 5,5,8 |
2 |
Let's evaluate the square difference from the mean of each term (Xi - μ)2:
(X1 - μ)2 = (2 - 16)2 = -142 = 196
(X2 - μ)2 = (15 - 16)2 = -12 = 1
(X3 - μ)2 = (15 - 16)2 = -12 = 1
(X4 - μ)2 = (18 - 16)2 = 22 = 4
(X5 - μ)2 = (30 - 16)2 = 142 = 196
ΣE(Xi - μ)2 = 196 + 1 + 1 + 4 + 196
ΣE(Xi - μ)2 = 398
Population | Sample | ||||||||
---|---|---|---|---|---|---|---|---|---|
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Variance: σp2 = 79.6 | Variance: σs2 = 99.5 | ||||||||
Standard Deviation: σp = √σp2 = √79.6 | Standard Deviation: σs = √σs2 = √99.5 | ||||||||
Standard Deviation: σp = 8.9219 | Standard Deviation: σs = 9.975 |
Population | Sample | ||||||||
---|---|---|---|---|---|---|---|---|---|
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|
|
|
|
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SEM = 3.99 | SEM = 4.461 |
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 = (2 - 16)3 = -143 = -2744
(X2 - μ)3 = (15 - 16)3 = -13 = -1
(X3 - μ)3 = (15 - 16)3 = -13 = -1
(X4 - μ)3 = (18 - 16)3 = 23 = 8
(X5 - μ)3 = (30 - 16)3 = 143 = 2744
ΣE(Xi - μ)3 = -2744 + -1 + -1 + 8 + 2744
ΣE(Xi - μ)3 = 6
Skewness = | E(Xi - μ)3 |
(n - 1)σ3 |
Skewness = | 6 |
(5 - 1)8.92193 |
Skewness = | 6 |
(4)710.18591309046 |
Skewness = | 6 |
2840.7436523618 |
Skewness = 0.0021121229981493
AD = | Σ|Xi - μ| |
n |
Evaluate the absolute value of the difference from the mean
|Xi - μ|:
|X1 - μ| = |2 - 16| = |-14| = 14
|X2 - μ| = |15 - 16| = |-1| = 1
|X3 - μ| = |15 - 16| = |-1| = 1
|X4 - μ| = |18 - 16| = |2| = 2
|X5 - μ| = |30 - 16| = |14| = 14
Σ|Xi - μ| = 14 + 1 + 1 + 2 + 14
Σ|Xi - μ| = 32
Calculate average deviation (mean absolute deviation)
AD = | Σ|Xi - μ| |
n |
AD = | 32 |
5 |
Average Deviation = 6.4
Range = Largest Number in the Number Set - Smallest Number in the Number Set
Range = 30 - 2
Range = 28
PSC1 = | μ - Mode |
σ |
PSC1 = | 3(16 - 15) |
8.9219 |
PSC1 = | 3 x 1 |
8.9219 |
PSC1 = | 3 |
8.9219 |
PSC1 = 0.3363
PSC2 = | μ - Median |
σ |
PSC1 = | 3(16 - 15) |
8.9219 |
PSC2 = | 3 x 1 |
8.9219 |
PSC2 = | 3 |
8.9219 |
PSC2 = 0.3363
Entropy = Ln(n)
Entropy = Ln(5)
Entropy = 1.6094379124341
Mid-Range = | Smallest Number in the Set + Largest Number in the Set |
2 |
Mid-Range = | 30 + 2 |
2 |
Mid-Range = | 32 |
2 |
Mid-Range = 16
We need to sort our number set from lowest to highest shown below:
{2,15,15,18,30}
V = | y(n + 1) |
100 |
V = | 75(5 + 1) |
100 |
V = | 75(6) |
100 |
V = | 450 |
100 |
V = 4 ← Rounded down to the nearest integer
Upper quartile (UQ) point = Point # 4 in the dataset which is 18
V = | y(n + 1) |
100 |
V = | 25(5 + 1) |
100 |
V = | 25(6) |
100 |
V = | 150 |
100 |
V = 2 ← Rounded up to the nearest integer
Lower quartile (LQ) point = Point # 2 in the dataset which is 15
2,15,15,18,30
IQR = UQ - LQ
IQR = 18 - 15
IQR = 3
Lower Inner Fence (LIF) = LQ - 1.5 x IQR
Lower Inner Fence (LIF) = 15 - 1.5 x 3
Lower Inner Fence (LIF) = 15 - 4.5
Lower Inner Fence (LIF) = 10.5
Upper Inner Fence (UIF) = UQ + 1.5 x IQR
Upper Inner Fence (UIF) = 18 + 1.5 x 3
Upper Inner Fence (UIF) = 18 + 4.5
Upper Inner Fence (UIF) = 22.5
Lower Outer Fence (LOF) = LQ - 3 x IQR
Lower Outer Fence (LOF) = 15 - 3 x 3
Lower Outer Fence (LOF) = 15 - 9
Lower Outer Fence (LOF) = 6
Upper Outer Fence (UOF) = UQ + 3 x IQR
Upper Outer Fence (UOF) = 18 + 3 x 3
Upper Outer Fence (UOF) = 18 + 9
Upper Outer Fence (UOF) = 27
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 6 < v < 10.5 and 22.5 < v < 27
2,15,15,18,30
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 < 6 or v > 27
2,15,15,18,30
2, 15, 15, 18, 30
Multiply each value by each probability amount
We do this by multiplying each Xi x pi to get a weighted score Y
Weighted Average = | X1p1 + X2p2 + X3p3 + X4p4 + X5p5 |
n |
Weighted Average = | 2 x 0.2 + 15 x 0.4 + 15 x 0.6 + 18 x 0.8 + 30 x 0.9 |
5 |
Weighted Average = | 0.4 + 6 + 9 + 14.4 + 27 |
5 |
Weighted Average = | 56.8 |
5 |
Weighted Average = 11.36
Show the freqency distribution table for this number set
2, 15, 15, 18, 30
Choose the smallest integer k such that 2k ≥ n where n = 5
For k = 1, we have 21 = 2
For k = 2, we have 22 = 4
For k = 3, we have 23 = 8 ← Use this since it is greater than our n value of 5
Therefore, we use 3 intervals
Our maximum value in our number set of 30 - 2 = 28
Each interval size is the difference of the maximum and minimum value divided by the number of intervals
Interval Size = | 28 |
3 |
Add 1 to this giving us 9 + 1 = 10
Class Limits | Class Boundaries | FD | CFD | RFD | CRFD |
---|---|---|---|---|---|
2 - 12 | 1.5 - 12.5 | 1 | 1 | 1/5 = 20% | 1/5 = 20% |
12 - 22 | 11.5 - 22.5 | 3 | 1 + 3 = 4 | 3/5 = 60% | 4/5 = 80% |
22 - 32 | 21.5 - 32.5 | 1 | 1 + 3 + 1 = 5 | 1/5 = 20% | 5/5 = 100% |
5 | 100% |
Go through our 5 numbers
Determine the ratio of each number to the next one
2:15 → 0.1333
15:15 → 1
15:18 → 0.8333
18:30 → 0.6
Successive Ratio = 2:15,15:15,15:18,18:30 or 0.1333,1,0.8333,0.6