You entered a number set X of {-6,4,3,-9,2,8}
-9, -6, 2, 3, 4, 8
8, 4, 3, 2, -6, -9
Sort our number set in ascending order
and assign a ranking to each number:
Number Set Value | -9 | -6 | 2 | 3 | 4 | 8 |
Rank | 1 | 2 | 3 | 4 | 5 | 6 |
Since we have 6 numbers in our original number set,
we assign ranks from lowest to highest (1 to 6)
Our original number set in unsorted order was -9,-6,2,3,4,8
Our respective ranked data set is 1,2,3,4,5,6
Root Mean Square = | √A |
√N |
where A = x12 + x22 + x32 + x42 + x52 + x62 and N = 6 number set items
A = -92 + -62 + 22 + 32 + 42 + 82
A = 81 + 36 + 4 + 9 + 16 + 64
A = 210
RMS = | √210 |
√6 |
RMS = | 14.491376746189 |
2.4494897427832 |
RMS = 5.9160797830996
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 = | -9 + -6 + 2 + 3 + 4 + 8 |
6 |
μ = | 2 |
6 |
μ = 0.33333333333333
Since our number set contains 6 elements which is an even number,
our median number is determined as follows
Number Set = (n1,n2,n3,n4,n5,n6)
Median Number 1 = ½(n)
Median Number 1 = ½(6)
Median Number 1 = Number Set Entry 3
Median Number 2 = Median Number 1 + 1
Median Number 2 = Number Set Entry 3 + 1
Median Number 2 = Number Set Entry 4
Median = ½(n3 + n4)
Our median is the average of entry 3 and entry 4 of our number set highlighted in red:
(-9,-6,2,3,4,8)
Median = ½(2 + 3)
Median = ½(5)
Median = 2.5
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/x1 + 1/x2 + 1/x3 + 1/x4 + 1/x5 + 1/x6 |
With N = 6 and each xi a member of the number set you entered, we have:
Harmonic Mean = | 6 |
1/-9 + 1/-6 + 1/2 + 1/3 + 1/4 + 1/8 |
Harmonic Mean = | 6 |
-0.11111111111111 + -0.16666666666667 + 0.5 + 0.33333333333333 + 0.25 + 0.125 |
Harmonic Mean = | 6 |
0.93055555555556 |
Harmonic Mean = 6.4477611940299
Geometric Mean = (x1 * x2 * x3 * x4 * x5 * x6)1/N
Geometric Mean = (-9 * -6 * 2 * 3 * 4 * 8)1/6
Geometric Mean = 103680.16666666666667
Geometric Mean = 4.6696302973561
Mid-Range = | Maximum Value in Number Set + Minimum Value in Number Set |
2 |
Mid-Range = | 8 + -9 |
2 |
Mid-Range = | -1 |
2 |
Mid-Range = -0.5
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
{8,4,3,2,-6,-9}
Stem | Leaf |
---|---|
8 | |
4 | |
3 | |
2 | |
- | 6,9 |
Mean, Variance, Standard Deviation, Median, Mode
μ = | Sum of your number Set |
Total Numbers Entered |
μ = | ΣXi |
n |
μ = | -9 + -6 + 2 + 3 + 4 + 8 |
6 |
μ = | 2 |
6 |
μ = 0.33333333333333
Let's evaluate the square difference from the mean of each term (Xi - μ)2:
(X1 - μ)2 = (-9 - 0.33333333333333)2 = -9.33333333333332 = 87.111111111111
(X2 - μ)2 = (-6 - 0.33333333333333)2 = -6.33333333333332 = 40.111111111111
(X3 - μ)2 = (2 - 0.33333333333333)2 = 1.66666666666672 = 2.7777777777778
(X4 - μ)2 = (3 - 0.33333333333333)2 = 2.66666666666672 = 7.1111111111111
(X5 - μ)2 = (4 - 0.33333333333333)2 = 3.66666666666672 = 13.444444444444
(X6 - μ)2 = (8 - 0.33333333333333)2 = 7.66666666666672 = 58.777777777778
ΣE(Xi - μ)2 = 87.111111111111 + 40.111111111111 + 2.7777777777778 + 7.1111111111111 + 13.444444444444 + 58.777777777778
ΣE(Xi - μ)2 = 209.33333333333
Population | Sample | ||||||||
---|---|---|---|---|---|---|---|---|---|
|
|
|
| ||||||
Variance: σp2 = 34.888888888889 | Variance: σs2 = 41.866666666667 | ||||||||
Standard Deviation: σp = √σp2 = √34.888888888889 | Standard Deviation: σs = √σs2 = √41.866666666667 | ||||||||
Standard Deviation: σp = 5.9067 | Standard Deviation: σs = 6.4704 |
Population | Sample | ||||||||
---|---|---|---|---|---|---|---|---|---|
|
|
|
|
|
| ||||
SEM = 2.4114 | SEM = 2.6415 |
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 = (-9 - 0.33333333333333)3 = -9.33333333333333 = -813.03703703704
(X2 - μ)3 = (-6 - 0.33333333333333)3 = -6.33333333333333 = -254.03703703704
(X3 - μ)3 = (2 - 0.33333333333333)3 = 1.66666666666673 = 4.6296296296296
(X4 - μ)3 = (3 - 0.33333333333333)3 = 2.66666666666673 = 18.962962962963
(X5 - μ)3 = (4 - 0.33333333333333)3 = 3.66666666666673 = 49.296296296296
(X6 - μ)3 = (8 - 0.33333333333333)3 = 7.66666666666673 = 450.62962962963
ΣE(Xi - μ)3 = -813.03703703704 + -254.03703703704 + 4.6296296296296 + 18.962962962963 + 49.296296296296 + 450.62962962963
ΣE(Xi - μ)3 = -543.55555555556
Skewness = | E(Xi - μ)3 |
(n - 1)σ3 |
Skewness = | -543.55555555556 |
(6 - 1)5.90673 |
Skewness = | -543.55555555556 |
(5)206.07947585376 |
Skewness = | -543.55555555556 |
1030.3973792688 |
Skewness = -0.52752032030717
AD = | Σ|Xi - μ| |
n |
Evaluate the absolute value of the difference from the mean
|Xi - μ|:
|X1 - μ| = |-9 - 0.33333333333333| = |-9.3333333333333| = 9.3333333333333
|X2 - μ| = |-6 - 0.33333333333333| = |-6.3333333333333| = 6.3333333333333
|X3 - μ| = |2 - 0.33333333333333| = |1.6666666666667| = 1.6666666666667
|X4 - μ| = |3 - 0.33333333333333| = |2.6666666666667| = 2.6666666666667
|X5 - μ| = |4 - 0.33333333333333| = |3.6666666666667| = 3.6666666666667
|X6 - μ| = |8 - 0.33333333333333| = |7.6666666666667| = 7.6666666666667
Σ|Xi - μ| = 9.3333333333333 + 6.3333333333333 + 1.6666666666667 + 2.6666666666667 + 3.6666666666667 + 7.6666666666667
Σ|Xi - μ| = 31.333333333333
Calculate average deviation (mean absolute deviation)
AD = | Σ|Xi - μ| |
n |
AD = | 31.333333333333 |
6 |
Average Deviation = 5.22222
Since our number set contains 6 elements which is an even number,
our median number is determined as follows
Number Set = (n1,n2,n3,n4,n5,n6)
Median Number 1 = ½(n)
Median Number 1 = ½(6)
Median Number 1 = Number Set Entry 3
Median Number 2 = Median Number 1 + 1
Median Number 2 = Number Set Entry 3 + 1
Median Number 2 = Number Set Entry 4
Median = ½(n3 + n4)
Our median is the average of entry 3 and entry 4 of our number set highlighted in red:
(-9,-6,2,3,4,8)
Median = ½(2 + 3)
Median = ½(5)
Median = 2.5
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 = 8 - -9
Range = 17
PSC1 = | μ - Mode |
σ |
PSC1 = | 3(0.33333333333333 - N/A) |
5.9067 |
Since no mode exists, we do not have a Pearsons Skewness Coefficient 1
PSC2 = | μ - Median |
σ |
PSC1 = | 3(0.33333333333333 - 2.5) |
5.9067 |
PSC2 = | 3 x -2.1666666666667 |
5.9067 |
PSC2 = | -6.5 |
5.9067 |
PSC2 = -1.1004
Entropy = Ln(n)
Entropy = Ln(6)
Entropy = 1.7917594692281
Mid-Range = | Smallest Number in the Set + Largest Number in the Set |
2 |
Mid-Range = | 8 + -9 |
2 |
Mid-Range = | -1 |
2 |
Mid-Range = -0.5
We need to sort our number set from lowest to highest shown below:
{-9,-6,2,3,4,8}
V = | y(n + 1) |
100 |
V = | 75(6 + 1) |
100 |
V = | 75(7) |
100 |
V = | 525 |
100 |
V = 5 ← Rounded down to the nearest integer
Upper quartile (UQ) point = Point # 5 in the dataset which is 4
-9,-6,2,3,4,8V = | y(n + 1) |
100 |
V = | 25(6 + 1) |
100 |
V = | 25(7) |
100 |
V = | 175 |
100 |
V = 2 ← Rounded up to the nearest integer
Lower quartile (LQ) point = Point # 2 in the dataset which is -6
-9,-6,2,3,4,8
IQR = UQ - LQ
IQR = 4 - -6
IQR = 10
Lower Inner Fence (LIF) = LQ - 1.5 x IQR
Lower Inner Fence (LIF) = -6 - 1.5 x 10
Lower Inner Fence (LIF) = -6 - 15
Lower Inner Fence (LIF) = -21
Upper Inner Fence (UIF) = UQ + 1.5 x IQR
Upper Inner Fence (UIF) = 4 + 1.5 x 10
Upper Inner Fence (UIF) = 4 + 15
Upper Inner Fence (UIF) = 19
Lower Outer Fence (LOF) = LQ - 3 x IQR
Lower Outer Fence (LOF) = -6 - 3 x 10
Lower Outer Fence (LOF) = -6 - 30
Lower Outer Fence (LOF) = -36
Upper Outer Fence (UOF) = UQ + 3 x IQR
Upper Outer Fence (UOF) = 4 + 3 x 10
Upper Outer Fence (UOF) = 4 + 30
Upper Outer Fence (UOF) = 34
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 -36 < v < -21 and 19 < v < 34
-9,-6,2,3,4,8
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 < -36 or v > 34
-9,-6,2,3,4,8
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
-9, -6, 2, 3, 4, 8
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 8 - -9 = 17
Each interval size is the difference of the maximum and minimum value divided by the number of intervals
Interval Size = | 17 |
0 |
Add 1 to this giving us INF + 1 = INF
Class Limits | Class Boundaries | FD | CFD | RFD | CRFD |
---|---|---|---|---|---|
0 | 100% |
Go through our 6 numbers
Determine the ratio of each number to the next one
-9:-6 → 1.5
-6:2 → -3
2:3 → 0.6667
3:4 → 0.75
4:8 → 0.5
Successive Ratio = -9:-6,-6:2,2:3,3:4,4:8 or 1.5,-3,0.6667,0.75,0.5