You entered a number set X of {95,91,80,67,88,42}
From the 6 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
42, 67, 80, 88, 91, 95
Rank Ascending
42 is the 1st lowest/smallest number
67 is the 2nd lowest/smallest number
80 is the 3rd lowest/smallest number
88 is the 4th lowest/smallest number
91 is the 5th lowest/smallest number
95 is the 6th lowest/smallest number
95, 91, 88, 80, 67, 42
Rank Descending
95 is the 1st highest/largest number
91 is the 2nd highest/largest number
88 is the 3rd highest/largest number
80 is the 4th highest/largest number
67 is the 5th highest/largest number
42 is the 6th highest/largest number
Sort our number set in ascending order
and assign a ranking to each number:
Number Set Value | 42 | 67 | 80 | 88 | 91 | 95 |
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 42,67,80,88,91,95
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 = 422 + 672 + 802 + 882 + 912 + 952
A = 1764 + 4489 + 6400 + 7744 + 8281 + 9025
A = 37703
RMS = | √37703 |
√6 |
RMS = | 194.17260362883 |
2.4494897427832 |
RMS = 79.270633486388
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 = | 42 + 67 + 80 + 88 + 91 + 95 |
6 |
μ = | 463 |
6 |
μ = 77.166666666667
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:
(42,67,80,88,91,95)
Median = ½(80 + 88)
Median = ½(168)
Median = 84
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 |
1/42 + 1/67 + 1/80 + 1/88 + 1/91 + 1/95 |
Harmonic Mean = | 6 |
0.023809523809524 + 0.014925373134328 + 0.0125 + 0.011363636363636 + 0.010989010989011 + 0.010526315789474 |
Harmonic Mean = | 6 |
0.084113860085973 |
Harmonic Mean = 71.33188268696
Geometric Mean = (x1 * x2 * x3 * x4 * x5 * x6)1/N
Geometric Mean = (42 * 67 * 80 * 88 * 91 * 95)1/6
Geometric Mean = 1712622912000.16666666666667
Geometric Mean = 74.520705590291
Mid-Range = | Maximum Value in Number Set + Minimum Value in Number Set |
2 |
Mid-Range = | 95 + 42 |
2 |
Mid-Range = | 137 |
2 |
Mid-Range = 68.5
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
{95,91,88,80,67,42}
Stem | Leaf |
---|---|
9 | 1,5 |
8 | 0,8 |
6 | 7 |
4 | 2 |
Mean, Variance, Standard Deviation, Median, Mode
μ = | Sum of your number Set |
Total Numbers Entered |
μ = | ΣXi |
n |
μ = | 42 + 67 + 80 + 88 + 91 + 95 |
6 |
μ = | 463 |
6 |
μ = 77.166666666667
Let's evaluate the square difference from the mean of each term (Xi - μ)2:
(X1 - μ)2 = (42 - 77.166666666667)2 = -35.1666666666672 = 1236.6944444444
(X2 - μ)2 = (67 - 77.166666666667)2 = -10.1666666666672 = 103.36111111111
(X3 - μ)2 = (80 - 77.166666666667)2 = 2.83333333333332 = 8.0277777777778
(X4 - μ)2 = (88 - 77.166666666667)2 = 10.8333333333332 = 117.36111111111
(X5 - μ)2 = (91 - 77.166666666667)2 = 13.8333333333332 = 191.36111111111
(X6 - μ)2 = (95 - 77.166666666667)2 = 17.8333333333332 = 318.02777777778
ΣE(Xi - μ)2 = 1236.6944444444 + 103.36111111111 + 8.0277777777778 + 117.36111111111 + 191.36111111111 + 318.02777777778
ΣE(Xi - μ)2 = 1974.8333333333
Population | Sample | ||||||||
---|---|---|---|---|---|---|---|---|---|
|
|
|
| ||||||
Variance: σp2 = 329.13888888889 | Variance: σs2 = 394.96666666667 | ||||||||
Standard Deviation: σp = √σp2 = √329.13888888889 | Standard Deviation: σs = √σs2 = √394.96666666667 | ||||||||
Standard Deviation: σp = 18.1422 | Standard Deviation: σs = 19.8738 |
Population | Sample | ||||||||
---|---|---|---|---|---|---|---|---|---|
|
|
|
|
|
| ||||
SEM = 7.4065 | SEM = 8.1134 |
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 = (42 - 77.166666666667)3 = -35.1666666666673 = -43490.421296296
(X2 - μ)3 = (67 - 77.166666666667)3 = -10.1666666666673 = -1050.837962963
(X3 - μ)3 = (80 - 77.166666666667)3 = 2.83333333333333 = 22.74537037037
(X4 - μ)3 = (88 - 77.166666666667)3 = 10.8333333333333 = 1271.412037037
(X5 - μ)3 = (91 - 77.166666666667)3 = 13.8333333333333 = 2647.162037037
(X6 - μ)3 = (95 - 77.166666666667)3 = 17.8333333333333 = 5671.4953703704
ΣE(Xi - μ)3 = -43490.421296296 + -1050.837962963 + 22.74537037037 + 1271.412037037 + 2647.162037037 + 5671.4953703704
ΣE(Xi - μ)3 = -34928.444444444
Skewness = | E(Xi - μ)3 |
(n - 1)σ3 |
Skewness = | -34928.444444444 |
(6 - 1)18.14223 |
Skewness = | -34928.444444444 |
(5)5971.3132007634 |
Skewness = | -34928.444444444 |
29856.566003817 |
Skewness = -1.1698748087767
AD = | Σ|Xi - μ| |
n |
Evaluate the absolute value of the difference from the mean
|Xi - μ|:
|X1 - μ| = |42 - 77.166666666667| = |-35.166666666667| = 35.166666666667
|X2 - μ| = |67 - 77.166666666667| = |-10.166666666667| = 10.166666666667
|X3 - μ| = |80 - 77.166666666667| = |2.8333333333333| = 2.8333333333333
|X4 - μ| = |88 - 77.166666666667| = |10.833333333333| = 10.833333333333
|X5 - μ| = |91 - 77.166666666667| = |13.833333333333| = 13.833333333333
|X6 - μ| = |95 - 77.166666666667| = |17.833333333333| = 17.833333333333
Σ|Xi - μ| = 35.166666666667 + 10.166666666667 + 2.8333333333333 + 10.833333333333 + 13.833333333333 + 17.833333333333
Σ|Xi - μ| = 90.666666666667
Calculate average deviation (mean absolute deviation)
AD = | Σ|Xi - μ| |
n |
AD = | 90.666666666667 |
6 |
Average Deviation = 15.11111
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:
(42,67,80,88,91,95)
Median = ½(80 + 88)
Median = ½(168)
Median = 84
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 = 95 - 42
Range = 53
PSC1 = | μ - Mode |
σ |
PSC1 = | 3(77.166666666667 - N/A) |
18.1422 |
Since no mode exists, we do not have a Pearsons Skewness Coefficient 1
PSC2 = | μ - Median |
σ |
PSC1 = | 3(77.166666666667 - 84) |
18.1422 |
PSC2 = | 3 x -6.8333333333333 |
18.1422 |
PSC2 = | -20.5 |
18.1422 |
PSC2 = -1.13
Entropy = Ln(n)
Entropy = Ln(6)
Entropy = 1.7917594692281
Mid-Range = | Smallest Number in the Set + Largest Number in the Set |
2 |
Mid-Range = | 95 + 42 |
2 |
Mid-Range = | 137 |
2 |
Mid-Range = 68.5
We need to sort our number set from lowest to highest shown below:
{42,67,80,88,91,95}
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 91
42,67,80,88,91,95V = | 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 67
42,67,80,88,91,95
IQR = UQ - LQ
IQR = 91 - 67
IQR = 24
Lower Inner Fence (LIF) = LQ - 1.5 x IQR
Lower Inner Fence (LIF) = 67 - 1.5 x 24
Lower Inner Fence (LIF) = 67 - 36
Lower Inner Fence (LIF) = 31
Upper Inner Fence (UIF) = UQ + 1.5 x IQR
Upper Inner Fence (UIF) = 91 + 1.5 x 24
Upper Inner Fence (UIF) = 91 + 36
Upper Inner Fence (UIF) = 127
Lower Outer Fence (LOF) = LQ - 3 x IQR
Lower Outer Fence (LOF) = 67 - 3 x 24
Lower Outer Fence (LOF) = 67 - 72
Lower Outer Fence (LOF) = -5
Upper Outer Fence (UOF) = UQ + 3 x IQR
Upper Outer Fence (UOF) = 91 + 3 x 24
Upper Outer Fence (UOF) = 91 + 72
Upper Outer Fence (UOF) = 163
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 -5 < v < 31 and 127 < v < 163
42,67,80,88,91,95
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 < -5 or v > 163
42,67,80,88,91,95
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
42, 67, 80, 88, 91, 95
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 95 - 42 = 53
Each interval size is the difference of the maximum and minimum value divided by the number of intervals
Interval Size = | 53 |
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
42:67 → 0.6269
67:80 → 0.8375
80:88 → 0.9091
88:91 → 0.967
91:95 → 0.9579
Successive Ratio = 42:67,67:80,80:88,88:91,91:95 or 0.6269,0.8375,0.9091,0.967,0.9579