l
You entered a number set X of {87.4,86.9,89.9,78.3,75.1,70.6}
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
70.6, 75.1, 78.3, 86.9, 87.4, 89.9
Rank Ascending
70.6 is the 1st lowest/smallest number
75.1 is the 2nd lowest/smallest number
78.3 is the 3rd lowest/smallest number
86.9 is the 4th lowest/smallest number
87.4 is the 5th lowest/smallest number
89.9 is the 6th lowest/smallest number
89.9, 87.4, 86.9, 78.3, 75.1, 70.6
Rank Descending
89.9 is the 1st highest/largest number
87.4 is the 2nd highest/largest number
86.9 is the 3rd highest/largest number
78.3 is the 4th highest/largest number
75.1 is the 5th highest/largest number
70.6 is the 6th highest/largest number
Sort our number set in ascending order
and assign a ranking to each number:
Number Set Value | 70.6 | 75.1 | 78.3 | 86.9 | 87.4 | 89.9 |
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 70.6,75.1,78.3,86.9,87.4,89.9
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 = 70.62 + 75.12 + 78.32 + 86.92 + 87.42 + 89.92
A = 4984.36 + 5640.01 + 6130.89 + 7551.61 + 7638.76 + 8082.01
A = 40027.64
RMS = | √40027.64 |
√6 |
RMS = | 200.0690880671 |
2.4494897427832 |
RMS = 81.677863178057
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 = | 70.6 + 75.1 + 78.3 + 86.9 + 87.4 + 89.9 |
6 |
μ = | 488.2 |
6 |
μ = 81.366666666667
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:
(70.6,75.1,78.3,86.9,87.4,89.9)
Median = ½(78.3 + 86.9)
Median = ½(165.2)
Median = 82.6
()
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 |
1/70.6 + 1/75.1 + 1/78.3 + 1/86.9 + 1/87.4 + 1/89.9 |
Harmonic Mean = | 6 |
0.014164305949009 + 0.013315579227696 + 0.012771392081737 + 0.01150747986191 + 0.011441647597254 + 0.011123470522803 |
Harmonic Mean = | 6 |
0.074323875240409 |
Harmonic Mean = 80.727760502158
Geometric Mean = (x1 * x2 * x3 * x4 * x5 * x6)1/N
Geometric Mean = (70.6 * 75.1 * 78.3 * 86.9 * 87.4 * 89.9)1/6
Geometric Mean = 283463601663.170.16666666666667
Geometric Mean = 81.049352596614
Mid-Range = | Maximum Value in Number Set + Minimum Value in Number Set |
2 |
Mid-Range = | 89.9 + 70.6 |
2 |
Mid-Range = | 160.5 |
2 |
Mid-Range = 80.25
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 |
---|---|
8 | 6.9,7.4,9.9 |
7 | 0.6,5.1,8.3 |
Let's evaluate the square difference from the mean of each term (Xi - μ)2:
(X1 - μ)2 = (70.6 - 81.366666666667)2 = -10.7666666666672 = 115.92111111111
(X2 - μ)2 = (75.1 - 81.366666666667)2 = -6.26666666666672 = 39.271111111111
(X3 - μ)2 = (78.3 - 81.366666666667)2 = -3.06666666666672 = 9.4044444444444
(X4 - μ)2 = (86.9 - 81.366666666667)2 = 5.53333333333332 = 30.617777777778
(X5 - μ)2 = (87.4 - 81.366666666667)2 = 6.03333333333332 = 36.401111111111
(X6 - μ)2 = (89.9 - 81.366666666667)2 = 8.53333333333332 = 72.817777777778
ΣE(Xi - μ)2 = 115.92111111111 + 39.271111111111 + 9.4044444444444 + 30.617777777778 + 36.401111111111 + 72.817777777778
ΣE(Xi - μ)2 = 304.43333333333
Population | Sample | ||||||||
---|---|---|---|---|---|---|---|---|---|
|
|
|
| ||||||
Variance: σp2 = 50.738888888889 | Variance: σs2 = 60.886666666667 | ||||||||
Standard Deviation: σp = √σp2 = √50.738888888889 | Standard Deviation: σs = √σs2 = √60.886666666667 | ||||||||
Standard Deviation: σp = 7.1231 | Standard Deviation: σs = 7.803 |
Population | Sample | ||||||||
---|---|---|---|---|---|---|---|---|---|
|
|
|
|
|
| ||||
SEM = 2.908 | SEM = 3.1856 |
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 = (70.6 - 81.366666666667)3 = -10.7666666666673 = -1248.083962963
(X2 - μ)3 = (75.1 - 81.366666666667)3 = -6.26666666666673 = -246.09896296296
(X3 - μ)3 = (78.3 - 81.366666666667)3 = -3.06666666666673 = -28.840296296296
(X4 - μ)3 = (86.9 - 81.366666666667)3 = 5.53333333333333 = 169.41837037037
(X5 - μ)3 = (87.4 - 81.366666666667)3 = 6.03333333333333 = 219.62003703704
(X6 - μ)3 = (89.9 - 81.366666666667)3 = 8.53333333333333 = 621.37837037037
ΣE(Xi - μ)3 = -1248.083962963 + -246.09896296296 + -28.840296296296 + 169.41837037037 + 219.62003703704 + 621.37837037037
ΣE(Xi - μ)3 = -512.60644444444
Skewness = | E(Xi - μ)3 |
(n - 1)σ3 |
Skewness = | -512.60644444444 |
(6 - 1)7.12313 |
Skewness = | -512.60644444444 |
(5)361.41579121939 |
Skewness = | -512.60644444444 |
1807.078956097 |
Skewness = -0.2836657705049
AD = | Σ|Xi - μ| |
n |
Evaluate the absolute value of the difference from the mean
|Xi - μ|:
|X1 - μ| = |70.6 - 81.366666666667| = |-10.766666666667| = 10.766666666667
|X2 - μ| = |75.1 - 81.366666666667| = |-6.2666666666667| = 6.2666666666667
|X3 - μ| = |78.3 - 81.366666666667| = |-3.0666666666667| = 3.0666666666667
|X4 - μ| = |86.9 - 81.366666666667| = |5.5333333333333| = 5.5333333333333
|X5 - μ| = |87.4 - 81.366666666667| = |6.0333333333333| = 6.0333333333333
|X6 - μ| = |89.9 - 81.366666666667| = |8.5333333333333| = 8.5333333333333
Σ|Xi - μ| = 10.766666666667 + 6.2666666666667 + 3.0666666666667 + 5.5333333333333 + 6.0333333333333 + 8.5333333333333
Σ|Xi - μ| = 40.2
Calculate average deviation (mean absolute deviation)
AD = | Σ|Xi - μ| |
n |
AD = | 40.2 |
6 |
Average Deviation = 6.7
Range = Largest Number in the Number Set - Smallest Number in the Number Set
Range = 89.9 - 70.6
Range = 19.3
Since no mode exists, we do not have a Pearsons Skewness Coefficient 1
PSC1 = | 3(81.366666666667 - N/A) |
7.1231 |
PSC2 = | μ - Median |
σ |
PSC1 = | 3(81.366666666667 - 82.6) |
7.1231 |
PSC2 = | 3 x -1.2333333333333 |
7.1231 |
PSC2 = | -3.7 |
7.1231 |
PSC2 = -0.5194
Entropy = Ln(n)
Entropy = Ln(6)
Entropy = 1.7917594692281
Mid-Range = | Smallest Number in the Set + Largest Number in the Set |
2 |
Mid-Range = | 89.9 + 70.6 |
2 |
Mid-Range = | 160.5 |
2 |
Mid-Range = 80.25
We need to sort our number set from lowest to highest shown below:
{70.6,75.1,78.3,86.9,87.4,89.9}
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 87.4
70.6,75.1,78.3,86.9,87.4,89.9
V = | 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 75.1
70.6,75.1,78.3,86.9,87.4,89.9
IQR = UQ - LQ
IQR = 87.4 - 75.1
IQR = 12.3
Lower Inner Fence (LIF) = LQ - 1.5 x IQR
Lower Inner Fence (LIF) = 75.1 - 1.5 x 12.3
Lower Inner Fence (LIF) = 75.1 - 18.45
Lower Inner Fence (LIF) = 56.65
Upper Inner Fence (UIF) = UQ + 1.5 x IQR
Upper Inner Fence (UIF) = 87.4 + 1.5 x 12.3
Upper Inner Fence (UIF) = 87.4 + 18.45
Upper Inner Fence (UIF) = 105.85
Lower Outer Fence (LOF) = LQ - 3 x IQR
Lower Outer Fence (LOF) = 75.1 - 3 x 12.3
Lower Outer Fence (LOF) = 75.1 - 36.9
Lower Outer Fence (LOF) = 38.2
Upper Outer Fence (UOF) = UQ + 3 x IQR
Upper Outer Fence (UOF) = 87.4 + 3 x 12.3
Upper Outer Fence (UOF) = 87.4 + 36.9
Upper Outer Fence (UOF) = 124.3
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 38.2 < v < 56.65 and 105.85 < v < 124.3
70.6,75.1,78.3,86.9,87.4,89.9
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 < 38.2 or v > 124.3
70.6,75.1,78.3,86.9,87.4,89.9
70.6, 75.1, 78.3, 86.9, 87.4, 89.9
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 + X6p6 |
n |
Weighted Average = | 70.6 x 0.2 + 75.1 x 0.4 + 78.3 x 0.6 + 86.9 x 0.8 + 87.4 x 0.9 + 89.9 x |
6 |
Weighted Average = | 14.12 + 30.04 + 46.98 + 69.52 + 78.66 + 0 |
6 |
Weighted Average = | 239.32 |
6 |
Weighted Average = 39.886666666667
Show the freqency distribution table for this number set
70.6, 75.1, 78.3, 86.9, 87.4, 89.9
Choose the smallest integer k such that 2k ≥ n where n = 6
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 6
Therefore, we use 3 intervals
Our maximum value in our number set of 89.9 - 70.6 = 19.3
Each interval size is the difference of the maximum and minimum value divided by the number of intervals
Interval Size = | 19.3 |
3 |
Add 1 to this giving us 6 + 1 = 7
Class Limits | Class Boundaries | FD | CFD | RFD | CRFD |
---|---|---|---|---|---|
70.6 - 77.6 | 70.1 - 78.1 | 2 | 2 | 2/6 = 33.33% | 2/6 = 33.33% |
77.6 - 84.6 | 77.1 - 85.1 | 1 | 2 + 1 = 3 | 1/6 = 16.67% | 3/6 = 50% |
84.6 - 91.6 | 84.1 - 92.1 | 3 | 2 + 1 + 3 = 6 | 3/6 = 50% | 6/6 = 100% |
6 | 100% |
Go through our 6 numbers
Determine the ratio of each number to the next one
70.6:75.1 → 0.9401
75.1:78.3 → 0.9591
78.3:86.9 → 0.901
86.9:87.4 → 0.9943
87.4:89.9 → 0.9722
Successive Ratio = 70.6:75.1,75.1:78.3,78.3:86.9,86.9:87.4,87.4:89.9 or 0.9401,0.9591,0.901,0.9943,0.9722