Calculate Number Set Basics from 2,4,6,8,10

<-- Enter Number Set
<-- Probabilities (or counts for Weighted Average), check box if you are using these →


You entered a number set X of {2,4,6,8,10}

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

Calculate Mean denoted as μ
μ =Sum of your number Set
Total Numbers Entered

μ =ΣXi
n

μ =2 + 4 + 6 + 8 + 10
5

μ =30
5

μ = 6

Calculate Variance denoted as σ2
Let's evaluate the square difference from the mean of each term (Xi - μ)2:
(X1 - μ)2 = (2 - 6)2 = -42 = 16
(X2 - μ)2 = (4 - 6)2 = -22 = 4
(X3 - μ)2 = (6 - 6)2 = 02 = 0
(X4 - μ)2 = (8 - 6)2 = 22 = 4
(X5 - μ)2 = (10 - 6)2 = 42 = 16

Adding our 5 sum of squared differences up, we have our variance numerator:
ΣE(Xi - μ)2 = 16 + 4 + 0 + 4 + 16
ΣE(Xi - μ)2 = 40

Now that we have the sum of squared differences from the means, calculate variance:
PopulationSample
σ2 =ΣE(Xi - μ)2
n

σ2 =ΣE(Xi - μ)2
n - 1

σ2 =40
5

σ2 =40
4

Variance: σp2 = 8Variance: σs2 = 10
Standard Deviation: σp = √σp2 = √8Standard Deviation: σs = √σs2 = √10
Standard Deviation: σp = 2.8284Standard Deviation: σs = 3.1623

Calculate the Standard Error of the Mean:
PopulationSample
SEM =σp
n

SEM =σs
n

SEM =2.8284
5

SEM =3.1623
5

SEM =2.8284
2.8

SEM =3.1623
2.8

SEM = 1.2649SEM = 1.4142

Calculate Skewness:
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 - 6)3 = -43 = -64
(X2 - μ)3 = (4 - 6)3 = -23 = -8
(X3 - μ)3 = (6 - 6)3 = 03 = 0
(X4 - μ)3 = (8 - 6)3 = 23 = 8
(X5 - μ)3 = (10 - 6)3 = 43 = 64

Adding our 5 sum of cubed differences up, we have our skewness numerator:
ΣE(Xi - μ)3 = -64 + -8 + 0 + 8 + 64
ΣE(Xi - μ)3 = 0

Now that we have the sum of cubed differences from the means, calculate skewness:
Skewness =E(Xi - μ)3
(n - 1)σ3

Skewness =0
(5 - 1)2.82843

Skewness =0
(4)22.

Skewness =0
90.

Skewness = 0

Calculate Average Deviation (Mean Absolute Deviation) denoted below:
AD =Σ|Xi - μ|
n

Let's evaluate the absolute value of the difference from the mean of each term |Xi - μ|:
|X1 - μ| = |2 - 6| = |-4| = 4
|X2 - μ| = |4 - 6| = |-2| = 2
|X3 - μ| = |6 - 6| = |0| = 0
|X4 - μ| = |8 - 6| = |2| = 2
|X5 - μ| = |10 - 6| = |4| = 4

Adding our 5 absolute value of differences from the mean, we have our average deviation numerator:
Σ|Xi - μ| = 4 + 2 + 0 + 2 + 4
Σ|Xi - μ| = 12

Now that we have the absolute value of the differences from the means, calculate average deviation (mean absolute deviation):
AD =Σ|Xi - μ|
n

AD =12
5

Average Deviation = 2.4

Calculate the Median (Middle Value)
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

Therefore, we sort our number set in ascending order and our median is entry 3 of our number set highlighted in red:
(2,4,6,8,10)
Median = 6

Calculate the Mode - Highest Frequency Number
The highest frequency of occurence in our number set is 1 times by the following numbers in green:
(2,4,6,8,10)
Since the maximum frequency of any number is 1, no mode exists.
Mode = N/A

Calculate the Range
Range = Largest Number in the Number Set - Smallest Number in the Number Set
Range = 10 - 2
Range = 8

Calculate Pearsons Skewness Coefficient 1:
PSC1 =μ - Mode
σ

PSC1 =3(6 - N/A)
2.8284

Since no mode exists, we do not have a Pearsons Skewness Coefficient 1

Calculate Pearsons Skewness Coefficient 2:
PSC2 =μ - Median
σ

PSC1 =3(6 - 6)
2.8284

PSC2 =3 x 0
2.8284

PSC2 =0
2.8284

PSC2 = 0

Calculate Entropy:
Entropy = Ln(n)
Entropy = Ln(5)
Entropy = 1

Calculate Mid-Range:
Mid-Range =Smallest Number in the Set + Largest Number in the Set
2

Mid-Range =10 + 2
2

Mid-Range =12
2

Mid-Range = 6