 # 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}
Do 3 things
Run statistical calculations on this
Treat ths as a vector and run calculations on this
Treat this as a sequence

## Order This Number Set

Order lowest to highest
2 < 4 < 6 < 8 < 10

Order highest to lowest
10 > 8 > 6 > 4 > 2

## Basic Stats Calculations

Mean, Variance, Standard Deviation, Median, Mode

## Calculate Mean (Average) 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

Evaluate the square difference from the mean of each term
(Xi - μ)2:

(X1 - μ)2 = (2 - μ = 6)2 = 22 = 4
(X2 - μ)2 = (4 - μ = 6)2 = 42 = 16
(X3 - μ)2 = (6 - μ = 6)2 = 62 = 36
(X4 - μ)2 = (8 - μ = 6)2 = 82 = 64
(X5 - μ)2 = (10 - μ = 6)2 = 102 = 100

## Add our 5 sum of squared differences up

ΣE(Xi - μ)2 = 4 + 16 + 36 + 64 + 100
ΣE(Xi - μ)2 = 220

## Variance Table

PopulationSample
 σ2  = ΣE(Xi - μ)2 n

 σ2  = ΣE(Xi - μ)2 n - 1

 σ2  = 220 5

 σ2  = 220 4

Variance: σp2 = 44Variance: σs2 = 55
Standard Deviation: σp = √σp2 = √44Standard Deviation: σs = √σs2 = √55
Standard Deviation: σp = 6.6332Standard Deviation: σs = 7.4162

## Calculate the Standard Error of the Mean:

PopulationSample
 SEM  = σp √n

 SEM  = σs √n

 SEM  = 6.6332 √5

 SEM  = 7.4162 √5

 SEM  = 6.6332 2.23607

 SEM  = 7.4162 2.23607

SEM = 2.9665SEM = 3.3166

## 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 - μ = 6)3 = 23 = 8
(X2 - μ)3 = (4 - μ = 6)3 = 43 = 64
(X3 - μ)3 = (6 - μ = 6)3 = 63 = 216
(X4 - μ)3 = (8 - μ = 6)3 = 83 = 512
(X5 - μ)3 = (10 - μ = 6)3 = 103 = 1000

## Adding our 5 sum of cubed differences

ΣE(Xi - μ)3 = 8 + 64 + 216 + 512 + 1000
ΣE(Xi - μ)3 = 1800

## Finish calculating skewness:

 Skewness  = E(Xi - μ)3 (n - 1)σ3

 Skewness  = 1800 (5 - 1)6.63323

 Skewness  = 1800 (4)291.85643694637

 Skewness  = 1800 1167.43

Skewness = 1.5418539495248

## 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 - μ = 6| = |2| = 2
|X2 - μ| = |4 - μ = 6| = |4| = 4
|X3 - μ| = |6 - μ = 6| = |6| = 6
|X4 - μ| = |8 - μ = 6| = |8| = 8
|X5 - μ| = |10 - μ = 6| = |10| = 10

## Add our 5 absolute value of differences from the mean

Σ|Xi - μ| = 2 + 4 + 6 + 8 + 10
Σ|Xi - μ| = 30

## Calculate average deviation (mean absolute deviation)

 AD  = Σ|Xi - μ| n

Average Deviation = 6

## 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

## Final Median Calculation

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.

## Calculate the Range

Range = Largest Number - Smallest Number
Range = 10 - 2
Range = 8

## 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

## Calculate Pearsons Skewness Coefficient 1:

 PSC1  = μ - Mode σ

 PSC1  = 3( μ = 6 - N/A) 6.6332

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

## Calculate Pearsons Skewness Coefficient 2:

 PSC2  = μ - Median σ

 PSC1  = 3( μ = 6 - 6) 6.6332

 PSC2  = 3 x -6 6.6332

 PSC2  = -18 6.6332

PSC2 = -2.7136

## Calculate Entropy:

Entropy = Ln(n)
Entropy = Ln(5)
Entropy = 1.6094379124341

## Calculate the Quartile Items

Sort our number set from lowest to highest
{2,4,6,8,10}

## Calculate Upper Quartile (UQ) when y = 75%:

 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 8
2,4,6,8,10

## Calculate Lower Quartile (LQ) when y = 25%:

 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 4

2,4,6,8,10

IQR = UQ - LQ
IQR = 8 - 4
IQR = 4

## Calculate Lower Inner Fence (LIF):

Lower Inner Fence (LIF) = LQ - 1.5 x IQR
Lower Inner Fence (LIF) = 4 - 1.5 x 4
Lower Inner Fence (LIF) = 4 - 6
Lower Inner Fence (LIF) = -2

## Calculate Upper Inner Fence (UIF):

Upper Inner Fence (UIF) = UQ + 1.5 x IQR
Upper Inner Fence (UIF) = 8 + 1.5 x 4
Upper Inner Fence (UIF) = 8 + 6
Upper Inner Fence (UIF) = 14

## Calculate Lower Outer Fence (LOF):

Lower Outer Fence (LOF) = LQ - 3 x IQR
Lower Outer Fence (LOF) = 4 - 3 x 4
Lower Outer Fence (LOF) = 4 - 12
Lower Outer Fence (LOF) = -8

## Calculate Upper Outer Fence (UOF):

Upper Outer Fence (UOF) = UQ + 3 x IQR
Upper Outer Fence (UOF) = 8 + 3 x 4
Upper Outer Fence (UOF) = 8 + 12
Upper Outer Fence (UOF) = 20

## Calculate Suspect Outliers:

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
-8 < v < -2 and 14 < v < 20
2,4,6,8,10

## Calculate Highly Suspect Outliers:

Highly Suspect Outliers are values outside the outer fences
Mark all values in our dataset (v) in red below such that
v < -8 or v > 20
2,4,6,8,10

## Stem and Leaf Plot

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

{10,8,6,4,2}

StemLeaf
10
8
6
4
2

## Ranked Data Calculation

Sort our number set in ascending order
and assign a ranking to each number:

## Ranked Data Table

 Number Set Value 2 4 6 8 10 Rank 1 2 3 4 5

## Step 2: Using original number set, assign the rank value:

Assign ranks from lowest to highest (1 to 5)
Our original number set in unsorted order was 2,4,6,8,10
Our respective ranked data set is 1,2,3,4,5

## Root Mean Square Calculation

 Root Mean Square  = √A √N

where A = x12 + x22 + x32 + x42 + x52 and N = 5 number set items

## Calculate A

A = 22 + 42 + 62 + 82 + 102
A = 4 + 16 + 36 + 64 + 100
A = 220

## Calculate Root Mean Square (RMS):

 RMS  = √220 √5

 RMS  = 14.8324 2.23607

RMS = 6.6332495807108

## Harmonic-Geometric-Mid Range Weighted Average

Calculate the harmonic mean,
geometric mean, mid-range, weighted-average:

## Calculate Harmonic Mean:

 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 + 1/4 + 1/6 + 1/8 + 1/10

 Harmonic Mean  = 5 0.5 + 0.25 + 0.16666666666667 + 0.125 + 0.1

 Harmonic Mean  = 5 1.14167

Harmonic Mean = 4.3795620437956

## Calculate Geometric Mean:

Geometric Mean = (x1 * x2 * x3 * x4 * x5)1/N
Geometric Mean = (2 * 4 * 6 * 8 * 10)1/5
Geometric Mean = 38400.2
Geometric Mean = 5.2103421693947

## Calcualte Mid-Range:

 Mid-Range  = Maximum Value in Number Set + Minimum Value in Number Set 2

 Mid-Range  = 10 + 2 2

 Mid-Range  = 12 2

Mid-Range = 6

2,4,6,8,10

## Weighted-Average Formula:

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 + 4 x 0.4 + 6 x 0.6 + 8 x 0.8 + 10 x 0.9 5

 Weighted Average  = 0.4 + 1.6 + 3.6 + 6.4 + 9 5

 Weighted Average  = 21 5

Weighted Average = 4.2

## Frequency Distribution Table

Show the freqency distribution table for this number set
2, 4, 6, 8, 10

## Determine the Number of Intervals using Sturges Rule:

We need to 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 10 - 2 = 8
Each interval size is the difference of the maximum and minimum value divided by the number of intervals
 Interval Size  = 8 3

Add 1 to this giving us 2 + 1 = 3

## Frequency Distribution Table

Class LimitsClass BoundariesFDCFDRFDCRFD
2 - 51.5 - 5.5222/5 = 40%2/5 = 40%
5 - 84.5 - 8.512 + 1 = 31/5 = 20%3/5 = 60%
8 - 117.5 - 11.522 + 1 + 2 = 52/5 = 40%5/5 = 100%
5 100%

## Successive Ratio Calculation

Go through our 5 numbers
Determine the ratio of each number to the next one

2:4 → 0.5

4:6 → 0.6667

6:8 → 0.75

## Successive Ratio 4: 2,4,6,8,10

8:10 → 0.8

Successive Ratio = 2:4,4:6,6:8,8:10 or 0.5,0.6667,0.75,0.8

## Treat this as a set

Calculate the power set and partitions
For set S = {2,4,6,8,10}, show:
Elements, cardinality, and power set

## List the elements of S

Elements = set objects
Use the ∈ symbol.
1. 2 ∈ S
2. 4 ∈ S
3. 6 ∈ S
4. 8 ∈ S
5. 10 ∈ S

## Cardinality of set S → |S|:

Cardinality = Number of set elements.
Since the set S contains 5 elements
|S| = 5

## Determine the power set P:

Power set = Set of all subsets of S
including S and ∅.

## Calculate power set subsets

S contains 5 terms
Power Set contains 25 = 32 items

## Build subsets of P

The subset A of a set B is
A set where all elements of A are in B.
#BinaryUse if 1Subset
0000002,4,6,8,10{}
1000012,4,6,8,10{10}
2000102,4,6,8,10{8}
3000112,4,6,8,10{8,10}
4001002,4,6,8,10{6}
5001012,4,6,8,10{6,10}
6001102,4,6,8,10{6,8}
7001112,4,6,8,10{6,8,10}
8010002,4,6,8,10{4}
9010012,4,6,8,10{4,10}
10010102,4,6,8,10{4,8}
11010112,4,6,8,10{4,8,10}
12011002,4,6,8,10{4,6}
13011012,4,6,8,10{4,6,10}
14011102,4,6,8,10{4,6,8}
15011112,4,6,8,10{4,6,8,10}
16100002,4,6,8,10{2}
17100012,4,6,8,10{2,10}
18100102,4,6,8,10{2,8}
19100112,4,6,8,10{2,8,10}
20101002,4,6,8,10{2,6}
21101012,4,6,8,10{2,6,10}
22101102,4,6,8,10{2,6,8}
23101112,4,6,8,10{2,6,8,10}
24110002,4,6,8,10{2,4}
25110012,4,6,8,10{2,4,10}
26110102,4,6,8,10{2,4,8}
27110112,4,6,8,10{2,4,8,10}
28111002,4,6,8,10{2,4,6}
29111012,4,6,8,10{2,4,6,10}
30111102,4,6,8,10{2,4,6,8}
31111112,4,6,8,10{2,4,6,8,10}

## List our Power Set P in notation form:

P = {{}, {2}, {4}, {6}, {8}, {10}, {2,10}, {2,4}, {2,6}, {2,8}, {4,10}, {4,6}, {4,8}, {6,10}, {6,8}, {8,10}, {2,4,10}, {2,4,6}, {2,4,8}, {2,6,10}, {2,6,8}, {2,8,10}, {4,6,10}, {4,6,8}, {4,8,10}, {6,8,10}, {2,4,6,10}, {2,4,6,8}, {2,4,8,10}, {2,6,8,10}, {4,6,8,10}, {2,4,6,8,10}}

{8,10},{2,4,6}

{8,10},{2,4,6}

{8,10},{2,4,6}

{6,10},

{6,10},

{6,10},

{6,8},

{6,8},

{6,8},

{6,8,10},{2,4}

{6,8,10},{2,4}

{4,10},{2,4,6}

{4,10},{2,4,6}

{4,10},{2,4,6}

{4,8},{2,4,6}

{4,8},{2,4,6}

{4,8},{2,4,6}

{4,8,10},

{4,8,10},

{4,6},

{4,6},

{4,6},

{4,6,10},

{4,6,10},

{4,6,8},

{4,6,8},

{4,6,8,10},{2}

{2,10},{2,4,6}

{2,10},{2,4,6}

{2,10},{2,4,6}

{2,8},{2,4,6}

{2,8},{2,4,6}

{2,8},{2,4,6}

{2,8,10},{2,4}

{2,8,10},{2,4}

{2,6},

{2,6},

{2,6},

{2,6,10},{2,4}

{2,6,10},{2,4}

{2,6,8},{2,4}

{2,6,8},{2,4}

{2,6,8,10},

{2,4},{2,4,6}

{2,4},{2,4,6}

{2,4},{2,4,6}

{2,4,10},

{2,4,10},

{2,4,8},

{2,4,8},

{2,4,8,10},

{2,4,6},

{2,4,6},

{2,4,6,10},

{2,4,6,8},

## Partition 56

{{2},{4},{6},{8},{10})

## Treat this as a Vector

(a1, a2, a3, a4, a5) = (2, 4, 6, 8, 10)

## Calculate the magnitude:

||A|| = Square Root(a12 + a22 + a32 + a42 + a52)
||A|| = Square Root(22 + 42 + 62 + 82 + 102)
||A|| = √4 + 16 + 36 + 64 + 100
||A|| = √220
||A|| = 14.832396974191

## Treat this as a sequence

Find the explicit formula and terms
Calculate the explicit formula
Calculate term number 10
And the Sum of the first 10 terms for:
2,4,6,8,10

## Explicit Formula

an = a1 + (n - 1)d

## Define d

d = Δ between consecutive terms
d = an - an - 1

We see a common difference = 2
We have a1 = 2
an = 2 + 2(n - 1)

## Calculate Term (6)

Plug in n = 6 and d = 2
a6 = 2 + 2(6 - 1)
a6 = 2 + 2(6 - 1)
a6 = 2 + 2(5)
a6 = 2 + 10
a6 = 12

## Calculate Term (7)

Plug in n = 7 and d = 2
a7 = 2 + 2(7 - 1)
a7 = 2 + 2(7 - 1)
a7 = 2 + 2(6)
a7 = 2 + 12
a7 = 14

## Calculate Term (8)

Plug in n = 8 and d = 2
a8 = 2 + 2(8 - 1)
a8 = 2 + 2(8 - 1)
a8 = 2 + 2(7)
a8 = 2 + 14
a8 = 16

## Calculate Term (9)

Plug in n = 9 and d = 2
a9 = 2 + 2(9 - 1)
a9 = 2 + 2(9 - 1)
a9 = 2 + 2(8)
a9 = 2 + 16
a9 = 18

## Calculate Term (10)

Plug in n = 10 and d = 2
a10 = 2 + 2(10 - 1)
a10 = 2 + 2(10 - 1)
a10 = 2 + 2(9)
a10 = 2 + 18
a10 = 20

## Calculate Sn:

Sn = Sum of the first n terms
 Sn  = n(a1 + an) 2

## Substituting n = 10, we get:

 S10  = 10(a1 + a10) 2

 S10  = 10(2 + 20) 2

 S10  = 10(22) 2

 S10  = 220 2

S10 = 110  Order lowest to highest = 2 < 4 < 6 < 8 < 10
Order highest to lowest = 10 > 8 > 6 > 4 > 2
μ = 6
σ2 = 44
σ = 6.6332
SEM = 3.3166
Skewness = 1.5418539495248
Average Deviation = 6
Median = 6
Mode = N/A
Range = 8
Mid-Range = 6
PSC1 = No Pearsons Skewness Coefficient
PSC2 = -2.7136
Entropy = 1.6094379124341
1,2,3,4,5
RMS = 6.6332495807108
Harmonic Mean = 4.3795620437956Geometric Mean = 5.2103421693947
Mid-Range = 6
Weighted Average = 4.2
Successive Ratio = Successive Ratio = 2:4,4:6,6:8,8:10 or 0.5,0.6667,0.75,0.8
P = {{}, {2}, {4}, {6}, {8}, {10}, {2,10}, {2,4}, {2,6}, {2,8}, {4,10}, {4,6}, {4,8}, {6,10}, {6,8}, {8,10}, {2,4,10}, {2,4,6}, {2,4,8}, {2,6,10}, {2,6,8}, {2,8,10}, {4,6,10}, {4,6,8}, {4,8,10}, {6,8,10}, {2,4,6,10}, {2,4,6,8}, {2,4,8,10}, {2,6,8,10}, {4,6,8,10}, {2,4,6,8,10}}
||A|| = 14.832396974191
S10 = 110

### How does the Basic Statistics Calculator work?

Free Basic Statistics Calculator - Given a number set, and an optional probability set, this calculates the following statistical items:
Expected Value
Mean = μ
Variance = σ2
Standard Deviation = σ
Standard Error of the Mean
Skewness
Mid-Range
Average Deviation (Mean Absolute Deviation)
Median
Mode
Range
Pearsons Skewness Coefficients
Entropy
Upper Quartile (hinge) (75th Percentile)
Lower Quartile (hinge) (25th Percentile)
InnerQuartile Range
Inner Fences (Lower Inner Fence and Upper Inner Fence)
Outer Fences (Lower Outer Fence and Upper Outer Fence)
Suspect Outliers
Highly Suspect Outliers
Stem and Leaf Plot
Ranked Data Set
Central Tendency Items such as Harmonic Mean and Geometric Mean and Mid-Range
Root Mean Square
Weighted Average (Weighted Mean)
Frequency Distribution
Successive Ratio
This calculator has 2 inputs.

### What 8 formulas are used for the Basic Statistics Calculator?

Root Mean Square = √A/√N
Successive Ratio = n1/n0
μ = ΣXi/n
Mode = Highest Frequency Number
Mid-Range = (Maximum Value in Number Set + Minimum Value in Number Set)/2
Quartile: V = y(n + 1)/100
σ2 = ΣE(Xi - μ)2/n

For more math formulas, check out our Formula Dossier

### What 20 concepts are covered in the Basic Statistics Calculator?

average deviation
Mean of the absolute values of the distance from the mean for each number in a number set
basic statistics
central tendency
a central or typical value for a probability distribution. Typical measures are the mode, median, mean
entropy
refers to disorder or uncertainty
expected value
predicted value of a variable or event
E(X) = ΣxI · P(x)
frequency distribution
frequency measurement of various outcomes
inner fence
ut-off values for upper and lower outliers in a dataset
mean
A statistical measurement also known as the average
median
the value separating the higher half from the lower half of a data sample,
mode
the number that occurs the most in a number set
outer fence
start with the IQR and multiply this number by 3. We then subtract this number from the first quartile and add it to the third quartile. These two numbers are our outer fences.
outlier
an observation that lies an abnormal distance from other values in a random sample from a population
quartile
1 of 4 equal groups in the distribution of a number set
range
Difference between the largest and smallest values in a number set
rank
the data transformation in which numerical or ordinal values are replaced by their rank when the data are sorted.
sample space
the set of all possible outcomes or results of that experiment.
standard deviation
a measure of the amount of variation or dispersion of a set of values. The square root of variance
stem and leaf plot
a technique used to classify either discrete or continuous variables. A stem and leaf plot is used to organize data as they are collected. A stem and leaf plot looks something like a bar graph. Each number in the data is broken down into a stem and a leaf, thus the name.
variance
How far a set of random numbers are spead out from the mean
weighted average
An average of numbers using probabilities for each event as a weighting