Calculate Root Mean Square from -6,4,3,-9,2,8

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

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You entered a number set X of {-6,4,3,-9,2,8}

From the 6 numbers you entered, we want to calculate the root mean square:

Ranked Data Calculation

Sort our number set in ascending order

and assign a ranking to each number:

Ranked Data Table

Number Set Value	-9	-6	2	3	4	8
Rank	1	2	3	4	5	6

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

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 -6,4,3,-9,2,8
Our respective ranked data set is 2,5,4,1,3,6

Root Mean Square Calculation

Root Mean Square  =	√A
	√N

where A = x₁² + x₂² + x₃² + x₄² + x₅² + x₆² and N = 6 number set items

Calculate A

A = -9² + -6² + 2² + 3² + 4² + 8²

A = 81 + 36 + 4 + 9 + 16 + 64

A = 210

Calculate Root Mean Square (RMS):

RMS  =	√210
	√6

RMS  =	14.491376746189
	2.4494897427832

RMS = 5.9160797830996

Central Tendency Calculation

Central tendency contains:
Mean, median, mode, harmonic mean,
geometric mean, mid-range, weighted-average:

Calculate Mean (Average) denoted as μ

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

Calculate the Median (Middle Value)
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

For an even number set, we average the 2 median number entries:
Median = ½(n₃ + n₄)

Therefore, we sort our number set in ascending order and 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

Calculate the Mode - Highest Frequency Number

The highest frequency of occurence in our number set is 1 times by the following numbers in green:
(-6,4,3,-9,2,8)
Since the maximum frequency of any number is 1, no mode exists.
Mode = N/A

Calculate Harmonic Mean:

Harmonic Mean  =	N
	1/x1 + 1/x2 + 1/x3 + 1/x4 + 1/x5 + 1/x6

With N = 6 and each x_i 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

Calculate Geometric Mean:

Geometric Mean = (x₁ * x₂ * x₃ * x₄ * x₅ * x₆)^1/N
Geometric Mean = (-9 * -6 * 2 * 3 * 4 * 8)^1/6
Geometric Mean = 10368^{0.16666666666667}
Geometric Mean = 4.6696302973561

Calcualte Mid-Range:

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

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

Sort our number set in descending order:

{8,4,3,2,-6,-9}

Stem	Leaf
8
4
3
2
-	6,9

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  =	-9 + -6 + 2 + 3 + 4 + 8
	6

μ  =	2
	6

μ = 0.33333333333333

Calculate Variance denoted as σ²
Let's evaluate the square difference from the mean of each term (X_i - 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

Adding our 6 sum of squared differences up, we have our variance numerator:
ΣE(X_i - μ)² = 87.111111111111 + 40.111111111111 + 2.7777777777778 + 7.1111111111111 + 13.444444444444 + 58.777777777778
ΣE(X_i - μ)² = 209.33333333333

Now that we have the sum of squared differences from the means, calculate variance:

Population	Sample
σ2  =   ΣE(Xi - μ)2    n	σ2  =   ΣE(Xi - μ)2    n - 1
σ2  =   209.33333333333    6	σ2  =   209.33333333333    5
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

Calculate the Standard Error of the Mean:

Population	Sample
SEM  =   σp    √n	SEM  =   σs    √n
SEM  =   5.9067    √6	SEM  =   6.4704    √6
SEM  =   5.9067    2.4494897427832	SEM  =   6.4704    2.4494897427832
SEM = 2.4114	SEM = 2.6415

Calculate Skewness:

Skewness  =	E(Xi - μ)3
	(n - 1)σ3

Let's evaluate the square difference from the mean of each term (X_i - 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

Adding our 6 sum of cubed differences up, we have our skewness numerator:
ΣE(X_i - μ)³ = -813.03703703704 + -254.03703703704 + 4.6296296296296 + 18.962962962963 + 49.296296296296 + 450.62962962963
ΣE(X_i - μ)³ = -543.55555555556

Now that we have the sum of cubed differences from the means, calculate skewness:

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

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 |X_i - 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

Adding our 6 absolute value of differences from the mean, we have our average deviation numerator:
Σ|X_i - μ| = 9.3333333333333 + 6.3333333333333 + 1.6666666666667 + 2.6666666666667 + 3.6666666666667 + 7.6666666666667
Σ|X_i - μ| = 31.333333333333

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  =	31.333333333333
	6

Average Deviation = 5.22222

Calculate the Median (Middle Value)
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

For an even number set, we average the 2 median number entries:
Median = ½(n₃ + n₄)

Therefore, we sort our number set in ascending order and 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

Calculate the Mode - Highest Frequency Number

The highest frequency of occurence in our number set is 1 times by the following numbers in green:
(-6,4,3,-9,2,8)
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 = 8 - -9
Range = 17

Calculate Pearsons Skewness Coefficient 1:

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

Calculate Pearsons Skewness Coefficient 2:

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

Calculate Entropy:

Entropy = Ln(n)
Entropy = Ln(6)
Entropy = 1.7917594692281

Calculate Mid-Range:

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

Calculate the Quartile Items

We need to sort our number set from lowest to highest shown below:
{-9,-6,2,3,4,8}

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

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,-6,2,3,4,8

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

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 -6
-9,-6,2,3,4,8

Calculate Inter-Quartile Range (IQR):

IQR = UQ - LQ
IQR = 4 - -6
IQR = 10

Calculate Lower Inner Fence (LIF):

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

Calculate Upper Inner Fence (UIF):

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

Calculate Lower Outer Fence (LOF):

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

Calculate Upper Outer Fence (UOF):

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

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 -36 < v < -21 and 19 < v < 34
-9,-6,2,3,4,8

Calculate Highly Suspect Outliers:

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

Calculate weighted average

-6,4,3,-9,2,8

Weighted-Average Formula:

Multiply each value by each probability amount

We do this by multiplying each X_i x p_i to get a weighted score Y

Weighted Average  =	X1p1 + X2p2 + X3p3 + X4p4 + X5p5 + X6p6
	n

Weighted Average  =	-6 x 0.2 + 4 x 0.4 + 3 x 0.6 + -9 x 0.8 + 2 x 0.9 + 8 x
	6

Weighted Average  =	-1.2 + 1.6 + 1.8 + -7.2 + 1.8 + 0
	6

Weighted Average  =	-3.2
	6

Weighted Average = -0.53333333333333

Frequency Distribution Table

Show the freqency distribution table for this number set

-9, -6, 2, 3, 4, 8

Determine the Number of Intervals using Sturges Rule:

We need to choose the smallest integer k such that 2^k ≥ 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 8 - -9 = 17

Each interval size is the difference of the maximum and minimum value divided by the number of intervals

Interval Size  =	17
	3

Add 1 to this giving us 5 + 1 = 6

Frequency Distribution Table

Class Limits	Class Boundaries	FD	CFD	RFD	CRFD
-9 - -3	-9.5 - -2.5	2	2	2/6 = 33.33%	2/6 = 33.33%
-3 - 3	-3.5 - 3.5	1	2 + 1 = 3	1/6 = 16.67%	3/6 = 50%
3 - 9	2.5 - 9.5	3	2 + 1 + 3 = 6	3/6 = 50%	6/6 = 100%
		6		100%

Successive Ratio Calculation

Go through our 6 numbers

Determine the ratio of each number to the next one

Successive Ratio 1: -9,-6,2,3,4,8

-9:-6 → 1.5

Successive Ratio 2: -9,-6,2,3,4,8

-6:2 → -3

Successive Ratio 3: -9,-6,2,3,4,8

2:3 → 0.6667

Successive Ratio 4: -9,-6,2,3,4,8

3:4 → 0.75

Successive Ratio 5: -9,-6,2,3,4,8

4:8 → 0.5

Successive Ratio Answer

Successive Ratio = -9:-6,-6:2,2:3,3:4,4:8 or 1.5,-3,0.6667,0.75,0.5

Final Answers

You have 2 free calculationss remaining

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Basic Statistics Calculator Video

Tags:

• average deviation
• basic statistics
• central tendency
• entropy
• expected value
• frequency distribution
• inner fence
• mean
• median
• mode
• outer fence
• outlier
• quartile
• range
• rank
• sample space
• standard deviation
• stem and leaf plot
• variance
• weighted average

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