Answer

↓Steps Explained:↓

You entered a number set X of {19,9,14,10,16,17}

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

Sort Ascending from Lowest to Highest

9, 10, 14, 16, 17, 19

Rank Ascending

9 is the 1st lowest/smallest number

10 is the 2nd lowest/smallest number

14 is the 3rd lowest/smallest number

16 is the 4th lowest/smallest number

17 is the 5th lowest/smallest number

19 is the 6th lowest/smallest number

Sort Descending from Highest to Lowest

19, 17, 16, 14, 10, 9

Rank Descending

19 is the 1st highest/largest number

17 is the 2nd highest/largest number

16 is the 3rd highest/largest number

14 is the 4th highest/largest number

10 is the 5th highest/largest number

9 is the 6th highest/largest number

Ranked Data Calculation

Sort our number set in ascending order

and assign a ranking to each number:

Ranked Data Table

Number Set Value	9	10	14	16	17	19
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 9,10,14,16,17,19

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

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 + 10² + 14² + 16² + 17² + 19²

A = 81 + 100 + 196 + 256 + 289 + 361

A = 1283

Calculate Root Mean Square (RMS):

RMS  =	√1283
	√6

RMS  =	35.818989377145
	2.4494897427832

RMS = 14.623041179363

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 + 10 + 14 + 16 + 17 + 19
	6

μ  =	85
	6

μ = 14.166666666667

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

Our median is the average of entry 3 and entry 4 of our number set highlighted in red:

(9,10,14,16,17,19)

Median = ½(14 + 16)

Median = ½(30)

Median = 15

Calculate the Mode - Highest Frequency Number

()

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 + 1/10 + 1/14 + 1/16 + 1/17 + 1/19

Harmonic Mean  =	6
	0.11111111111111 + 0.1 + 0.071428571428571 + 0.0625 + 0.058823529411765 + 0.052631578947368

Harmonic Mean  =	6
	0.45649479089882

Harmonic Mean = 13.143633004412

Calculate Geometric Mean:

Geometric Mean = (x₁ * x₂ * x₃ * x₄ * x₅ * x₆)^1/N

Geometric Mean = (9 * 10 * 14 * 16 * 17 * 19)^1/6

Geometric Mean = 6511680^{0.16666666666667}

Geometric Mean = 13.665184772838

Calculate Mid-Range:

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

Mid-Range  =	19 + 9
	2

Mid-Range  =	28
	2

Mid-Range = 14

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:

Stem	Leaf
1	0,4,6,7,9
9

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 - 14.166666666667)² = -5.1666666666667² = 26.694444444444

(X₂ - μ)² = (10 - 14.166666666667)² = -4.1666666666667² = 17.361111111111

(X₃ - μ)² = (14 - 14.166666666667)² = -0.16666666666667² = 0.027777777777778

(X₄ - μ)² = (16 - 14.166666666667)² = 1.8333333333333² = 3.3611111111111

(X₅ - μ)² = (17 - 14.166666666667)² = 2.8333333333333² = 8.0277777777778

(X₆ - μ)² = (19 - 14.166666666667)² = 4.8333333333333² = 23.361111111111

Adding our 6 sum of squared differences up

ΣE(X_i - μ)² = 26.694444444444 + 17.361111111111 + 0.027777777777778 + 3.3611111111111 + 8.0277777777778 + 23.361111111111

ΣE(X_i - μ)² = 78.833333333333

Use the sum of squared differences to calculate variance

Population	Sample
σ2  =   ΣE(Xi - μ)2    n	σ2  =   ΣE(Xi - μ)2    n - 1
σ2  =   78.833333333333    6	σ2  =   78.833333333333    5
Variance: σp2 = 13.138888888889	Variance: σs2 = 15.766666666667
Standard Deviation: σp = √σp2 = √13.138888888889	Standard Deviation: σs = √σs2 = √15.766666666667
Standard Deviation: σp = 3.6248	Standard Deviation: σs = 3.9707

Calculate the Standard Error of the Mean:

Population	Sample
SEM  =   σp    √n	SEM  =   σs    √n
SEM  =   3.6248    √6	SEM  =   3.9707    √6
SEM  =   3.6248    2.4494897427832	SEM  =   3.9707    2.4494897427832
SEM = 1.4798	SEM = 1.621

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 - 14.166666666667)³ = -5.1666666666667³ = -137.9212962963

(X₂ - μ)³ = (10 - 14.166666666667)³ = -4.1666666666667³ = -72.337962962963

(X₃ - μ)³ = (14 - 14.166666666667)³ = -0.16666666666667³ = -0.0046296296296296

(X₄ - μ)³ = (16 - 14.166666666667)³ = 1.8333333333333³ = 6.162037037037

(X₅ - μ)³ = (17 - 14.166666666667)³ = 2.8333333333333³ = 22.74537037037

(X₆ - μ)³ = (19 - 14.166666666667)³ = 4.8333333333333³ = 112.91203703704

Add our 6 sum of cubed differences up

ΣE(X_i - μ)³ = -137.9212962963 + -72.337962962963 + -0.0046296296296296 + 6.162037037037 + 22.74537037037 + 112.91203703704

ΣE(X_i - μ)³ = -68.444444444444

Calculate skewnes

Skewness  =	E(Xi - μ)3
	(n - 1)σ3

Skewness  =	-68.444444444444
	(6 - 1)3.62483

Skewness  =	-68.444444444444
	(5)47.626881684992

Skewness  =	-68.444444444444
	238.13440842496

Skewness = -0.28741938175647

Calculate Average Deviation (Mean Absolute Deviation) denoted below:

AD  =	Σ|Xi - μ|
	n

Evaluate the absolute value of the difference from the mean

|X_i - μ|:

|X₁ - μ| = |9 - 14.166666666667| = |-5.1666666666667| = 5.1666666666667

|X₂ - μ| = |10 - 14.166666666667| = |-4.1666666666667| = 4.1666666666667

|X₃ - μ| = |14 - 14.166666666667| = |-0.16666666666667| = 0.16666666666667

|X₄ - μ| = |16 - 14.166666666667| = |1.8333333333333| = 1.8333333333333

|X₅ - μ| = |17 - 14.166666666667| = |2.8333333333333| = 2.8333333333333

|X₆ - μ| = |19 - 14.166666666667| = |4.8333333333333| = 4.8333333333333

Average deviation numerator:

Σ|X_i - μ| = 5.1666666666667 + 4.1666666666667 + 0.16666666666667 + 1.8333333333333 + 2.8333333333333 + 4.8333333333333

Σ|X_i - μ| = 19

Calculate average deviation (mean absolute deviation)

AD  =	Σ|Xi - μ|
	n

AD  =	19
	6

Average Deviation = 3.16667

Calculate the Range

Range = Largest Number in the Number Set - Smallest Number in the Number Set

Range = 19 - 9

Range = 10

Calculate Pearsons Skewness Coefficient 1:

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

PSC1  =	3(14.166666666667 - N/A)
	3.6248

Calculate Pearsons Skewness Coefficient 2:

PSC2  =	μ - Median
	σ

PSC1  =	3(14.166666666667 - 15)
	3.6248

PSC2  =	3 x -0.83333333333333
	3.6248

PSC2  =	-2.5
	3.6248

PSC2 = -0.6897

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  =	19 + 9
	2

Mid-Range  =	28
	2

Mid-Range = 14

Calculate the Quartile Items

We need to sort our number set from lowest to highest shown below:

{9,10,14,16,17,19}

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 17

9,10,14,16,17,19

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 10

9,10,14,16,17,19

Calculate Inter-Quartile Range (IQR):

IQR = UQ - LQ

IQR = 17 - 10

IQR = 7

Calculate Lower Inner Fence (LIF):

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

Lower Inner Fence (LIF) = 10 - 1.5 x 7

Lower Inner Fence (LIF) = 10 - 10.5

Lower Inner Fence (LIF) = -0.5

Calculate Upper Inner Fence (UIF):

Upper Inner Fence (UIF) = UQ + 1.5 x IQR

Upper Inner Fence (UIF) = 17 + 1.5 x 7

Upper Inner Fence (UIF) = 17 + 10.5

Upper Inner Fence (UIF) = 27.5

Calculate Lower Outer Fence (LOF):

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

Lower Outer Fence (LOF) = 10 - 3 x 7

Lower Outer Fence (LOF) = 10 - 21

Lower Outer Fence (LOF) = -11

Calculate Upper Outer Fence (UOF):

Upper Outer Fence (UOF) = UQ + 3 x IQR

Upper Outer Fence (UOF) = 17 + 3 x 7

Upper Outer Fence (UOF) = 17 + 21

Upper Outer Fence (UOF) = 38

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 -11 < v < -0.5 and 27.5 < v < 38

9,10,14,16,17,19

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 < -11 or v > 38

9,10,14,16,17,19

Calculate weighted average

9, 10, 14, 16, 17, 19

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  =	9 x 0.2 + 10 x 0.4 + 14 x 0.6 + 16 x 0.8 + 17 x 0.9 + 19 x
	6

Weighted Average  =	1.8 + 4 + 8.4 + 12.8 + 15.3 + 0
	6

Weighted Average  =	42.3
	6

Weighted Average = 7.05

Frequency Distribution Table

Show the freqency distribution table for this number set

9, 10, 14, 16, 17, 19

Determine the Number of Intervals using Sturges Rule:

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 19 - 9 = 10

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

Interval Size  =	10
	3

Add 1 to this giving us 3 + 1 = 4

Frequency Distribution Table

Class Limits	Class Boundaries	FD	CFD	RFD	CRFD
9 - 13	8.5 - 13.5	2	2	2/6 = 33.33%	2/6 = 33.33%
13 - 17	12.5 - 17.5	2	2 + 2 = 4	2/6 = 33.33%	4/6 = 66.67%
17 - 21	16.5 - 21.5	2	2 + 2 + 2 = 6	2/6 = 33.33%	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: 9,10,14,16,17,19

9:10 → 0.9

Successive Ratio 2: 9,10,14,16,17,19

10:14 → 0.7143

Successive Ratio 3: 9,10,14,16,17,19

14:16 → 0.875

Successive Ratio 4: 9,10,14,16,17,19

16:17 → 0.9412

Successive Ratio 5: 9,10,14,16,17,19

17:19 → 0.8947

Successive Ratio Answer

Successive Ratio = 9:10,10:14,14:16,16:17,17:19 or 0.9,0.7143,0.875,0.9412,0.8947

Final Answers

1,2,3,4,5,6
RMS = 14.623041179363
μ = 14.166666666667
Median = 15
Mode = N/A
Harmonic Mean = 13.143633004412Geometric Mean = 13.665184772838
Mid-Range = 14
σ_s² = 13.138888888889
σ = 3.6248
Weighted Average = 7.05
Successive Ratio = Successive Ratio = 9:10,10:14,14:16,16:17,17:19 or 0.9,0.7143,0.875,0.9412,0.8947

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