Answer

1,2,4,4,5,6,7,8,9,10,11,12,13,14
RMS = 34.882610895074
μ = 32.291428571429
Median = 33.735
Mode = 16.58
Harmonic Mean = 26.224356816288Geometric Mean = 29.294540856618
Mid-Range = 35.38
σ_s² = 174.06018367347
σ = 13.1932
Weighted Average = 3.9077857142857
Successive Ratio = Successive Ratio = 13.29:14.70,14.70:16.58,16.58:16.58,16.58:25.51,25.51:27.86,27.86:33.03,33.03:34.44,34.44:39.14,39.14:39.61,39.61:42.43,42.43:44.31,44.31:47.13,47.13:57.47 or 0.9041,0.8866,1,0.6499,0.9156,0.8435,0.9591,0.8799,0.9881,0.9335,0.9576,0.9402,0.8201

↓Steps Explained:↓

You entered a number set X of {27.86,13.29,33.03,44.31,16.58,42.43,39.61,25.51,39.14,16.58,47.13,14.70,57.47,34.44}

From the 14 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

13.29, 14.70, 16.58, 16.58, 25.51, 27.86, 33.03, 34.44, 39.14, 39.61, 42.43, 44.31, 47.13, 57.47

Rank Ascending

13.29 is the 1st lowest/smallest number

14.70 is the 2nd lowest/smallest number

16.58 is the 3rd lowest/smallest number

16.58 is the 4th lowest/smallest number

25.51 is the 5th lowest/smallest number

27.86 is the 6th lowest/smallest number

33.03 is the 7th lowest/smallest number

34.44 is the 8th lowest/smallest number

39.14 is the 9th lowest/smallest number

39.61 is the 10th lowest/smallest number

42.43 is the 11th lowest/smallest number

44.31 is the 12th lowest/smallest number

47.13 is the 13th lowest/smallest number

57.47 is the 14th lowest/smallest number

Sort Descending from Highest to Lowest

57.47, 47.13, 44.31, 42.43, 39.61, 39.14, 34.44, 33.03, 27.86, 25.51, 16.58, 16.58, 14.70, 13.29

Rank Descending

57.47 is the 1st highest/largest number

47.13 is the 2nd highest/largest number

44.31 is the 3rd highest/largest number

42.43 is the 4th highest/largest number

39.61 is the 5th highest/largest number

39.14 is the 6th highest/largest number

34.44 is the 7th highest/largest number

33.03 is the 8th highest/largest number

27.86 is the 9th highest/largest number

25.51 is the 10th highest/largest number

16.58 is the 11th highest/largest number

16.58 is the 12th highest/largest number

14.70 is the 13th highest/largest number

13.29 is the 14th 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 13.29 14.70 16.58 16.58 25.51 27.86 33.03 34.44 39.14 39.61 42.43 44.31 47.13 57.47
Rank 1 2 3 4 5 6 7 8 9 10 11 12 13 14

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

Since we have 14 numbers in our original number set,
we assign ranks from lowest to highest (1 to 14)

Our original number set in unsorted order was 13.29,14.70,16.58,16.58,25.51,27.86,33.03,34.44,39.14,39.61,42.43,44.31,47.13,57.47

Our respective ranked data set is 1,2,4,4,5,6,7,8,9,10,11,12,13,14

Root Mean Square Calculation

Root Mean Square =	√A
	√N

where A = x₁² + x₂² + x₃² + x₄² + x₅² + x₆² + x₇² + x₈² + x₉² + x₁₀² + x₁₁² + x₁₂² + x₁₃² + x₁₄² and N = 14 number set items

Calculate A

A = 13.29² + 14.70² + 16.58² + 16.58² + 25.51² + 27.86² + 33.03² + 34.44² + 39.14² + 39.61² + 42.43² + 44.31² + 47.13² + 57.47²

A = 176.6241 + 216.09 + 274.8964 + 274.8964 + 650.7601 + 776.1796 + 1090.9809 + 1186.1136 + 1531.9396 + 1568.9521 + 1800.3049 + 1963.3761 + 2221.2369 + 3302.8009

A = 17035.1516

Calculate Root Mean Square (RMS):

RMS =	√17035.1516
	√14

RMS =	130.51877872552
	3.7416573867739

RMS = 34.882610895074

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 =	ΣX_i
	n

μ =	13.29 + 14.70 + 16.58 + 16.58 + 25.51 + 27.86 + 33.03 + 34.44 + 39.14 + 39.61 + 42.43 + 44.31 + 47.13 + 57.47
	14

μ =	452.08
	14

μ = 32.291428571429

Calculate the Median (Middle Value)

Since our number set contains 14 elements which is an even number,
our median number is determined as follows

Number Set = (n₁,n₂,n₃,n₄,n₅,n₆,n₇,n₈,n₉,n₁₀,n₁₁,n₁₂,n₁₃,n₁₄)

Median Number 1 = ½(n)

Median Number 1 = ½(14)

Median Number 1 = Number Set Entry 7

Median Number 2 = Median Number 1 + 1

Median Number 2 = Number Set Entry 7 + 1

Median Number 2 = Number Set Entry 8

For an even number set, we average the 2 median number entries:

Median = ½(n₇ + n₈)

Therefore, we sort our number set in ascending order

Our median is the average of entry 7 and entry 8 of our number set highlighted in red:

(13.29,14.70,16.58,16.58,25.51,27.86,33.03,34.44,39.14,39.61,42.43,44.31,47.13,57.47)

Median = ½(33.03 + 34.44)

Median = ½(67.47)

Median = 33.735

Calculate the Mode - Highest Frequency Number

()

Our mode is denoted as: 16.58

Mode = 16.58

Calculate Harmonic Mean:

Harmonic Mean =	N
	1/x₁ + 1/x₂ + 1/x₃ + 1/x₄ + 1/x₅ + 1/x₆ + 1/x₇ + 1/x₈ + 1/x₉ + 1/x₁₀ + 1/x₁₁ + 1/x₁₂ + 1/x₁₃ + 1/x₁₄

With N = 14 and each x_i a member of the number set you entered, we have:

Harmonic Mean =	14
	1/13.29 + 1/14.70 + 1/16.58 + 1/16.58 + 1/25.51 + 1/27.86 + 1/33.03 + 1/34.44 + 1/39.14 + 1/39.61 + 1/42.43 + 1/44.31 + 1/47.13 + 1/57.47

Harmonic Mean =	14
	0.075244544770504 + 0.068027210884354 + 0.060313630880579 + 0.060313630880579 + 0.039200313602509 + 0.035893754486719 + 0.030275507114744 + 0.029036004645761 + 0.025549310168625 + 0.025246149962131 + 0.023568230025925 + 0.022568269013767 + 0.02121790791428 + 0.017400382808422

Harmonic Mean =	14
	0.5338548471589

Harmonic Mean = 26.224356816288

Calculate Geometric Mean:

Geometric Mean = (x₁ * x₂ * x₃ * x₄ * x₅ * x₆ * x₇ * x₈ * x₉ * x₁₀ * x₁₁ * x₁₂ * x₁₃ * x₁₄)^1/N

Geometric Mean = (13.29 * 14.70 * 16.58 * 16.58 * 25.51 * 27.86 * 33.03 * 34.44 * 39.14 * 39.61 * 42.43 * 44.31 * 47.13 * 57.47)^1/14

Geometric Mean = 3.4277862923141E+20^{0.071428571428571}

Geometric Mean = 29.294540856618

Calculate Mid-Range:

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

Mid-Range =	57.47 + 13.29
	2

Mid-Range =	70.76
	2

Mid-Range = 35.38

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
5 7.47
4 2.43,4.31,7.13
3 3.03,4.44,9.14,9.61
2 5.51,7.86
1 3.29,4.70,6.58,6.58

Calculate Variance denoted as σ²

Let's evaluate the square difference from the mean of each term (X_i - μ)²:

(X₁ - μ)² = (13.29 - 32.291428571429)² = -19.001428571429² = 361.0542877551

(X₂ - μ)² = (14.70 - 32.291428571429)² = -17.591428571429² = 309.45835918367

(X₃ - μ)² = (16.58 - 32.291428571429)² = -15.711428571429² = 246.8489877551

(X₄ - μ)² = (16.58 - 32.291428571429)² = -15.711428571429² = 246.8489877551

(X₅ - μ)² = (25.51 - 32.291428571429)² = -6.7814285714286² = 45.987773469388

(X₆ - μ)² = (27.86 - 32.291428571429)² = -4.4314285714286² = 19.637559183674

(X₇ - μ)² = (33.03 - 32.291428571429)² = 0.73857142857143² = 0.54548775510204

(X₈ - μ)² = (34.44 - 32.291428571429)² = 2.1485714285714² = 4.6163591836734

(X₉ - μ)² = (39.14 - 32.291428571429)² = 6.8485714285714² = 46.902930612245

(X₁₀ - μ)² = (39.61 - 32.291428571429)² = 7.3185714285714² = 53.561487755102

(X₁₁ - μ)² = (42.43 - 32.291428571429)² = 10.138571428571² = 102.79063061224

(X₁₂ - μ)² = (44.31 - 32.291428571429)² = 12.018571428571² = 144.44605918367

(X₁₃ - μ)² = (47.13 - 32.291428571429)² = 14.838571428571² = 220.18320204082

(X₁₄ - μ)² = (57.47 - 32.291428571429)² = 25.178571428571² = 633.96045918367

Adding our 14 sum of squared differences up

ΣE(X_i - μ)² = 361.0542877551 + 309.45835918367 + 246.8489877551 + 246.8489877551 + 45.987773469388 + 19.637559183674 + 0.54548775510204 + 4.6163591836734 + 46.902930612245 + 53.561487755102 + 102.79063061224 + 144.44605918367 + 220.18320204082 + 633.96045918367

ΣE(X_i - μ)² = 2436.8425714286

Use the sum of squared differences to calculate variance

Population Sample
σ²  =   ΣE(X_i - μ)²
   n

σ²  =   ΣE(X_i - μ)²
   n - 1

σ²  =   2436.8425714286
   14

σ²  =   2436.8425714286
   13

Variance: σ_p² = 174.06018367347 Variance: σ_s² = 187.44942857143
Standard Deviation: σ_p = √σ_p² = √174.06018367347 Standard Deviation: σ_s = √σ_s² = √187.44942857143
Standard Deviation: σ_p = 13.1932 Standard Deviation: σ_s = 13.6912

Calculate the Standard Error of the Mean:

Population Sample
SEM  =   σ_p
   √n

SEM  =   σ_s
   √n

SEM  =   13.1932
   √14

SEM  =   13.6912
   √14

SEM  =   13.1932
   3.7416573867739

SEM  =   13.6912
   3.7416573867739

SEM = 3.526 SEM = 3.6591

Calculate Skewness:

Skewness =	E(X_i - μ)³
	(n - 1)σ³

Let's evaluate the square difference from the mean of each term (X_i - μ)³:

(X₁ - μ)³ = (13.29 - 32.291428571429)³ = -19.001428571429³ = -6860.5472591866

(X₂ - μ)³ = (14.70 - 32.291428571429)³ = -17.591428571429³ = -5443.8146214111

(X₃ - μ)³ = (16.58 - 32.291428571429)³ = -15.711428571429³ = -3878.3502390437

(X₄ - μ)³ = (16.58 - 32.291428571429)³ = -15.711428571429³ = -3878.3502390437

(X₅ - μ)³ = (25.51 - 32.291428571429)³ = -6.7814285714286³ = -311.86280094169

(X₆ - μ)³ = (27.86 - 32.291428571429)³ = -4.4314285714286³ = -87.02244083965

(X₇ - μ)³ = (33.03 - 32.291428571429)³ = 0.73857142857143³ = 0.40288167055393

(X₈ - μ)³ = (34.44 - 32.291428571429)³ = 2.1485714285714³ = 9.9185774460641

(X₉ - μ)³ = (39.14 - 32.291428571429)³ = 6.8485714285714³ = 321.21807050729

(X₁₀ - μ)³ = (39.61 - 32.291428571429)³ = 7.3185714285714³ = 391.99357395627

(X₁₁ - μ)³ = (42.43 - 32.291428571429)³ = 10.138571428571³ = 1042.1501506501

(X₁₂ - μ)³ = (44.31 - 32.291428571429)³ = 12.018571428571³ = 1736.0352798746

(X₁₃ - μ)³ = (47.13 - 32.291428571429)³ = 14.838571428571³ = 3267.2041708542

(X₁₄ - μ)³ = (57.47 - 32.291428571429)³ = 25.178571428571³ = 15962.218704446

Add our 14 sum of cubed differences up

ΣE(X_i - μ)³ = -6860.5472591866 + -5443.8146214111 + -3878.3502390437 + -3878.3502390437 + -311.86280094169 + -87.02244083965 + 0.40288167055393 + 9.9185774460641 + 321.21807050729 + 391.99357395627 + 1042.1501506501 + 1736.0352798746 + 3267.2041708542 + 15962.218704446

ΣE(X_i - μ)³ = 2271.1938089388

Calculate skewnes

Skewness =	E(X_i - μ)³
	(n - 1)σ³

Skewness =	2271.1938089388
	(14 - 1)13.1932³

Skewness =	2271.1938089388
	(13)2296.4153347896

Skewness =	2271.1938089388
	29853.399352264

Skewness = 0.076078230895554

Calculate Average Deviation (Mean Absolute Deviation) denoted below:

AD =	Σ\|X_i - μ\|
	n

Evaluate the absolute value of the difference from the mean

|X_i - μ|:

|X₁ - μ| = |13.29 - 32.291428571429| = |-19.001428571429| = 19.001428571429

|X₂ - μ| = |14.70 - 32.291428571429| = |-17.591428571429| = 17.591428571429

|X₃ - μ| = |16.58 - 32.291428571429| = |-15.711428571429| = 15.711428571429

|X₄ - μ| = |16.58 - 32.291428571429| = |-15.711428571429| = 15.711428571429

|X₅ - μ| = |25.51 - 32.291428571429| = |-6.7814285714286| = 6.7814285714286

|X₆ - μ| = |27.86 - 32.291428571429| = |-4.4314285714286| = 4.4314285714286

|X₇ - μ| = |33.03 - 32.291428571429| = |0.73857142857143| = 0.73857142857143

|X₈ - μ| = |34.44 - 32.291428571429| = |2.1485714285714| = 2.1485714285714

|X₉ - μ| = |39.14 - 32.291428571429| = |6.8485714285714| = 6.8485714285714

|X₁₀ - μ| = |39.61 - 32.291428571429| = |7.3185714285714| = 7.3185714285714

|X₁₁ - μ| = |42.43 - 32.291428571429| = |10.138571428571| = 10.138571428571

|X₁₂ - μ| = |44.31 - 32.291428571429| = |12.018571428571| = 12.018571428571

|X₁₃ - μ| = |47.13 - 32.291428571429| = |14.838571428571| = 14.838571428571

|X₁₄ - μ| = |57.47 - 32.291428571429| = |25.178571428571| = 25.178571428571

Average deviation numerator:

Σ|X_i - μ| = 19.001428571429 + 17.591428571429 + 15.711428571429 + 15.711428571429 + 6.7814285714286 + 4.4314285714286 + 0.73857142857143 + 2.1485714285714 + 6.8485714285714 + 7.3185714285714 + 10.138571428571 + 12.018571428571 + 14.838571428571 + 25.178571428571

Σ|X_i - μ| = 158.45714285714

Calculate average deviation (mean absolute deviation)

AD =	Σ\|X_i - μ\|
	n

AD =	158.45714285714
	14

Average Deviation = 11.31837

Calculate the Range

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

Range = 57.47 - 13.29

Range = 44.18

Calculate Pearsons Skewness Coefficient 1:

PSC1 =	μ - Mode
	σ

PSC1 =	3(32.291428571429 - 16.58)
	13.1932

PSC1 =	3 x 15.711428571429
	13.1932

PSC1 =	47.134285714286
	13.1932

PSC1 = 3.5726

Calculate Pearsons Skewness Coefficient 2:

PSC2 =	μ - Median
	σ

PSC1 =	3(32.291428571429 - 33.735)
	13.1932

PSC2 =	3 x -1.4435714285714
	13.1932

PSC2 =	-4.3307142857143
	13.1932

PSC2 = -0.3283

Calculate Entropy:

Entropy = Ln(n)

Entropy = Ln(14)

Entropy = 2.6390573296153

Calculate Mid-Range:

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

Mid-Range =	57.47 + 13.29
	2

Mid-Range =	70.76
	2

Mid-Range = 35.38

Calculate the Quartile Items

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

{13.29,14.70,16.58,16.58,25.51,27.86,33.03,34.44,39.14,39.61,42.43,44.31,47.13,57.47}

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

V =	y(n + 1)
	100

V =	75(14 + 1)
	100

V =	75(15)
	100

V =	1125
	100

V = 11 ← Rounded down to the nearest integer

Upper quartile (UQ) point = Point # 11 in the dataset which is 42.43

13.29,14.70,16.58,16.58,25.51,27.86,33.03,34.44,39.14,39.61,42.43,44.31,47.13,57.47

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

V =	y(n + 1)
	100

V =	25(14 + 1)
	100

V =	25(15)
	100

V =	375
	100

V = 4 ← Rounded up to the nearest integer

Lower quartile (LQ) point = Point # 4 in the dataset which is 16.58

13.29,14.70,16.58,16.58,25.51,27.86,33.03,34.44,39.14,39.61,42.43,44.31,47.13,57.47

Calculate Inter-Quartile Range (IQR):

IQR = UQ - LQ

IQR = 42.43 - 16.58

IQR = 25.85

Calculate Lower Inner Fence (LIF):

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

Lower Inner Fence (LIF) = 16.58 - 1.5 x 25.85

Lower Inner Fence (LIF) = 16.58 - 38.775

Lower Inner Fence (LIF) = -22.195

Calculate Upper Inner Fence (UIF):

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

Upper Inner Fence (UIF) = 42.43 + 1.5 x 25.85

Upper Inner Fence (UIF) = 42.43 + 38.775

Upper Inner Fence (UIF) = 81.205

Calculate Lower Outer Fence (LOF):

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

Lower Outer Fence (LOF) = 16.58 - 3 x 25.85

Lower Outer Fence (LOF) = 16.58 - 77.55

Lower Outer Fence (LOF) = -60.97

Calculate Upper Outer Fence (UOF):

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

Upper Outer Fence (UOF) = 42.43 + 3 x 25.85

Upper Outer Fence (UOF) = 42.43 + 77.55

Upper Outer Fence (UOF) = 119.98

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 -60.97 < v < -22.195 and 81.205 < v < 119.98

13.29,14.70,16.58,16.58,25.51,27.86,33.03,34.44,39.14,39.61,42.43,44.31,47.13,57.47

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 < -60.97 or v > 119.98

13.29,14.70,16.58,16.58,25.51,27.86,33.03,34.44,39.14,39.61,42.43,44.31,47.13,57.47

Calculate weighted average

13.29, 14.70, 16.58, 16.58, 25.51, 27.86, 33.03, 34.44, 39.14, 39.61, 42.43, 44.31, 47.13, 57.47

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 + X₇p₇ + X₈p₈ + X₉p₉ + X₁₀p₁₀ + X₁₁p₁₁ + X₁₂p₁₂ + X₁₃p₁₃ + X₁₄p₁₄
	n

Weighted Average =	13.29 x 0.2 + 14.70 x 0.4 + 16.58 x 0.6 + 16.58 x 0.8 + 25.51 x 0.9 + 27.86 x + 33.03 x + 34.44 x + 39.14 x + 39.61 x + 42.43 x + 44.31 x + 47.13 x + 57.47 x
	14

Weighted Average =	2.658 + 5.88 + 9.948 + 13.264 + 22.959 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0
	14

Weighted Average =	54.709
	14

Weighted Average = 3.9077857142857

Frequency Distribution Table

Show the freqency distribution table for this number set

13.29, 14.70, 16.58, 16.58, 25.51, 27.86, 33.03, 34.44, 39.14, 39.61, 42.43, 44.31, 47.13, 57.47

Determine the Number of Intervals using Sturges Rule:

Choose the smallest integer k such that 2^k ≥ n where n = 14

For k = 1, we have 2¹ = 2

For k = 2, we have 2² = 4

For k = 3, we have 2³ = 8

For k = 4, we have 2⁴ = 16 ← Use this since it is greater than our n value of 14

Therefore, we use 4 intervals

Our maximum value in our number set of 57.47 - 13.29 = 44.18

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

Interval Size =	44.18
	4

Add 1 to this giving us 11 + 1 = 12

Frequency Distribution Table

Class Limits Class Boundaries FD CFD RFD CRFD
13.29 - 25.29 12.79 - 25.79 4 4 4/14 = 28.57% 4/14 = 28.57%
25.29 - 37.29 24.79 - 37.79 4 4 + 4 = 8 4/14 = 28.57% 8/14 = 57.14%
37.29 - 49.29 36.79 - 49.79 5 4 + 4 + 5 = 13 5/14 = 35.71% 13/14 = 92.86%
49.29 - 61.29 48.79 - 61.79 1 4 + 4 + 5 + 1 = 14 1/14 = 7.14% 14/14 = 100%
14 100%

Successive Ratio Calculation

Go through our 14 numbers

Determine the ratio of each number to the next one

Successive Ratio 1: 13.29,14.70,16.58,16.58,25.51,27.86,33.03,34.44,39.14,39.61,42.43,44.31,47.13,57.47

13.29:14.70 → 0.9041

Successive Ratio 2: 13.29,14.70,16.58,16.58,25.51,27.86,33.03,34.44,39.14,39.61,42.43,44.31,47.13,57.47

14.70:16.58 → 0.8866

Successive Ratio 3: 13.29,14.70,16.58,16.58,25.51,27.86,33.03,34.44,39.14,39.61,42.43,44.31,47.13,57.47

16.58:16.58 → 1

Successive Ratio 4: 13.29,14.70,16.58,16.58,25.51,27.86,33.03,34.44,39.14,39.61,42.43,44.31,47.13,57.47

16.58:25.51 → 0.6499

Successive Ratio 5: 13.29,14.70,16.58,16.58,25.51,27.86,33.03,34.44,39.14,39.61,42.43,44.31,47.13,57.47

25.51:27.86 → 0.9156

Successive Ratio 6: 13.29,14.70,16.58,16.58,25.51,27.86,33.03,34.44,39.14,39.61,42.43,44.31,47.13,57.47

27.86:33.03 → 0.8435

Successive Ratio 7: 13.29,14.70,16.58,16.58,25.51,27.86,33.03,34.44,39.14,39.61,42.43,44.31,47.13,57.47

33.03:34.44 → 0.9591

Successive Ratio 8: 13.29,14.70,16.58,16.58,25.51,27.86,33.03,34.44,39.14,39.61,42.43,44.31,47.13,57.47

34.44:39.14 → 0.8799

Successive Ratio 9: 13.29,14.70,16.58,16.58,25.51,27.86,33.03,34.44,39.14,39.61,42.43,44.31,47.13,57.47

39.14:39.61 → 0.9881

Successive Ratio 10: 13.29,14.70,16.58,16.58,25.51,27.86,33.03,34.44,39.14,39.61,42.43,44.31,47.13,57.47

39.61:42.43 → 0.9335

Successive Ratio 11: 13.29,14.70,16.58,16.58,25.51,27.86,33.03,34.44,39.14,39.61,42.43,44.31,47.13,57.47

42.43:44.31 → 0.9576

Successive Ratio 12: 13.29,14.70,16.58,16.58,25.51,27.86,33.03,34.44,39.14,39.61,42.43,44.31,47.13,57.47

44.31:47.13 → 0.9402

Successive Ratio 13: 13.29,14.70,16.58,16.58,25.51,27.86,33.03,34.44,39.14,39.61,42.43,44.31,47.13,57.47

47.13:57.47 → 0.8201

Successive Ratio Answer

Successive Ratio = 13.29:14.70,14.70:16.58,16.58:16.58,16.58:25.51,25.51:27.86,27.86:33.03,33.03:34.44,34.44:39.14,39.14:39.61,39.61:42.43,42.43:44.31,44.31:47.13,47.13:57.47 or 0.9041,0.8866,1,0.6499,0.9156,0.8435,0.9591,0.8799,0.9881,0.9335,0.9576,0.9402,0.8201

Number Set Value	13.29	14.70	16.58	16.58	25.51	27.86	33.03	34.44	39.14	39.61	42.43	44.31	47.13	57.47
Rank	1	2	3	4	5	6	7	8	9	10	11	12	13	14

Stem	Leaf
5	7.47
4	2.43,4.31,7.13
3	3.03,4.44,9.14,9.61
2	5.51,7.86
1	3.29,4.70,6.58,6.58

Class Limits	Class Boundaries	FD	CFD	RFD	CRFD
13.29 - 25.29	12.79 - 25.79	4	4	4/14 = 28.57%	4/14 = 28.57%
25.29 - 37.29	24.79 - 37.79	4	4 + 4 = 8	4/14 = 28.57%	8/14 = 57.14%
37.29 - 49.29	36.79 - 49.79	5	4 + 4 + 5 = 13	5/14 = 35.71%	13/14 = 92.86%
49.29 - 61.29	48.79 - 61.79	1	4 + 4 + 5 + 1 = 14	1/14 = 7.14%	14/14 = 100%
		14		100%

Answer

↓Steps Explained:↓

Sort Ascending from Lowest to Highest

Sort Descending from Highest to Lowest

Ranked Data Calculation

Ranked Data Table

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

Root Mean Square Calculation

Calculate A

Calculate Root Mean Square (RMS):

Central Tendency Calculation

Calculate Mean (Average) denoted as μ

Calculate the Median (Middle Value)

For an even number set, we average the 2 median number entries:

Therefore, we sort our number set in ascending order

Calculate the Mode - Highest Frequency Number

Calculate Harmonic Mean:

Calculate Geometric Mean:

Calculate Mid-Range:

Stem and Leaf Plot

Sort our number set in descending order:

Calculate Variance denoted as σ2

Adding our 14 sum of squared differences up

Use the sum of squared differences to calculate variance

Calculate the Standard Error of the Mean:

Calculate Skewness:

Add our 14 sum of cubed differences up

Calculate skewnes

Calculate Average Deviation (Mean Absolute Deviation) denoted below:

Average deviation numerator:

Calculate the Range

Calculate Pearsons Skewness Coefficient 1:

Calculate Pearsons Skewness Coefficient 2:

Calculate Entropy:

Calculate Mid-Range:

Calculate the Quartile Items

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

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

Calculate Inter-Quartile Range (IQR):

Calculate Lower Inner Fence (LIF):

Calculate Upper Inner Fence (UIF):

Calculate Lower Outer Fence (LOF):

Calculate Upper Outer Fence (UOF):

Calculate Suspect Outliers:

Calculate Highly Suspect Outliers:

Calculate weighted average

Weighted-Average Formula:

Frequency Distribution Table

Determine the Number of Intervals using Sturges Rule:

Frequency Distribution Table

Successive Ratio Calculation

Successive Ratio 1: 13.29,14.70,16.58,16.58,25.51,27.86,33.03,34.44,39.14,39.61,42.43,44.31,47.13,57.47

Successive Ratio 2: 13.29,14.70,16.58,16.58,25.51,27.86,33.03,34.44,39.14,39.61,42.43,44.31,47.13,57.47

Successive Ratio 3: 13.29,14.70,16.58,16.58,25.51,27.86,33.03,34.44,39.14,39.61,42.43,44.31,47.13,57.47

Successive Ratio 4: 13.29,14.70,16.58,16.58,25.51,27.86,33.03,34.44,39.14,39.61,42.43,44.31,47.13,57.47

Successive Ratio 5: 13.29,14.70,16.58,16.58,25.51,27.86,33.03,34.44,39.14,39.61,42.43,44.31,47.13,57.47

Successive Ratio 6: 13.29,14.70,16.58,16.58,25.51,27.86,33.03,34.44,39.14,39.61,42.43,44.31,47.13,57.47

Successive Ratio 7: 13.29,14.70,16.58,16.58,25.51,27.86,33.03,34.44,39.14,39.61,42.43,44.31,47.13,57.47

Successive Ratio 8: 13.29,14.70,16.58,16.58,25.51,27.86,33.03,34.44,39.14,39.61,42.43,44.31,47.13,57.47

Successive Ratio 9: 13.29,14.70,16.58,16.58,25.51,27.86,33.03,34.44,39.14,39.61,42.43,44.31,47.13,57.47

Successive Ratio 10: 13.29,14.70,16.58,16.58,25.51,27.86,33.03,34.44,39.14,39.61,42.43,44.31,47.13,57.47

Successive Ratio 11: 13.29,14.70,16.58,16.58,25.51,27.86,33.03,34.44,39.14,39.61,42.43,44.31,47.13,57.47

Successive Ratio 12: 13.29,14.70,16.58,16.58,25.51,27.86,33.03,34.44,39.14,39.61,42.43,44.31,47.13,57.47

Successive Ratio 13: 13.29,14.70,16.58,16.58,25.51,27.86,33.03,34.44,39.14,39.61,42.43,44.31,47.13,57.47

Successive Ratio Answer

Final Answers

Calculate Variance denoted as σ²