Answer

1,2,3,4,5
RMS = 110.96936514192
Harmonic Mean = 106.44538844784Geometric Mean = 107.88399507889
Mid-Range = 113.5
Weighted Average = 67.92
Successive Ratio = Successive Ratio = 87:99,99:101,101:120,120:140 or 0.8788,0.9802,0.8417,0.8571

Steps Explained:

You entered a number set X of {120,99,101,87,140}

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

Sort Ascending from Lowest to Highest

87, 99, 101, 120, 140

Rank Ascending

87 is the 1st lowest/smallest number

99 is the 2nd lowest/smallest number

101 is the 3rd lowest/smallest number

120 is the 4th lowest/smallest number

140 is the 5th lowest/smallest number

Sort Descending from Highest to Lowest

140, 120, 101, 99, 87

Rank Descending

140 is the 1st highest/largest number

120 is the 2nd highest/largest number

101 is the 3rd highest/largest number

99 is the 4th highest/largest number

87 is the 5th 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 87 99 101 120 140
Rank 1 2 3 4 5

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

Since we have 5 numbers in our original number set,
we assign ranks from lowest to highest (1 to 5)

Our original number set in unsorted order was 87,99,101,120,140

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 = 87² + 99² + 101² + 120² + 140²

A = 7569 + 9801 + 10201 + 14400 + 19600

A = 61571

Calculate Root Mean Square (RMS):

RMS =	√61571
	√5

RMS =	248.13504387732
	2.2360679774998

RMS = 110.96936514192

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 =	87 + 99 + 101 + 120 + 140
	5

μ =	547
	5

μ = 109.4

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

Our median is entry 3 of our number set highlighted in red:

(87,99,101,120,140)

Median = 101

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

With N = 5 and each x_i a member of the number set you entered, we have:

Harmonic Mean =	5
	1/87 + 1/99 + 1/101 + 1/120 + 1/140

Harmonic Mean =	5
	0.011494252873563 + 0.01010101010101 + 0.0099009900990099 + 0.0083333333333333 + 0.0071428571428571

Harmonic Mean =	5
	0.046972443549774

Harmonic Mean = 106.44538844784

Calculate Geometric Mean:

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

Geometric Mean = (87 * 99 * 101 * 120 * 140)^1/5

Geometric Mean = 14614538400^0.2

Geometric Mean = 107.88399507889

Calculate Mid-Range:

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

Mid-Range =	140 + 87
	2

Mid-Range =	227
	2

Mid-Range = 113.5

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:

Stem Leaf
1 01,20,40
9 9
8 7

Calculate Variance denoted as σ²

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

(X₁ - μ)² = (87 - 109.4)² = -22.4² = 501.76

(X₂ - μ)² = (99 - 109.4)² = -10.4² = 108.16

(X₃ - μ)² = (101 - 109.4)² = -8.4² = 70.56

(X₄ - μ)² = (120 - 109.4)² = 10.6² = 112.36

(X₅ - μ)² = (140 - 109.4)² = 30.6² = 936.36

Adding our 5 sum of squared differences up

ΣE(X_i - μ)² = 501.76 + 108.16 + 70.56 + 112.36 + 936.36

ΣE(X_i - μ)² = 1729.2

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  =   1729.2
   5

σ²  =   1729.2
   4

Variance: σ_p² = 345.84 Variance: σ_s² = 432.3
Standard Deviation: σ_p = √σ_p² = √345.84 Standard Deviation: σ_s = √σ_s² = √432.3
Standard Deviation: σ_p = 18.5968 Standard Deviation: σ_s = 20.7918

Calculate the Standard Error of the Mean:

Population Sample
SEM  =   σ_p
   √n

SEM  =   σ_s
   √n

SEM  =   18.5968
   √5

SEM  =   20.7918
   √5

SEM  =   18.5968
   2.2360679774998

SEM  =   20.7918
   2.2360679774998

SEM = 8.3167 SEM = 9.2984

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 = (87 - 109.4)³ = -22.4³ = -11239.424

(X₂ - μ)³ = (99 - 109.4)³ = -10.4³ = -1124.864

(X₃ - μ)³ = (101 - 109.4)³ = -8.4³ = -592.704

(X₄ - μ)³ = (120 - 109.4)³ = 10.6³ = 1191.016

(X₅ - μ)³ = (140 - 109.4)³ = 30.6³ = 28652.616

Add our 5 sum of cubed differences up

ΣE(X_i - μ)³ = -11239.424 + -1124.864 + -592.704 + 1191.016 + 28652.616

ΣE(X_i - μ)³ = 16886.64

Calculate skewnes

Skewness =	E(X_i - μ)³
	(n - 1)σ³

Skewness =	16886.64
	(5 - 1)18.5968³

Skewness =	16886.64
	(4)6431.5353553592

Skewness =	16886.64
	25726.141421437

Skewness = 0.6564000299683

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| = |87 - 109.4| = |-22.4| = 22.4

|X₂ - μ| = |99 - 109.4| = |-10.4| = 10.4

|X₃ - μ| = |101 - 109.4| = |-8.4| = 8.4

|X₄ - μ| = |120 - 109.4| = |10.6| = 10.6

|X₅ - μ| = |140 - 109.4| = |30.6| = 30.6

Average deviation numerator:

Σ|X_i - μ| = 22.4 + 10.4 + 8.4 + 10.6 + 30.6

Σ|X_i - μ| = 82.4

Calculate average deviation (mean absolute deviation)

AD =	Σ\|X_i - μ\|
	n

AD =	82.4
	5

Average Deviation = 16.48

Calculate the Range

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

Range = 140 - 87

Range = 53

Calculate Pearsons Skewness Coefficient 1:

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

PSC1 =	3(109.4 - N/A)
	18.5968

Calculate Pearsons Skewness Coefficient 2:

PSC2 =	μ - Median
	σ

PSC1 =	3(109.4 - 101)
	18.5968

PSC2 =	3 x 8.4
	18.5968

PSC2 =	25.2
	18.5968

PSC2 = 1.3551

Calculate Entropy:

Entropy = Ln(n)

Entropy = Ln(5)

Entropy = 1.6094379124341

Calculate Mid-Range:

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

Mid-Range =	140 + 87
	2

Mid-Range =	227
	2

Mid-Range = 113.5

Calculate the Quartile Items

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

{87,99,101,120,140}

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 120

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 99

87,99,101,120,140

Calculate Inter-Quartile Range (IQR):

IQR = UQ - LQ

IQR = 120 - 99

IQR = 21

Calculate Lower Inner Fence (LIF):

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

Lower Inner Fence (LIF) = 99 - 1.5 x 21

Lower Inner Fence (LIF) = 99 - 31.5

Lower Inner Fence (LIF) = 67.5

Calculate Upper Inner Fence (UIF):

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

Upper Inner Fence (UIF) = 120 + 1.5 x 21

Upper Inner Fence (UIF) = 120 + 31.5

Upper Inner Fence (UIF) = 151.5

Calculate Lower Outer Fence (LOF):

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

Lower Outer Fence (LOF) = 99 - 3 x 21

Lower Outer Fence (LOF) = 99 - 63

Lower Outer Fence (LOF) = 36

Calculate Upper Outer Fence (UOF):

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

Upper Outer Fence (UOF) = 120 + 3 x 21

Upper Outer Fence (UOF) = 120 + 63

Upper Outer Fence (UOF) = 183

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 < 67.5 and 151.5 < v < 183

87,99,101,120,140

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 > 183

87,99,101,120,140

Calculate weighted average

87, 99, 101, 120, 140

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
	n

Weighted Average =	87 x 0.2 + 99 x 0.4 + 101 x 0.6 + 120 x 0.8 + 140 x 0.9
	5

Weighted Average =	17.4 + 39.6 + 60.6 + 96 + 126
	5

Weighted Average =	339.6
	5

Weighted Average = 67.92

Frequency Distribution Table

Show the freqency distribution table for this number set

87, 99, 101, 120, 140

Determine the Number of Intervals using Sturges Rule:

Choose the smallest integer k such that 2^k ≥ 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 140 - 87 = 53

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

Interval Size =	53
	3

Add 1 to this giving us 17 + 1 = 18

Frequency Distribution Table

Class Limits Class Boundaries FD CFD RFD CRFD
87 - 105 86.5 - 105.5 3 3 3/5 = 60% 3/5 = 60%
105 - 123 104.5 - 123.5 1 3 + 1 = 4 1/5 = 20% 4/5 = 80%
123 - 141 122.5 - 141.5 1 3 + 1 + 1 = 5 1/5 = 20% 5/5 = 100%
5 100%

Successive Ratio Calculation

Go through our 5 numbers

Determine the ratio of each number to the next one

Successive Ratio 1: 87,99,101,120,140

87:99 → 0.8788

Successive Ratio 2: 87,99,101,120,140

99:101 → 0.9802

Successive Ratio 3: 87,99,101,120,140

101:120 → 0.8417

Successive Ratio 4: 87,99,101,120,140

120:140 → 0.8571

Successive Ratio Answer

Successive Ratio = 87:99,99:101,101:120,120:140 or 0.8788,0.9802,0.8417,0.8571

Class Limits	Class Boundaries	FD	CFD	RFD	CRFD
87 - 105	86.5 - 105.5	3	3	3/5 = 60%	3/5 = 60%
105 - 123	104.5 - 123.5	1	3 + 1 = 4	1/5 = 20%	4/5 = 80%
123 - 141	122.5 - 141.5	1	3 + 1 + 1 = 5	1/5 = 20%	5/5 = 100%
		5		100%

Calculate Number Set Basics from 120,99,101,87,140

Enter Number Set (Comma Separated)

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)

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 5 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 5 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: 87,99,101,120,140

Successive Ratio 2: 87,99,101,120,140

Successive Ratio 3: 87,99,101,120,140

Successive Ratio 4: 87,99,101,120,140

Successive Ratio Answer

Final Answers

Calculate Variance denoted as σ²