Answer

1,2,3,4,5,6
RMS = 81.677863178057
μ = 81.366666666667
Median = 82.6
Mode = N/A
Harmonic Mean = 80.727760502158Geometric Mean = 81.049352596614
Mid-Range = 80.25
σ_s² = 50.738888888889
σ = 7.1231
Weighted Average = 39.886666666667
Successive Ratio = Successive Ratio = 70.6:75.1,75.1:78.3,78.3:86.9,86.9:87.4,87.4:89.9 or 0.9401,0.9591,0.901,0.9943,0.9722

↓Steps Explained:↓

You entered a number set X of {87.4,86.9,89.9,78.3,75.1,70.6}

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

70.6, 75.1, 78.3, 86.9, 87.4, 89.9

Rank Ascending

70.6 is the 1st lowest/smallest number

75.1 is the 2nd lowest/smallest number

78.3 is the 3rd lowest/smallest number

86.9 is the 4th lowest/smallest number

87.4 is the 5th lowest/smallest number

89.9 is the 6th lowest/smallest number

Sort Descending from Highest to Lowest

89.9, 87.4, 86.9, 78.3, 75.1, 70.6

Rank Descending

89.9 is the 1st highest/largest number

87.4 is the 2nd highest/largest number

86.9 is the 3rd highest/largest number

78.3 is the 4th highest/largest number

75.1 is the 5th highest/largest number

70.6 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 70.6 75.1 78.3 86.9 87.4 89.9
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 70.6,75.1,78.3,86.9,87.4,89.9

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 = 70.6² + 75.1² + 78.3² + 86.9² + 87.4² + 89.9²

A = 4984.36 + 5640.01 + 6130.89 + 7551.61 + 7638.76 + 8082.01

A = 40027.64

Calculate Root Mean Square (RMS):

RMS =	√40027.64
	√6

RMS =	200.0690880671
	2.4494897427832

RMS = 81.677863178057

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 =	70.6 + 75.1 + 78.3 + 86.9 + 87.4 + 89.9
	6

μ =	488.2
	6

μ = 81.366666666667

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:

(70.6,75.1,78.3,86.9,87.4,89.9)

Median = ½(78.3 + 86.9)

Median = ½(165.2)

Median = 82.6

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
	1/70.6 + 1/75.1 + 1/78.3 + 1/86.9 + 1/87.4 + 1/89.9

Harmonic Mean =	6
	0.014164305949009 + 0.013315579227696 + 0.012771392081737 + 0.01150747986191 + 0.011441647597254 + 0.011123470522803

Harmonic Mean =	6
	0.074323875240409

Harmonic Mean = 80.727760502158

Calculate Geometric Mean:

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

Geometric Mean = (70.6 * 75.1 * 78.3 * 86.9 * 87.4 * 89.9)^1/6

Geometric Mean = 283463601663.17^{0.16666666666667}

Geometric Mean = 81.049352596614

Calculate Mid-Range:

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

Mid-Range =	89.9 + 70.6
	2

Mid-Range =	160.5
	2

Mid-Range = 80.25

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
8 6.9,7.4,9.9
7 0.6,5.1,8.3

Calculate Variance denoted as σ²

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

(X₁ - μ)² = (70.6 - 81.366666666667)² = -10.766666666667² = 115.92111111111

(X₂ - μ)² = (75.1 - 81.366666666667)² = -6.2666666666667² = 39.271111111111

(X₃ - μ)² = (78.3 - 81.366666666667)² = -3.0666666666667² = 9.4044444444444

(X₄ - μ)² = (86.9 - 81.366666666667)² = 5.5333333333333² = 30.617777777778

(X₅ - μ)² = (87.4 - 81.366666666667)² = 6.0333333333333² = 36.401111111111

(X₆ - μ)² = (89.9 - 81.366666666667)² = 8.5333333333333² = 72.817777777778

Adding our 6 sum of squared differences up

ΣE(X_i - μ)² = 115.92111111111 + 39.271111111111 + 9.4044444444444 + 30.617777777778 + 36.401111111111 + 72.817777777778

ΣE(X_i - μ)² = 304.43333333333

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  =   304.43333333333
   6

σ²  =   304.43333333333
   5

Variance: σ_p² = 50.738888888889 Variance: σ_s² = 60.886666666667
Standard Deviation: σ_p = √σ_p² = √50.738888888889 Standard Deviation: σ_s = √σ_s² = √60.886666666667
Standard Deviation: σ_p = 7.1231 Standard Deviation: σ_s = 7.803

Calculate the Standard Error of the Mean:

Population Sample
SEM  =   σ_p
   √n

SEM  =   σ_s
   √n

SEM  =   7.1231
   √6

SEM  =   7.803
   √6

SEM  =   7.1231
   2.4494897427832

SEM  =   7.803
   2.4494897427832

SEM = 2.908 SEM = 3.1856

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 = (70.6 - 81.366666666667)³ = -10.766666666667³ = -1248.083962963

(X₂ - μ)³ = (75.1 - 81.366666666667)³ = -6.2666666666667³ = -246.09896296296

(X₃ - μ)³ = (78.3 - 81.366666666667)³ = -3.0666666666667³ = -28.840296296296

(X₄ - μ)³ = (86.9 - 81.366666666667)³ = 5.5333333333333³ = 169.41837037037

(X₅ - μ)³ = (87.4 - 81.366666666667)³ = 6.0333333333333³ = 219.62003703704

(X₆ - μ)³ = (89.9 - 81.366666666667)³ = 8.5333333333333³ = 621.37837037037

Add our 6 sum of cubed differences up

ΣE(X_i - μ)³ = -1248.083962963 + -246.09896296296 + -28.840296296296 + 169.41837037037 + 219.62003703704 + 621.37837037037

ΣE(X_i - μ)³ = -512.60644444444

Calculate skewnes

Skewness =	E(X_i - μ)³
	(n - 1)σ³

Skewness =	-512.60644444444
	(6 - 1)7.1231³

Skewness =	-512.60644444444
	(5)361.41579121939

Skewness =	-512.60644444444
	1807.078956097

Skewness = -0.2836657705049

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| = |70.6 - 81.366666666667| = |-10.766666666667| = 10.766666666667

|X₂ - μ| = |75.1 - 81.366666666667| = |-6.2666666666667| = 6.2666666666667

|X₃ - μ| = |78.3 - 81.366666666667| = |-3.0666666666667| = 3.0666666666667

|X₄ - μ| = |86.9 - 81.366666666667| = |5.5333333333333| = 5.5333333333333

|X₅ - μ| = |87.4 - 81.366666666667| = |6.0333333333333| = 6.0333333333333

|X₆ - μ| = |89.9 - 81.366666666667| = |8.5333333333333| = 8.5333333333333

Average deviation numerator:

Σ|X_i - μ| = 10.766666666667 + 6.2666666666667 + 3.0666666666667 + 5.5333333333333 + 6.0333333333333 + 8.5333333333333

Σ|X_i - μ| = 40.2

Calculate average deviation (mean absolute deviation)

AD =	Σ\|X_i - μ\|
	n

AD =	40.2
	6

Average Deviation = 6.7

Calculate the Range

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

Range = 89.9 - 70.6

Range = 19.3

Calculate Pearsons Skewness Coefficient 1:

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

PSC1 =	3(81.366666666667 - N/A)
	7.1231

Calculate Pearsons Skewness Coefficient 2:

PSC2 =	μ - Median
	σ

PSC1 =	3(81.366666666667 - 82.6)
	7.1231

PSC2 =	3 x -1.2333333333333
	7.1231

PSC2 =	-3.7
	7.1231

PSC2 = -0.5194

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 =	89.9 + 70.6
	2

Mid-Range =	160.5
	2

Mid-Range = 80.25

Calculate the Quartile Items

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

{70.6,75.1,78.3,86.9,87.4,89.9}

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 87.4

70.6,75.1,78.3,86.9,87.4,89.9

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 75.1

70.6,75.1,78.3,86.9,87.4,89.9

Calculate Inter-Quartile Range (IQR):

IQR = UQ - LQ

IQR = 87.4 - 75.1

IQR = 12.3

Calculate Lower Inner Fence (LIF):

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

Lower Inner Fence (LIF) = 75.1 - 1.5 x 12.3

Lower Inner Fence (LIF) = 75.1 - 18.45

Lower Inner Fence (LIF) = 56.65

Calculate Upper Inner Fence (UIF):

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

Upper Inner Fence (UIF) = 87.4 + 1.5 x 12.3

Upper Inner Fence (UIF) = 87.4 + 18.45

Upper Inner Fence (UIF) = 105.85

Calculate Lower Outer Fence (LOF):

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

Lower Outer Fence (LOF) = 75.1 - 3 x 12.3

Lower Outer Fence (LOF) = 75.1 - 36.9

Lower Outer Fence (LOF) = 38.2

Calculate Upper Outer Fence (UOF):

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

Upper Outer Fence (UOF) = 87.4 + 3 x 12.3

Upper Outer Fence (UOF) = 87.4 + 36.9

Upper Outer Fence (UOF) = 124.3

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 38.2 < v < 56.65 and 105.85 < v < 124.3

70.6,75.1,78.3,86.9,87.4,89.9

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 < 38.2 or v > 124.3

70.6,75.1,78.3,86.9,87.4,89.9

Calculate weighted average

70.6, 75.1, 78.3, 86.9, 87.4, 89.9

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 =	70.6 x 0.2 + 75.1 x 0.4 + 78.3 x 0.6 + 86.9 x 0.8 + 87.4 x 0.9 + 89.9 x
	6

Weighted Average =	14.12 + 30.04 + 46.98 + 69.52 + 78.66 + 0
	6

Weighted Average =	239.32
	6

Weighted Average = 39.886666666667

Frequency Distribution Table

Show the freqency distribution table for this number set

70.6, 75.1, 78.3, 86.9, 87.4, 89.9

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 89.9 - 70.6 = 19.3

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

Interval Size =	19.3
	3

Add 1 to this giving us 6 + 1 = 7

Frequency Distribution Table

Class Limits Class Boundaries FD CFD RFD CRFD
70.6 - 77.6 70.1 - 78.1 2 2 2/6 = 33.33% 2/6 = 33.33%
77.6 - 84.6 77.1 - 85.1 1 2 + 1 = 3 1/6 = 16.67% 3/6 = 50%
84.6 - 91.6 84.1 - 92.1 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: 70.6,75.1,78.3,86.9,87.4,89.9

70.6:75.1 → 0.9401

Successive Ratio 2: 70.6,75.1,78.3,86.9,87.4,89.9

75.1:78.3 → 0.9591

Successive Ratio 3: 70.6,75.1,78.3,86.9,87.4,89.9

78.3:86.9 → 0.901

Successive Ratio 4: 70.6,75.1,78.3,86.9,87.4,89.9

86.9:87.4 → 0.9943

Successive Ratio 5: 70.6,75.1,78.3,86.9,87.4,89.9

87.4:89.9 → 0.9722

Successive Ratio Answer

Successive Ratio = 70.6:75.1,75.1:78.3,78.3:86.9,86.9:87.4,87.4:89.9 or 0.9401,0.9591,0.901,0.9943,0.9722

Class Limits	Class Boundaries	FD	CFD	RFD	CRFD
70.6 - 77.6	70.1 - 78.1	2	2	2/6 = 33.33%	2/6 = 33.33%
77.6 - 84.6	77.1 - 85.1	1	2 + 1 = 3	1/6 = 16.67%	3/6 = 50%
84.6 - 91.6	84.1 - 92.1	3	2 + 1 + 3 = 6	3/6 = 50%	6/6 = 100%
		6		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 6 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 6 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: 70.6,75.1,78.3,86.9,87.4,89.9

Successive Ratio 2: 70.6,75.1,78.3,86.9,87.4,89.9

Successive Ratio 3: 70.6,75.1,78.3,86.9,87.4,89.9

Successive Ratio 4: 70.6,75.1,78.3,86.9,87.4,89.9

Successive Ratio 5: 70.6,75.1,78.3,86.9,87.4,89.9

Successive Ratio Answer

Final Answers

Calculate Variance denoted as σ²