Calculate Number Set Basics from 87.4,86.9,89.9,78.3,75.1,70.6

<-- 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 {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

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 87.4,86.9,89.9,78.3,75.1,70.6
Our respective ranked data set is 5,4,6,3,2,1

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  =	ΣXi
	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 and 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

The highest frequency of occurence in our number set is 1 times by the following numbers in green:
(87.4,86.9,89.9,78.3,75.1,70.6)
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

Calcualte 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:

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

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

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

μ  =	488.2
	6

μ = 81.366666666667

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, we have our variance numerator:
ΣE(X_i - μ)² = 115.92111111111 + 39.271111111111 + 9.4044444444444 + 30.617777777778 + 36.401111111111 + 72.817777777778
ΣE(X_i - μ)² = 304.43333333333

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  =   304.43333333333    6	σ2  =   304.43333333333    5
Variance: σp2 = 50.738888888889	Variance: σs2 = 60.886666666667
Standard Deviation: σp = √σp2 = √50.738888888889	Standard Deviation: σs = √σs2 = √60.886666666667
Standard Deviation: σp = 7.1231	Standard Deviation: σs = 7.803

Calculate the Standard Error of the Mean:

Population	Sample
SEM  =   σp    √n	SEM  =   σs    √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(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 = (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

Adding our 6 sum of cubed differences up, we have our skewness numerator:
ΣE(X_i - μ)³ = -1248.083962963 + -246.09896296296 + -28.840296296296 + 169.41837037037 + 219.62003703704 + 621.37837037037
ΣE(X_i - μ)³ = -512.60644444444

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

Skewness  =	E(Xi - 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|Xi - μ|
	n

Let's evaluate the absolute value of the difference from the mean of each term |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

Adding our 6 absolute value of differences from the mean, we have our 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

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  =	40.2
	6

Average Deviation = 6.7

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:
(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

The highest frequency of occurence in our number set is 1 times by the following numbers in green:
(87.4,86.9,89.9,78.3,75.1,70.6)
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 = 89.9 - 70.6
Range = 19.3

Calculate Pearsons Skewness Coefficient 1:

PSC1  =	μ - Mode
	σ

PSC1  =	3(81.366666666667 - N/A)
	7.1231

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

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

87.4,86.9,89.9,78.3,75.1,70.6

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

Weighted Average  =	17.48 + 34.76 + 53.94 + 62.64 + 67.59 + 0
	6

Weighted Average  =	236.41
	6

Weighted Average = 39.401666666667

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:

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 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

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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