Answer
Success!
1,2,3,4,5,6,7
RMS = 24.796313089997
μ = 13.428571428571
Median = 7
Mode = N/A
Harmonic Mean = 3.1419347656685Geometric Mean = 5.691826851234
Mid-Range = 32.5
σs2 = 434.5306122449
σ = 20.8454
Weighted Average = 2.2285714285714
Successive Ratio = Successive Ratio = 1:2,2:3,3:7,7:8,8:9,9:64 or 0.5,0.6667,0.4286,0.875,0.8889,0.1406

↓Steps Explained:↓

You entered a number set X of {9,8,7,64,3,2,1}

From the 7 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

1, 2, 3, 7, 8, 9, 64

Rank Ascending

1 is the 1st lowest/smallest number

2 is the 2nd lowest/smallest number

3 is the 3rd lowest/smallest number

7 is the 4th lowest/smallest number

8 is the 5th lowest/smallest number

9 is the 6th lowest/smallest number

64 is the 7th lowest/smallest number

Sort Descending from Highest to Lowest

64, 9, 8, 7, 3, 2, 1

Rank Descending

64 is the 1st highest/largest number

9 is the 2nd highest/largest number

8 is the 3rd highest/largest number

7 is the 4th highest/largest number

3 is the 5th highest/largest number

2 is the 6th highest/largest number

1 is the 7th 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 Value12378964
Rank1234567

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

Since we have 7 numbers in our original number set,
we assign ranks from lowest to highest (1 to 7)

Our original number set in unsorted order was 1,2,3,7,8,9,64

Our respective ranked data set is 1,2,3,4,5,6,7

Root Mean Square Calculation

Root Mean Square  =  A
  N

where A = x12 + x22 + x32 + x42 + x52 + x62 + x72 and N = 7 number set items

Calculate A

A = 12 + 22 + 32 + 72 + 82 + 92 + 642

A = 1 + 4 + 9 + 49 + 64 + 81 + 4096

A = 4304

Calculate Root Mean Square (RMS):

RMS  =  4304
  7

RMS  =  65.604877867427
  2.6457513110646

RMS = 24.796313089997

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  =  1 + 2 + 3 + 7 + 8 + 9 + 64
  7

μ  =  94
  7

μ = 13.428571428571

Calculate the Median (Middle Value)

Since our number set contains 7 elements which is an odd number,
our median number is determined as follows:

Number Set = (n1,n2,n3,n4,n5,n6,n7)

Median Number = Entry ½(n + 1)

Median Number = Entry ½(8)

Median Number = n4

Therefore, we sort our number set in ascending order

Our median is entry 4 of our number set highlighted in red:

(1,2,3,7,8,9,64)

Median = 7

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 + 1/x7

With N = 7 and each xi a member of the number set you entered, we have:

Harmonic Mean  =  7
  1/1 + 1/2 + 1/3 + 1/7 + 1/8 + 1/9 + 1/64

Harmonic Mean  =  7
  1 + 0.5 + 0.33333333333333 + 0.14285714285714 + 0.125 + 0.11111111111111 + 0.015625

Harmonic Mean  =  7
  2.2279265873016

Harmonic Mean = 3.1419347656685

Calculate Geometric Mean:

Geometric Mean = (x1 * x2 * x3 * x4 * x5 * x6 * x7)1/N

Geometric Mean = (1 * 2 * 3 * 7 * 8 * 9 * 64)1/7

Geometric Mean = 1935360.14285714285714

Geometric Mean = 5.691826851234

Calculate Mid-Range:

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

Mid-Range  =  64 + 1
  2

Mid-Range  =  65
  2

Mid-Range = 32.5

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:

StemLeaf
64
9
8
7
3
2
1

Calculate Variance denoted as σ2

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

(X1 - μ)2 = (1 - 13.428571428571)2 = -12.4285714285712 = 154.4693877551

(X2 - μ)2 = (2 - 13.428571428571)2 = -11.4285714285712 = 130.61224489796

(X3 - μ)2 = (3 - 13.428571428571)2 = -10.4285714285712 = 108.75510204082

(X4 - μ)2 = (7 - 13.428571428571)2 = -6.42857142857142 = 41.326530612245

(X5 - μ)2 = (8 - 13.428571428571)2 = -5.42857142857142 = 29.469387755102

(X6 - μ)2 = (9 - 13.428571428571)2 = -4.42857142857142 = 19.612244897959

(X7 - μ)2 = (64 - 13.428571428571)2 = 50.5714285714292 = 2557.4693877551

Adding our 7 sum of squared differences up

ΣE(Xi - μ)2 = 154.4693877551 + 130.61224489796 + 108.75510204082 + 41.326530612245 + 29.469387755102 + 19.612244897959 + 2557.4693877551

ΣE(Xi - μ)2 = 3041.7142857143

Use the sum of squared differences to calculate variance

PopulationSample

σ2  =  ΣE(Xi - μ)2
  n

σ2  =  ΣE(Xi - μ)2
  n - 1

σ2  =  3041.7142857143
  7

σ2  =  3041.7142857143
  6

Variance: σp2 = 434.5306122449Variance: σs2 = 506.95238095238
Standard Deviation: σp = √σp2 = √434.5306122449Standard Deviation: σs = √σs2 = √506.95238095238
Standard Deviation: σp = 20.8454Standard Deviation: σs = 22.5156

Calculate the Standard Error of the Mean:

PopulationSample

SEM  =  σp
  n

SEM  =  σs
  n

SEM  =  20.8454
  7

SEM  =  22.5156
  7

SEM  =  20.8454
  2.6457513110646

SEM  =  22.5156
  2.6457513110646

SEM = 7.8788SEM = 8.5101

Calculate Skewness:

Skewness  =  E(Xi - μ)3
  (n - 1)σ3

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

(X1 - μ)3 = (1 - 13.428571428571)3 = -12.4285714285713 = -1919.833819242

(X2 - μ)3 = (2 - 13.428571428571)3 = -11.4285714285713 = -1492.7113702624

(X3 - μ)3 = (3 - 13.428571428571)3 = -10.4285714285713 = -1134.1603498542

(X4 - μ)3 = (7 - 13.428571428571)3 = -6.42857142857143 = -265.67055393586

(X5 - μ)3 = (8 - 13.428571428571)3 = -5.42857142857143 = -159.97667638484

(X6 - μ)3 = (9 - 13.428571428571)3 = -4.42857142857143 = -86.854227405248

(X7 - μ)3 = (64 - 13.428571428571)3 = 50.5714285714293 = 129334.88046647

Add our 7 sum of cubed differences up

ΣE(Xi - μ)3 = -1919.833819242 + -1492.7113702624 + -1134.1603498542 + -265.67055393586 + -159.97667638484 + -86.854227405248 + 129334.88046647

ΣE(Xi - μ)3 = 124275.67346939

Calculate skewnes

Skewness  =  E(Xi - μ)3
  (n - 1)σ3

Skewness  =  124275.67346939
  (7 - 1)20.84543

Skewness  =  124275.67346939
  (6)9057.9662779607

Skewness  =  124275.67346939
  54347.797667764

Skewness = 2.2866735875684

Calculate Average Deviation (Mean Absolute Deviation) denoted below:

AD  =  Σ|Xi - μ|
  n

Evaluate the absolute value of the difference from the mean

|Xi - μ|:

|X1 - μ| = |1 - 13.428571428571| = |-12.428571428571| = 12.428571428571

|X2 - μ| = |2 - 13.428571428571| = |-11.428571428571| = 11.428571428571

|X3 - μ| = |3 - 13.428571428571| = |-10.428571428571| = 10.428571428571

|X4 - μ| = |7 - 13.428571428571| = |-6.4285714285714| = 6.4285714285714

|X5 - μ| = |8 - 13.428571428571| = |-5.4285714285714| = 5.4285714285714

|X6 - μ| = |9 - 13.428571428571| = |-4.4285714285714| = 4.4285714285714

|X7 - μ| = |64 - 13.428571428571| = |50.571428571429| = 50.571428571429

Average deviation numerator:

Σ|Xi - μ| = 12.428571428571 + 11.428571428571 + 10.428571428571 + 6.4285714285714 + 5.4285714285714 + 4.4285714285714 + 50.571428571429

Σ|Xi - μ| = 101.14285714286

Calculate average deviation (mean absolute deviation)

AD  =  Σ|Xi - μ|
  n

AD  =  101.14285714286
  7

Average Deviation = 14.44898

Calculate the Range

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

Range = 64 - 1

Range = 63

Calculate Pearsons Skewness Coefficient 1:

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

PSC1  =  3(13.428571428571 - N/A)
  20.8454

Calculate Pearsons Skewness Coefficient 2:

PSC2  =  μ - Median
  σ

PSC1  =  3(13.428571428571 - 7)
  20.8454

PSC2  =  3 x 6.4285714285714
  20.8454

PSC2  =  19.285714285714
  20.8454

PSC2 = 0.9252

Calculate Entropy:

Entropy = Ln(n)

Entropy = Ln(7)

Entropy = 1.9459101490553

Calculate Mid-Range:

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

Mid-Range  =  64 + 1
  2

Mid-Range  =  65
  2

Mid-Range = 32.5

Calculate the Quartile Items

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

{1,2,3,7,8,9,64}

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

V  =  y(n + 1)
  100

V  =  75(7 + 1)
  100

V  =  75(8)
  100

V  =  600
  100

V = 6 ← Rounded down to the nearest integer

Upper quartile (UQ) point = Point # 6 in the dataset which is 9

1,2,3,7,8,9,64

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

V  =  y(n + 1)
  100

V  =  25(7 + 1)
  100

V  =  25(8)
  100

V  =  200
  100

V = 2 ← Rounded up to the nearest integer

Lower quartile (LQ) point = Point # 2 in the dataset which is 2

1,2,3,7,8,9,64

Calculate Inter-Quartile Range (IQR):

IQR = UQ - LQ

IQR = 9 - 2

IQR = 7

Calculate Lower Inner Fence (LIF):

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

Lower Inner Fence (LIF) = 2 - 1.5 x 7

Lower Inner Fence (LIF) = 2 - 10.5

Lower Inner Fence (LIF) = -8.5

Calculate Upper Inner Fence (UIF):

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

Upper Inner Fence (UIF) = 9 + 1.5 x 7

Upper Inner Fence (UIF) = 9 + 10.5

Upper Inner Fence (UIF) = 19.5

Calculate Lower Outer Fence (LOF):

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

Lower Outer Fence (LOF) = 2 - 3 x 7

Lower Outer Fence (LOF) = 2 - 21

Lower Outer Fence (LOF) = -19

Calculate Upper Outer Fence (UOF):

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

Upper Outer Fence (UOF) = 9 + 3 x 7

Upper Outer Fence (UOF) = 9 + 21

Upper Outer Fence (UOF) = 30

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 -19 < v < -8.5 and 19.5 < v < 30

1,2,3,7,8,9,64

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 < -19 or v > 30

1,2,3,7,8,9,64

Calculate weighted average

1, 2, 3, 7, 8, 9, 64

Weighted-Average Formula:

Multiply each value by each probability amount

We do this by multiplying each Xi x pi to get a weighted score Y

Weighted Average  =  X1p1 + X2p2 + X3p3 + X4p4 + X5p5 + X6p6 + X7p7
  n

Weighted Average  =  1 x 0.2 + 2 x 0.4 + 3 x 0.6 + 7 x 0.8 + 8 x 0.9 + 9 x + 64 x
  7

Weighted Average  =  0.2 + 0.8 + 1.8 + 5.6 + 7.2 + 0 + 0
  7

Weighted Average  =  15.6
  7

Weighted Average = 2.2285714285714

Frequency Distribution Table

Show the freqency distribution table for this number set

1, 2, 3, 7, 8, 9, 64

Determine the Number of Intervals using Sturges Rule:

Choose the smallest integer k such that 2k ≥ n where n = 7

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 7

Therefore, we use 3 intervals

Our maximum value in our number set of 64 - 1 = 63

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

Interval Size  =  63
  3

Add 1 to this giving us 21 + 1 = 22

Frequency Distribution Table

Class LimitsClass BoundariesFDCFDRFDCRFD
1 - 230.5 - 23.5666/7 = 85.71%6/7 = 85.71%
23 - 4522.5 - 45.56 + = 6/7 = 0%6/7 = 85.71%
45 - 6744.5 - 67.516 + + 1 = 71/7 = 14.29%7/7 = 100%
  7 100% 

Successive Ratio Calculation

Go through our 7 numbers

Determine the ratio of each number to the next one

Successive Ratio 1: 1,2,3,7,8,9,64

1:2 → 0.5

Successive Ratio 2: 1,2,3,7,8,9,64

2:3 → 0.6667

Successive Ratio 3: 1,2,3,7,8,9,64

3:7 → 0.4286

Successive Ratio 4: 1,2,3,7,8,9,64

7:8 → 0.875

Successive Ratio 5: 1,2,3,7,8,9,64

8:9 → 0.8889

Successive Ratio 6: 1,2,3,7,8,9,64

9:64 → 0.1406

Successive Ratio Answer

Successive Ratio = 1:2,2:3,3:7,7:8,8:9,9:64 or 0.5,0.6667,0.4286,0.875,0.8889,0.1406

Final Answers

1,2,3,4,5,6,7
RMS = 24.796313089997
μ = 13.428571428571
Median = 7
Mode = N/A
Harmonic Mean = 3.1419347656685Geometric Mean = 5.691826851234
Mid-Range = 32.5
σs2 = 434.5306122449
σ = 20.8454
Weighted Average = 2.2285714285714
Successive Ratio = Successive Ratio = 1:2,2:3,3:7,7:8,8:9,9:64 or 0.5,0.6667,0.4286,0.875,0.8889,0.1406
Take the Quiz


Related Calculators:  Bernoulli Trials  |  Binomial Distribution  |  Geometric Distribution