l Calculate Number Set Basics from 2,15,15,18,30
Enter Number Set (Comma Separated)

2,15,15,18,30

Answer
1,3,3,4,5
RMS = 18.319388636087
Harmonic Mean = 6.9230769230769Geometric Mean = 11.943215116605
Mid-Range = 16
Weighted Average = 11.36
Successive Ratio = Successive Ratio = 2:15,15:15,15:18,18:30 or 0.1333,1,0.8333,0.6


Steps Explained:

You entered a number set X of {2,15,15,18,30}

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

2, 15, 15, 18, 30

Rank Ascending

2 is the 1st lowest/smallest number

15 is the 2nd lowest/smallest number

15 is the 3rd lowest/smallest number

18 is the 4th lowest/smallest number

30 is the 5th lowest/smallest number

Sort Descending from Highest to Lowest

30, 18, 15, 15, 2

Rank Descending

30 is the 1st highest/largest number

18 is the 2nd highest/largest number

15 is the 3rd highest/largest number

15 is the 4th highest/largest number

2 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 Value215151830
Rank12345

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 2,15,15,18,30

Our respective ranked data set is 1,3,3,4,5

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 = 22 + 152 + 152 + 182 + 302

A = 4 + 225 + 225 + 324 + 900

A = 1678

Calculate Root Mean Square (RMS):

RMS  =  1678
  5

RMS  =  40.963398296528
  2.2360679774998

RMS = 18.319388636087

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  =  2 + 15 + 15 + 18 + 30
  5

μ  =  80
  5

μ = 16

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:

(2,15,15,18,30)

Median = 15

Calculate the Mode - Highest Frequency Number

()

Our mode is denoted as: 15

Mode = 15

Calculate Harmonic Mean:

Harmonic Mean  =  N
  1/x1 + 1/x2 + 1/x3 + 1/x4 + 1/x5

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

Harmonic Mean  =  5
  1/2 + 1/15 + 1/15 + 1/18 + 1/30

Harmonic Mean  =  5
  0.5 + 0.066666666666667 + 0.066666666666667 + 0.055555555555556 + 0.033333333333333

Harmonic Mean  =  5
  0.72222222222222

Harmonic Mean = 6.9230769230769

Calculate Geometric Mean:

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

Geometric Mean = (2 * 15 * 15 * 18 * 30)1/5

Geometric Mean = 2430000.2

Geometric Mean = 11.943215116605

Calculate Mid-Range:

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

Mid-Range  =  30 + 2
  2

Mid-Range  =  32
  2

Mid-Range = 16

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
30
15,5,8
2

Calculate Variance denoted as σ2

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

(X1 - μ)2 = (2 - 16)2 = -142 = 196

(X2 - μ)2 = (15 - 16)2 = -12 = 1

(X3 - μ)2 = (15 - 16)2 = -12 = 1

(X4 - μ)2 = (18 - 16)2 = 22 = 4

(X5 - μ)2 = (30 - 16)2 = 142 = 196

Adding our 5 sum of squared differences up

ΣE(Xi - μ)2 = 196 + 1 + 1 + 4 + 196

ΣE(Xi - μ)2 = 398

Use the sum of squared differences to calculate variance

PopulationSample

σ2  =  ΣE(Xi - μ)2
  n

σ2  =  ΣE(Xi - μ)2
  n - 1

σ2  =  398
  5

σ2  =  398
  4

Variance: σp2 = 79.6Variance: σs2 = 99.5
Standard Deviation: σp = √σp2 = √79.6Standard Deviation: σs = √σs2 = √99.5
Standard Deviation: σp = 8.9219Standard Deviation: σs = 9.975

Calculate the Standard Error of the Mean:

PopulationSample

SEM  =  σp
  n

SEM  =  σs
  n

SEM  =  8.9219
  5

SEM  =  9.975
  5

SEM  =  8.9219
  2.2360679774998

SEM  =  9.975
  2.2360679774998

SEM = 3.99SEM = 4.461

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 = (2 - 16)3 = -143 = -2744

(X2 - μ)3 = (15 - 16)3 = -13 = -1

(X3 - μ)3 = (15 - 16)3 = -13 = -1

(X4 - μ)3 = (18 - 16)3 = 23 = 8

(X5 - μ)3 = (30 - 16)3 = 143 = 2744

Add our 5 sum of cubed differences up

ΣE(Xi - μ)3 = -2744 + -1 + -1 + 8 + 2744

ΣE(Xi - μ)3 = 6

Calculate skewnes

Skewness  =  E(Xi - μ)3
  (n - 1)σ3

Skewness  =  6
  (5 - 1)8.92193

Skewness  =  6
  (4)710.18591309046

Skewness  =  6
  2840.7436523618

Skewness = 0.0021121229981493

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| = |2 - 16| = |-14| = 14

|X2 - μ| = |15 - 16| = |-1| = 1

|X3 - μ| = |15 - 16| = |-1| = 1

|X4 - μ| = |18 - 16| = |2| = 2

|X5 - μ| = |30 - 16| = |14| = 14

Average deviation numerator:

Σ|Xi - μ| = 14 + 1 + 1 + 2 + 14

Σ|Xi - μ| = 32

Calculate average deviation (mean absolute deviation)

AD  =  Σ|Xi - μ|
  n

AD  =  32
  5

Average Deviation = 6.4

Calculate the Range

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

Range = 30 - 2

Range = 28

Calculate Pearsons Skewness Coefficient 1:

PSC1  =  μ - Mode
  σ

PSC1  =  3(16 - 15)
  8.9219

PSC1  =  3 x 1
  8.9219

PSC1  =  3
  8.9219

PSC1 = 0.3363

Calculate Pearsons Skewness Coefficient 2:

PSC2  =  μ - Median
  σ

PSC1  =  3(16 - 15)
  8.9219

PSC2  =  3 x 1
  8.9219

PSC2  =  3
  8.9219

PSC2 = 0.3363

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  =  30 + 2
  2

Mid-Range  =  32
  2

Mid-Range = 16

Calculate the Quartile Items

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

{2,15,15,18,30}

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 18

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 15

2,15,15,18,30

Calculate Inter-Quartile Range (IQR):

IQR = UQ - LQ

IQR = 18 - 15

IQR = 3

Calculate Lower Inner Fence (LIF):

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

Lower Inner Fence (LIF) = 15 - 1.5 x 3

Lower Inner Fence (LIF) = 15 - 4.5

Lower Inner Fence (LIF) = 10.5

Calculate Upper Inner Fence (UIF):

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

Upper Inner Fence (UIF) = 18 + 1.5 x 3

Upper Inner Fence (UIF) = 18 + 4.5

Upper Inner Fence (UIF) = 22.5

Calculate Lower Outer Fence (LOF):

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

Lower Outer Fence (LOF) = 15 - 3 x 3

Lower Outer Fence (LOF) = 15 - 9

Lower Outer Fence (LOF) = 6

Calculate Upper Outer Fence (UOF):

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

Upper Outer Fence (UOF) = 18 + 3 x 3

Upper Outer Fence (UOF) = 18 + 9

Upper Outer Fence (UOF) = 27

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 6 < v < 10.5 and 22.5 < v < 27

2,15,15,18,30

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 < 6 or v > 27

2,15,15,18,30

Calculate weighted average

2, 15, 15, 18, 30

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
  n

Weighted Average  =  2 x 0.2 + 15 x 0.4 + 15 x 0.6 + 18 x 0.8 + 30 x 0.9
  5

Weighted Average  =  0.4 + 6 + 9 + 14.4 + 27
  5

Weighted Average  =  56.8
  5

Weighted Average = 11.36

Frequency Distribution Table

Show the freqency distribution table for this number set

2, 15, 15, 18, 30

Determine the Number of Intervals using Sturges Rule:

Choose the smallest integer k such that 2k ≥ 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 30 - 2 = 28

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

Interval Size  =  28
  3

Add 1 to this giving us 9 + 1 = 10

Frequency Distribution Table

Class LimitsClass BoundariesFDCFDRFDCRFD
2 - 121.5 - 12.5111/5 = 20%1/5 = 20%
12 - 2211.5 - 22.531 + 3 = 43/5 = 60%4/5 = 80%
22 - 3221.5 - 32.511 + 3 + 1 = 51/5 = 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: 2,15,15,18,30

2:15 → 0.1333

Successive Ratio 2: 2,15,15,18,30

15:15 → 1

Successive Ratio 3: 2,15,15,18,30

15:18 → 0.8333

Successive Ratio 4: 2,15,15,18,30

18:30 → 0.6

Successive Ratio Answer

Successive Ratio = 2:15,15:15,15:18,18:30 or 0.1333,1,0.8333,0.6

Final Answers

1,3,3,4,5
RMS = 18.319388636087
Harmonic Mean = 6.9230769230769Geometric Mean = 11.943215116605
Mid-Range = 16
Weighted Average = 11.36
Successive Ratio = Successive Ratio = 2:15,15:15,15:18,18:30 or 0.1333,1,0.8333,0.6

Related Calculators:  Bernoulli Trials  |  Binomial Distribution  |  Geometric Distribution