Calculate Number Set Basics from 110,220,5

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You entered a number set X of {110,220,5}

From the 3 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

5, 110, 220

Rank Ascending

5 is the 1st lowest/smallest number

110 is the 2nd lowest/smallest number

220 is the 3rd lowest/smallest number

Sort Descending from Highest to Lowest

220, 110, 5

Rank Descending

220 is the 1st highest/largest number

110 is the 2nd highest/largest number

5 is the 3rd highest/largest number

Ranked Data Calculation

Sort our number set in ascending order

and assign a ranking to each number:

Ranked Data Table

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

Since we have 3 numbers in our original number set,
we assign ranks from lowest to highest (1 to 3)

Our original number set in unsorted order was 5,110,220

Our respective ranked data set is 1,2,3

Root Mean Square Calculation

where A = x₁² + x₂² + x₃² and N = 3 number set items

Calculate A

A = 5² + 110² + 220²

A = 25 + 12100 + 48400

A = 60525

Calculate Root Mean Square (RMS):

RMS = 142.03872711342

Central Tendency Calculation

Central tendency contains:
Mean, median, mode, harmonic mean,
geometric mean, mid-range, weighted-average:

Calculate Mean (Average) denoted as μ

μ = 111.66666666667

Calculate the Median (Middle Value)

Since our number set contains 3 elements which is an odd number,
our median number is determined as follows:

Number Set = (n₁,n₂,n₃)

Median Number = Entry ½(n + 1)

Median Number = Entry ½(4)

Median Number = n₂

Therefore, we sort our number set in ascending order

Our median is entry 2 of our number set highlighted in red:

(5,110,220)

Median = 110

Calculate the Mode - Highest Frequency Number

The highest frequency of occurence in our number set is 1 times
by the following numbers in green:

()

Since the maximum frequency of any number is 1, no mode exists.

Mode = N/A

Calculate Harmonic Mean:

With N = 3 and each x_i a member of the number set you entered, we have:

Harmonic Mean = 14.042553191489

Calculate Geometric Mean:

Geometric Mean = (x₁ * x₂ * x₃)^1/N

Geometric Mean = (5 * 110 * 220)^1/3

Geometric Mean = 121000^{0.33333333333333}

Geometric Mean = 49.460874432487

Calculate Mid-Range:

Mid-Range = 112.5

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:

{220,110,5}

Calculate Variance denoted as σ²

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

(X₁ - μ)² = (5 - 111.66666666667)² = -106.66666666667² = 11377.777777778

(X₂ - μ)² = (110 - 111.66666666667)² = -1.6666666666667² = 2.7777777777778

(X₃ - μ)² = (220 - 111.66666666667)² = 108.33333333333² = 11736.111111111

Adding our 3 sum of squared differences up

ΣE(X_i - μ)² = 11377.777777778 + 2.7777777777778 + 11736.111111111

ΣE(X_i - μ)² = 23116.666666667

Use the sum of squared differences to calculate variance

Calculate the Standard Error of the Mean:

Calculate Skewness:

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

(X₁ - μ)³ = (5 - 111.66666666667)³ = -106.66666666667³ = -1213629.6296296

(X₂ - μ)³ = (110 - 111.66666666667)³ = -1.6666666666667³ = -4.6296296296297

(X₃ - μ)³ = (220 - 111.66666666667)³ = 108.33333333333³ = 1271412.037037

Add our 3 sum of cubed differences up

ΣE(X_i - μ)³ = -1213629.6296296 + -4.6296296296297 + 1271412.037037

ΣE(X_i - μ)³ = 57777.777777777

Calculate skewnes

Skewness = 0.042709532118099

Calculate Average Deviation (Mean Absolute Deviation) denoted below:

Evaluate the absolute value of the difference from the mean

|X_i - μ|:

|X₁ - μ| = |5 - 111.66666666667| = |-106.66666666667| = 106.66666666667

|X₂ - μ| = |110 - 111.66666666667| = |-1.6666666666667| = 1.6666666666667

|X₃ - μ| = |220 - 111.66666666667| = |108.33333333333| = 108.33333333333

Average deviation numerator:

Σ|X_i - μ| = 106.66666666667 + 1.6666666666667 + 108.33333333333

Σ|X_i - μ| = 216.66666666667

Calculate average deviation (mean absolute deviation)

Average Deviation = 72.22222

Calculate the Range

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

Range = 220 - 5

Range = 215

Calculate Pearsons Skewness Coefficient 1:

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

Calculate Pearsons Skewness Coefficient 2:

PSC2 = 0.057

Calculate Entropy:

Entropy = Ln(n)

Entropy = Ln(3)

Entropy = 1.0986122886681

Calculate Mid-Range:

Mid-Range = 112.5

Calculate the Quartile Items

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

{5,110,220}

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

V = 3 ← Rounded down to the nearest integer

Upper quartile (UQ) point = Point # 3 in the dataset which is 220

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

V = 1 ← Rounded up to the nearest integer

Lower quartile (LQ) point = Point # 1 in the dataset which is 5

5,110,220

Calculate Inter-Quartile Range (IQR):

IQR = UQ - LQ

IQR = 220 - 5

IQR = 215

Calculate Lower Inner Fence (LIF):

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

Lower Inner Fence (LIF) = 5 - 1.5 x 215

Lower Inner Fence (LIF) = 5 - 322.5

Lower Inner Fence (LIF) = -317.5

Calculate Upper Inner Fence (UIF):

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

Upper Inner Fence (UIF) = 220 + 1.5 x 215

Upper Inner Fence (UIF) = 220 + 322.5

Upper Inner Fence (UIF) = 542.5

Calculate Lower Outer Fence (LOF):

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

Lower Outer Fence (LOF) = 5 - 3 x 215

Lower Outer Fence (LOF) = 5 - 645

Lower Outer Fence (LOF) = -640

Calculate Upper Outer Fence (UOF):

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

Upper Outer Fence (UOF) = 220 + 3 x 215

Upper Outer Fence (UOF) = 220 + 645

Upper Outer Fence (UOF) = 865

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 -640 < v < -317.5 and 542.5 < v < 865

5,110,220

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 < -640 or v > 865

5,110,220

Calculate weighted average

5, 110, 220

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 = 59

Frequency Distribution Table

Show the freqency distribution table for this number set

5, 110, 220

Determine the Number of Intervals using Sturges Rule:

We need to choose the smallest integer k such that 2^k ≥ n where n = 3

For k = 1, we have 2¹ = 2

For k = 2, we have 2² = 4 ← Use this since it is greater than our n value of 3

Therefore, we use 2 intervals

Our maximum value in our number set of 220 - 5 = 215

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

Add 1 to this giving us 107 + 1 = 108

Frequency Distribution Table

Successive Ratio Calculation

Go through our 3 numbers

Determine the ratio of each number to the next one

Successive Ratio 1: 5,110,220

5:110 → 0.0455

Successive Ratio 2: 5,110,220

110:220 → 0.5

Successive Ratio Answer

Successive Ratio = 5:110,110:220 or 0.0455,0.5

Final Answers