Calculate Number Set Basics from -6,4,3,-9,2,8

Crop Image

×

Crop

You entered a number set X of {-6,4,3,-9,2,8}

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

-9, -6, 2, 3, 4, 8

Rank Ascending

-9 is the 1st lowest/smallest number

-6 is the 2nd lowest/smallest number

2 is the 3rd lowest/smallest number

3 is the 4th lowest/smallest number

4 is the 5th lowest/smallest number

8 is the 6th lowest/smallest number

Sort Descending from Highest to Lowest

8, 4, 3, 2, -6, -9

Rank Descending

8 is the 1st highest/largest number

4 is the 2nd highest/largest number

3 is the 3rd highest/largest number

2 is the 4th highest/largest number

-6 is the 5th highest/largest number

-9 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

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 -9,-6,2,3,4,8

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

Root Mean Square Calculation

where A = x₁² + x₂² + x₃² + x₄² + x₅² + x₆² and N = 6 number set items

Calculate A

A = -9² + -6² + 2² + 3² + 4² + 8²

A = 81 + 36 + 4 + 9 + 16 + 64

A = 210

Calculate Root Mean Square (RMS):

RMS = 5.9160797830996

Central Tendency Calculation

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

Calculate Mean (Average) denoted as μ

μ = 0.33333333333333

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:

(-9,-6,2,3,4,8)

Median = ½(2 + 3)

Median = ½(5)

Median = 2.5

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 = 6 and each x_i a member of the number set you entered, we have:

Harmonic Mean = 6.4477611940299

Calculate Geometric Mean:

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

Geometric Mean = (-9 * -6 * 2 * 3 * 4 * 8)^1/6

Geometric Mean = 10368^{0.16666666666667}

Geometric Mean = 4.6696302973561

Calculate Mid-Range:

Mid-Range = -0.5

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:

{8,4,3,2,-6,-9}

Basic Stats Calculations

Mean, Variance, Standard Deviation, Median, Mode

Calculate Mean (Average) denoted as μ

μ = 0.33333333333333

Calculate Variance denoted as σ²

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

(X₁ - μ)² = (-9 - 0.33333333333333)² = -9.3333333333333² = 87.111111111111

(X₂ - μ)² = (-6 - 0.33333333333333)² = -6.3333333333333² = 40.111111111111

(X₃ - μ)² = (2 - 0.33333333333333)² = 1.6666666666667² = 2.7777777777778

(X₄ - μ)² = (3 - 0.33333333333333)² = 2.6666666666667² = 7.1111111111111

(X₅ - μ)² = (4 - 0.33333333333333)² = 3.6666666666667² = 13.444444444444

(X₆ - μ)² = (8 - 0.33333333333333)² = 7.6666666666667² = 58.777777777778

Adding our 6 sum of squared differences up

ΣE(X_i - μ)² = 87.111111111111 + 40.111111111111 + 2.7777777777778 + 7.1111111111111 + 13.444444444444 + 58.777777777778

ΣE(X_i - μ)² = 209.33333333333

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 = (-9 - 0.33333333333333)³ = -9.3333333333333³ = -813.03703703704

(X₂ - μ)³ = (-6 - 0.33333333333333)³ = -6.3333333333333³ = -254.03703703704

(X₃ - μ)³ = (2 - 0.33333333333333)³ = 1.6666666666667³ = 4.6296296296296

(X₄ - μ)³ = (3 - 0.33333333333333)³ = 2.6666666666667³ = 18.962962962963

(X₅ - μ)³ = (4 - 0.33333333333333)³ = 3.6666666666667³ = 49.296296296296

(X₆ - μ)³ = (8 - 0.33333333333333)³ = 7.6666666666667³ = 450.62962962963

Add our 6 sum of cubed differences up

ΣE(X_i - μ)³ = -813.03703703704 + -254.03703703704 + 4.6296296296296 + 18.962962962963 + 49.296296296296 + 450.62962962963

ΣE(X_i - μ)³ = -543.55555555556

Calculate skewnes

Skewness = -0.52752032030717

Calculate Average Deviation (Mean Absolute Deviation) denoted below:

Evaluate the absolute value of the difference from the mean

|X_i - μ|:

|X₁ - μ| = |-9 - 0.33333333333333| = |-9.3333333333333| = 9.3333333333333

|X₂ - μ| = |-6 - 0.33333333333333| = |-6.3333333333333| = 6.3333333333333

|X₃ - μ| = |2 - 0.33333333333333| = |1.6666666666667| = 1.6666666666667

|X₄ - μ| = |3 - 0.33333333333333| = |2.6666666666667| = 2.6666666666667

|X₅ - μ| = |4 - 0.33333333333333| = |3.6666666666667| = 3.6666666666667

|X₆ - μ| = |8 - 0.33333333333333| = |7.6666666666667| = 7.6666666666667

Average deviation numerator:

Σ|X_i - μ| = 9.3333333333333 + 6.3333333333333 + 1.6666666666667 + 2.6666666666667 + 3.6666666666667 + 7.6666666666667

Σ|X_i - μ| = 31.333333333333

Calculate average deviation (mean absolute deviation)

Average Deviation = 5.22222

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:

(-9,-6,2,3,4,8)

Median = ½(2 + 3)

Median = ½(5)

Median = 2.5

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 the Range

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

Range = 8 - -9

Range = 17

Calculate Pearsons Skewness Coefficient 1:

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

Calculate Pearsons Skewness Coefficient 2:

PSC2 = -1.1004

Calculate Entropy:

Entropy = Ln(n)

Entropy = Ln(6)

Entropy = 1.7917594692281

Calculate Mid-Range:

Mid-Range = -0.5

Calculate the Quartile Items

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

{-9,-6,2,3,4,8}

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

V = 5 ← Rounded down to the nearest integer

Upper quartile (UQ) point = Point # 5 in the dataset which is 4

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

V = 2 ← Rounded up to the nearest integer

Lower quartile (LQ) point = Point # 2 in the dataset which is -6

-9,-6,2,3,4,8

Calculate Inter-Quartile Range (IQR):

IQR = UQ - LQ

IQR = 4 - -6

IQR = 10

Calculate Lower Inner Fence (LIF):

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

Lower Inner Fence (LIF) = -6 - 1.5 x 10

Lower Inner Fence (LIF) = -6 - 15

Lower Inner Fence (LIF) = -21

Calculate Upper Inner Fence (UIF):

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

Upper Inner Fence (UIF) = 4 + 1.5 x 10

Upper Inner Fence (UIF) = 4 + 15

Upper Inner Fence (UIF) = 19

Calculate Lower Outer Fence (LOF):

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

Lower Outer Fence (LOF) = -6 - 3 x 10

Lower Outer Fence (LOF) = -6 - 30

Lower Outer Fence (LOF) = -36

Calculate Upper Outer Fence (UOF):

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

Upper Outer Fence (UOF) = 4 + 3 x 10

Upper Outer Fence (UOF) = 4 + 30

Upper Outer Fence (UOF) = 34

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 -36 < v < -21 and 19 < v < 34

-9,-6,2,3,4,8

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 < -36 or v > 34

-9,-6,2,3,4,8

Calculate weighted average

-9, -6, 2, 3, 4, 8

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 = 0.5

Frequency Distribution Table

Show the freqency distribution table for this number set

-9, -6, 2, 3, 4, 8

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 8 - -9 = 17

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

Add 1 to this giving us 5 + 1 = 6

Frequency Distribution Table

Successive Ratio Calculation

Go through our 6 numbers

Determine the ratio of each number to the next one

Successive Ratio 1: -9,-6,2,3,4,8

-9:-6 → 1.5

Successive Ratio 2: -9,-6,2,3,4,8

-6:2 → -3

Successive Ratio 3: -9,-6,2,3,4,8

2:3 → 0.6667

Successive Ratio 4: -9,-6,2,3,4,8

3:4 → 0.75

Successive Ratio 5: -9,-6,2,3,4,8

4:8 → 0.5

Successive Ratio Answer

Successive Ratio = -9:-6,-6:2,2:3,3:4,4:8 or 1.5,-3,0.6667,0.75,0.5

Final Answers