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

<-- Enter Number Set
<-- Probabilities (or counts for Weighted Average), check box if you are using these →

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

## Calculate Mean (Average) denoted as μ

 μ  = Sum of your number Set Total Numbers Entered

 μ  = ΣXi n

 μ  = -6 + 4 + 3 + -9 + 2 + 8 6

 μ  = 2 6

μ = 0.33333333333333

Calculate Variance denoted as σ2
Let's evaluate the square difference from the mean of each term (Xi - μ)2:
(X1 - μ)2 = (-6 - 0.33333333333333)2 = -6.33333333333332 = 40.111111111111
(X2 - μ)2 = (4 - 0.33333333333333)2 = 3.66666666666672 = 13.444444444444
(X3 - μ)2 = (3 - 0.33333333333333)2 = 2.66666666666672 = 7.1111111111111
(X4 - μ)2 = (-9 - 0.33333333333333)2 = -9.33333333333332 = 87.111111111111
(X5 - μ)2 = (2 - 0.33333333333333)2 = 1.66666666666672 = 2.7777777777778
(X6 - μ)2 = (8 - 0.33333333333333)2 = 7.66666666666672 = 58.777777777778

Adding our 6 sum of squared differences up, we have our variance numerator:
ΣE(Xi - μ)2 = 40.111111111111 + 13.444444444444 + 7.1111111111111 + 87.111111111111 + 2.7777777777778 + 58.777777777778
ΣE(Xi - μ)2 = 209.33333333333

Now that we have the sum of squared differences from the means, calculate variance:
PopulationSample
 σ2  = ΣE(Xi - μ)2 n

 σ2  = ΣE(Xi - μ)2 n - 1

 σ2  = 209.333 6

 σ2  = 209.333 5

Variance: σp2 = 34.888888888889Variance: σs2 = 41.866666666667
Standard Deviation: σp = √σp2 = √34.888888888889Standard Deviation: σs = √σs2 = √41.866666666667
Standard Deviation: σp = 5.9067Standard Deviation: σs = 6.4704

## Calculate the Standard Error of the Mean:

PopulationSample
 SEM  = σp √n

 SEM  = σs √n

 SEM  = 5.9067 √6

 SEM  = 6.4704 √6

 SEM  = 5.9067 2.44949

 SEM  = 6.4704 2.44949

SEM = 2.4114SEM = 2.6415

## 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 = (-6 - 0.33333333333333)3 = -6.33333333333333 = -254.03703703704
(X2 - μ)3 = (4 - 0.33333333333333)3 = 3.66666666666673 = 49.296296296296
(X3 - μ)3 = (3 - 0.33333333333333)3 = 2.66666666666673 = 18.962962962963
(X4 - μ)3 = (-9 - 0.33333333333333)3 = -9.33333333333333 = -813.03703703704
(X5 - μ)3 = (2 - 0.33333333333333)3 = 1.66666666666673 = 4.6296296296296
(X6 - μ)3 = (8 - 0.33333333333333)3 = 7.66666666666673 = 450.62962962963

Adding our 6 sum of cubed differences up, we have our skewness numerator:
ΣE(Xi - μ)3 = -254.03703703704 + 49.296296296296 + 18.962962962963 + -813.03703703704 + 4.6296296296296 + 450.62962962963
ΣE(Xi - μ)3 = -543.55555555556

Now that we have the sum of cubed differences from the means, calculate skewness:
 Skewness  = E(Xi - μ)3 (n - 1)σ3

 Skewness  = -543.55555555556 (6 - 1)5.90673

 Skewness  = -543.55555555556 (5)206.07947585376

 Skewness  = -543.556 1030.4

Skewness = -0.52752032030717

## 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 |Xi - μ|:
|X1 - μ| = |-6 - 0.33333333333333| = |-6.3333333333333| = 6.3333333333333
|X2 - μ| = |4 - 0.33333333333333| = |3.6666666666667| = 3.6666666666667
|X3 - μ| = |3 - 0.33333333333333| = |2.6666666666667| = 2.6666666666667
|X4 - μ| = |-9 - 0.33333333333333| = |-9.3333333333333| = 9.3333333333333
|X5 - μ| = |2 - 0.33333333333333| = |1.6666666666667| = 1.6666666666667
|X6 - μ| = |8 - 0.33333333333333| = |7.6666666666667| = 7.6666666666667

Adding our 6 absolute value of differences from the mean, we have our average deviation numerator:
Σ|Xi - μ| = 6.3333333333333 + 3.6666666666667 + 2.6666666666667 + 9.3333333333333 + 1.6666666666667 + 7.6666666666667
Σ|Xi - μ| = 31.333333333333

Now that we have the absolute value of the differences from the means, calculate average deviation (mean absolute deviation):
 AD  = Σ|Xi - μ| n

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

 PSC1  = μ - Mode σ

 PSC1  = 3(0.33333333333333 - N/A) 5.9067

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

## Calculate Pearsons Skewness Coefficient 2:

 PSC2  = μ - Median σ

 PSC1  = 3(0.33333333333333 - 2.5) 5.9067

 PSC2  = 3 x -2.1666666666667 5.9067

 PSC2  = -6.5 5.9067

PSC2 = -1.1004

## 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  = 8 + -9 2

 Mid-Range  = -1 2

Mid-Range = -0.5

### How does the Basic Statistics Calculator work?

Free Basic Statistics Calculator - Given a number set, and an optional probability set, this calculates the following statistical items:
Expected Value
Mean = μ
Variance = σ2
Standard Deviation = σ
Standard Error of the Mean
Skewness
Mid-Range
Average Deviation (Mean Absolute Deviation)
Median
Mode
Range
Pearsons Skewness Coefficients
Entropy
Upper Quartile (hinge) (75th Percentile)
Lower Quartile (hinge) (25th Percentile)
InnerQuartile Range
Inner Fences (Lower Inner Fence and Upper Inner Fence)
Outer Fences (Lower Outer Fence and Upper Outer Fence)
Suspect Outliers
Highly Suspect Outliers
Stem and Leaf Plot
Ranked Data Set
Central Tendency Items such as Harmonic Mean and Geometric Mean and Mid-Range
Root Mean Square
Weighted Average (Weighted Mean)
Frequency Distribution
Successive Ratio
This calculator has 2 inputs.

### What 8 formulas are used for the Basic Statistics Calculator?

1. Root Mean Square = √A/√N
2. Successive Ratio = n1/n0
3. μ = ΣXi/n
4. Mode = Highest Frequency Number
5. Mid-Range = (Maximum Value in Number Set + Minimum Value in Number Set)/2
6. Quartile: V = y(n + 1)/100
7. σ2 = ΣE(Xi - μ)2/n

For more math formulas, check out our Formula Dossier

### What 20 concepts are covered in the Basic Statistics Calculator?

average deviation
Mean of the absolute values of the distance from the mean for each number in a number set
basic statistics
central tendency
a central or typical value for a probability distribution. Typical measures are the mode, median, mean
entropy
refers to disorder or uncertainty
expected value
predicted value of a variable or event
E(X) = ΣxI · P(x)
frequency distribution
frequency measurement of various outcomes
inner fence
ut-off values for upper and lower outliers in a dataset
mean
A statistical measurement also known as the average
median
the value separating the higher half from the lower half of a data sample,
mode
the number that occurs the most in a number set
outer fence
start with the IQR and multiply this number by 3. We then subtract this number from the first quartile and add it to the third quartile. These two numbers are our outer fences.
outlier
an observation that lies an abnormal distance from other values in a random sample from a population
quartile
1 of 4 equal groups in the distribution of a number set
range
Difference between the largest and smallest values in a number set
rank
the data transformation in which numerical or ordinal values are replaced by their rank when the data are sorted.
sample space
the set of all possible outcomes or results of that experiment.
standard deviation
a measure of the amount of variation or dispersion of a set of values. The square root of variance
stem and leaf plot
a technique used to classify either discrete or continuous variables. A stem and leaf plot is used to organize data as they are collected. A stem and leaf plot looks something like a bar graph. Each number in the data is broken down into a stem and a leaf, thus the name.
variance
How far a set of random numbers are spead out from the mean
weighted average
An average of numbers using probabilities for each event as a weighting