Calculate Number Set Basics from 2,15,15,18,30

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

Calculate Mean (Average) denoted as μ

μ =	Sum of your number Set
	Total Numbers Entered

μ =	ΣX_i
	n

μ =	2 + 15 + 15 + 18 + 30
	5

μ =	80
	5

μ = 16

Calculate Variance denoted as σ²
Let's evaluate the square difference from the mean of each term (X_i - 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, we have our variance numerator:
ΣE(X_i - μ)² = 196 + 1 + 1 + 4 + 196
ΣE(X_i - μ)² = 398

Now that we have the sum of squared differences from the means, calculate variance:

Population

Sample

σ² =	ΣE(X_i - μ)²
	n

σ² =	ΣE(X_i - μ)²
	n - 1

σ² =	398
	5

σ² =	398
	4

Variance: σ_p² = 79.6

Variance: σ_s² = 99.5

Standard Deviation: σ_p = √σ_p² = √79.6

Standard Deviation: σ_s = √σ_s² = √99.5

Standard Deviation: σ_p = 8.9219

Standard Deviation: σ_s = 9.975

Calculate the Standard Error of the Mean:

Sample

SEM =	σ_p
	√n

SEM =	σ_s
	√n

SEM =	8.9219
	√5

SEM =	9.975
	√5

SEM =	8.9219
	2.2360679774998

SEM =	9.975
	2.2360679774998

SEM = 3.99

SEM = 4.461

Calculate Skewness:

Skewness =	E(X_i - μ)³
	(n - 1)σ³

Let's evaluate the square difference from the mean of each term (X_i - 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

Adding our 5 sum of cubed differences up, we have our skewness numerator:
ΣE(X_i - μ)³ = -2744 + -1 + -1 + 8 + 2744
ΣE(X_i - μ)³ = 6

Now that we have the sum of cubed differences from the means, calculate skewness:

Skewness =	E(X_i - 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\|X_i - μ\|
	n

Let's evaluate the absolute value of the difference from the mean of each term |X_i - 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

Adding our 5 absolute value of differences from the mean, we have our average deviation numerator:
Σ|X_i - μ| = 14 + 1 + 1 + 2 + 14
Σ|X_i - μ| = 32

Now that we have the absolute value of the differences from the means, calculate average deviation (mean absolute deviation):

AD =	Σ\|X_i - μ\|
	n

AD =	32
	5

Average Deviation = 6.4

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 and 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

The highest frequency of occurence in our number set is 2 times by the following numbers in green:
(2,15,15,18,30)
Our mode is denoted as: 15
Mode = 15

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

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?

Root Mean Square = √A/√N
Successive Ratio = n₁/n₀
μ = ΣX_i/n
Mode = Highest Frequency Number
Mid-Range = (Maximum Value in Number Set + Minimum Value in Number Set)/2
Quartile: V = y(n + 1)/100
σ² = ΣE(X_i - 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) = Σx_I · 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

Example calculations for the Basic Statistics Calculator

Basic Statistics Calculator Video

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