You entered a number set X of {65,39,65,88,14,42}

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

Ranked Data Calculation

Sort our number set in ascending order

and assign a ranking to each number:

Ranked Data Table

Number Set Value	14	39	42	65	65	88
Rank	1	2	3	4	5	6

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 65,39,65,88,14,42
Our respective ranked data set is 5,2,5,6,1,3

Root Mean Square Calculation

Root Mean Square  =	√A
	√N

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

Calculate A

A = 142 + 392 + 422 + 652 + 652 + 882

A = 196 + 1521 + 1764 + 4225 + 4225 + 7744

A = 19675

Calculate Root Mean Square (RMS):

RMS  =	√19675
	√6

RMS  =	140.26760139106
	2.4494897427832

RMS = 57.264008475365

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  =	14 + 39 + 42 + 65 + 65 + 88
	6

μ  =	313
	6

μ = 52.166666666667

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:
(14,39,42,65,65,88)
Median = ½(42 + 65)
Median = ½(107)
Median = 53.5

Calculate the Mode - Highest Frequency Number

The highest frequency of occurence in our number set is 2 times by the following numbers in green:
(65,39,65,88,14,42)
Our mode is denoted as: 65
Mode = 65

Calculate Harmonic Mean:

Harmonic Mean  =	N
	1/x1 + 1/x2 + 1/x3 + 1/x4 + 1/x5 + 1/x6

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

Harmonic Mean  =	6
	1/14 + 1/39 + 1/42 + 1/65 + 1/65 + 1/88

Harmonic Mean  =	6
	0.071428571428571 + 0.025641025641026 + 0.023809523809524 + 0.015384615384615 + 0.015384615384615 + 0.011363636363636

Harmonic Mean  =	6
	0.16301198801199

Harmonic Mean = 36.807108932128

Calculate Geometric Mean:

Geometric Mean = (x1 * x2 * x3 * x4 * x5 * x6)1/N
Geometric Mean = (14 * 39 * 42 * 65 * 65 * 88)1/6
Geometric Mean = 85261176000.16666666666667
Geometric Mean = 45.198624639682

Calcualte Mid-Range:

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

Mid-Range  =	88 + 14
	2

Mid-Range  =	102
	2

Mid-Range = 51

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:

{88,65,65,42,39,14}

Stem	Leaf
8	8
6	5,5
4	2
3	9
1	4

Basic Stats Calculations

Mean, Variance, Standard Deviation, Median, Mode

Calculate Mean (Average) denoted as μ

μ  =	Sum of your number Set
	Total Numbers Entered

μ  =	ΣXi
	n

μ  =	14 + 39 + 42 + 65 + 65 + 88
	6

μ  =	313
	6

μ = 52.166666666667

Calculate Variance denoted as σ2
Let's evaluate the square difference from the mean of each term (Xi - μ)2:
(X1 - μ)2 = (14 - 52.166666666667)2 = -38.1666666666672 = 1456.6944444444
(X2 - μ)2 = (39 - 52.166666666667)2 = -13.1666666666672 = 173.36111111111
(X3 - μ)2 = (42 - 52.166666666667)2 = -10.1666666666672 = 103.36111111111
(X4 - μ)2 = (65 - 52.166666666667)2 = 12.8333333333332 = 164.69444444444
(X5 - μ)2 = (65 - 52.166666666667)2 = 12.8333333333332 = 164.69444444444
(X6 - μ)2 = (88 - 52.166666666667)2 = 35.8333333333332 = 1284.0277777778

Adding our 6 sum of squared differences up, we have our variance numerator:
ΣE(Xi - μ)2 = 1456.6944444444 + 173.36111111111 + 103.36111111111 + 164.69444444444 + 164.69444444444 + 1284.0277777778
ΣE(Xi - μ)2 = 3346.8333333333

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

Population	Sample
σ2  =   ΣE(Xi - μ)2    n	σ2  =   ΣE(Xi - μ)2    n - 1
σ2  =   3346.8333333333    6	σ2  =   3346.8333333333    5
Variance: σp2 = 557.80555555556	Variance: σs2 = 669.36666666667
Standard Deviation: σp = √σp2 = √557.80555555556	Standard Deviation: σs = √σs2 = √669.36666666667
Standard Deviation: σp = 23.6179	Standard Deviation: σs = 25.8721

Calculate the Standard Error of the Mean:

Population	Sample
SEM  =   σp    √n	SEM  =   σs    √n
SEM  =   23.6179    √6	SEM  =   25.8721    √6	SEM  =   23.6179    2.4494897427832	SEM  =   25.8721    2.4494897427832
SEM = 9.642	SEM = 10.5622

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 = (14 - 52.166666666667)3 = -38.1666666666673 = -55597.171296296
(X2 - μ)3 = (39 - 52.166666666667)3 = -13.1666666666673 = -2282.587962963
(X3 - μ)3 = (42 - 52.166666666667)3 = -10.1666666666673 = -1050.837962963
(X4 - μ)3 = (65 - 52.166666666667)3 = 12.8333333333333 = 2113.5787037037
(X5 - μ)3 = (65 - 52.166666666667)3 = 12.8333333333333 = 2113.5787037037
(X6 - μ)3 = (88 - 52.166666666667)3 = 35.8333333333333 = 46010.99537037

Adding our 6 sum of cubed differences up, we have our skewness numerator:
ΣE(Xi - μ)3 = -55597.171296296 + -2282.587962963 + -1050.837962963 + 2113.5787037037 + 2113.5787037037 + 46010.99537037
ΣE(Xi - μ)3 = -8692.4444444444

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

Skewness  =	E(Xi - μ)3
	(n - 1)σ3

Skewness  =	-8692.4444444444
	(6 - 1)23.61793

Skewness  =	-8692.4444444444
	(5)13174.187442763

Skewness  =	-8692.4444444444
	65870.937213817

Skewness = -0.13196175448709

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| = |14 - 52.166666666667| = |-38.166666666667| = 38.166666666667
|X2 - μ| = |39 - 52.166666666667| = |-13.166666666667| = 13.166666666667
|X3 - μ| = |42 - 52.166666666667| = |-10.166666666667| = 10.166666666667
|X4 - μ| = |65 - 52.166666666667| = |12.833333333333| = 12.833333333333
|X5 - μ| = |65 - 52.166666666667| = |12.833333333333| = 12.833333333333
|X6 - μ| = |88 - 52.166666666667| = |35.833333333333| = 35.833333333333

Adding our 6 absolute value of differences from the mean, we have our average deviation numerator:
Σ|Xi - μ| = 38.166666666667 + 13.166666666667 + 10.166666666667 + 12.833333333333 + 12.833333333333 + 35.833333333333
Σ|Xi - μ| = 123

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

AD  =	Σ|Xi - μ|
	n

AD  =	123
	6

Average Deviation = 20.5

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:
(14,39,42,65,65,88)
Median = ½(42 + 65)
Median = ½(107)
Median = 53.5

Calculate the Mode - Highest Frequency Number

The highest frequency of occurence in our number set is 2 times by the following numbers in green:
(65,39,65,88,14,42)
Our mode is denoted as: 65
Mode = 65

Calculate the Range

Range = Largest Number in the Number Set - Smallest Number in the Number Set
Range = 88 - 14
Range = 74

Calculate Pearsons Skewness Coefficient 1:

PSC1  =	μ - Mode
	σ

PSC1  =	3(52.166666666667 - 65)
	23.6179

PSC1  =	3 x -12.833333333333
	23.6179

PSC1  =	-38.5
	23.6179

PSC1 = -1.6301

Calculate Pearsons Skewness Coefficient 2:

PSC2  =	μ - Median
	σ

PSC1  =	3(52.166666666667 - 53.5)
	23.6179

PSC2  =	3 x -1.3333333333333
	23.6179

PSC2  =	-4
	23.6179

PSC2 = -0.1694Entropy = Ln(n)
Entropy = Ln(6)
Entropy = 1.7917594692281

Mid-Range  =	Smallest Number in the Set + Largest Number in the Set
	2

Mid-Range  =	88 + 14
	2

Mid-Range  =	102
	2

Mid-Range = 51

Calculate the Quartile Items

We need to sort our number set from lowest to highest shown below:
{14,39,42,65,65,88}

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

V  =	y(n + 1)
	100

V  =	75(6 + 1)
	100

V  =	75(7)
	100

V  =	525
	100

V = 5 ← Rounded down to the nearest integer

Upper quartile (UQ) point = Point # 5 in the dataset which is 65
14,39,42,65,65,88

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

V  =	y(n + 1)
	100

V  =	25(6 + 1)
	100

V  =	25(7)
	100

V  =	175
	100

V = 2 ← Rounded up to the nearest integer

Lower quartile (LQ) point = Point # 2 in the dataset which is 39
14,39,42,65,65,88

Calculate Inter-Quartile Range (IQR):

IQR = UQ - LQ
IQR = 65 - 39
IQR = 26

Calculate Lower Inner Fence (LIF):

Lower Inner Fence (LIF) = LQ - 1.5 x IQR
Lower Inner Fence (LIF) = 39 - 1.5 x 26
Lower Inner Fence (LIF) = 39 - 39
Lower Inner Fence (LIF) = 0

Calculate Upper Inner Fence (UIF):

Upper Inner Fence (UIF) = UQ + 1.5 x IQR
Upper Inner Fence (UIF) = 65 + 1.5 x 26
Upper Inner Fence (UIF) = 65 + 39
Upper Inner Fence (UIF) = 104

Calculate Lower Outer Fence (LOF):

Lower Outer Fence (LOF) = LQ - 3 x IQR
Lower Outer Fence (LOF) = 39 - 3 x 26
Lower Outer Fence (LOF) = 39 - 78
Lower Outer Fence (LOF) = -39

Calculate Upper Outer Fence (UOF):

Upper Outer Fence (UOF) = UQ + 3 x IQR
Upper Outer Fence (UOF) = 65 + 3 x 26
Upper Outer Fence (UOF) = 65 + 78
Upper Outer Fence (UOF) = 143

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 -39 < v < 0 and 104 < v < 143
14,39,42,65,65,88

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 < -39 or v > 143
14,39,42,65,65,88

Calculate weighted average

65,39,65,88,14,42

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 + X6p6
	n

Weighted Average  =	65 x + 39 x + 65 x + 88 x + 14 x + 42 x
	6

Weighted Average  =	0 + 0 + 0 + 0 + 0 + 0
	6

Weighted Average  =	0
	6

Weighted Average = 0

Frequency Distribution Table

Show the freqency distribution table for this number set

14, 39, 42, 65, 65, 88

Determine the Number of Intervals using Sturges Rule:

We need to choose the smallest integer k such that 2k ≥ 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 88 - 14 = 74

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

Interval Size  =	74
	3

Add 1 to this giving us 24 + 1 = 25

Frequency Distribution Table

Class Limits	Class Boundaries	FD	CFD	RFD	CRFD
14 - 39	13.5 - 39.5	1	1	1/6 = 16.67%	1/6 = 16.67%
39 - 64	38.5 - 64.5	2	1 + 2 = 3	2/6 = 33.33%	3/6 = 50%
64 - 89	63.5 - 89.5	3	1 + 2 + 3 = 6	3/6 = 50%	6/6 = 100%
		6		100%

Successive Ratio Calculation

Go through our 6 numbers

Determine the ratio of each number to the next one

Successive Ratio 1: 14,39,42,65,65,88

14:39 → 0.359

Successive Ratio 2: 14,39,42,65,65,88

39:42 → 0.9286

Successive Ratio 3: 14,39,42,65,65,88

42:65 → 0.6462

Successive Ratio 4: 14,39,42,65,65,88

65:65 → 1

Successive Ratio 5: 14,39,42,65,65,88

65:88 → 0.7386

Successive Ratio Answer

Successive Ratio = 14:39,39:42,42:65,65:65,65:88 or 0.359,0.9286,0.6462,1,0.7386

Final Answers

5,2,5,6,1,3
RMS = 57.264008475365
Harmonic Mean = 36.807108932128Geometric Mean = 45.198624639682
Mid-Range = 51
Weighted Average = 0
Successive Ratio = Successive Ratio = 14:39,39:42,42:65,65:65,65:88 or 0.359,0.9286,0.6462,1,0.7386

You have 1 free calculations remaining

What is the Answer?

5,2,5,6,1,3
RMS = 57.264008475365
Harmonic Mean = 36.807108932128Geometric Mean = 45.198624639682
Mid-Range = 51
Weighted Average = 0
Successive Ratio = Successive Ratio = 14:39,39:42,42:65,65:65,65:88 or 0.359,0.9286,0.6462,1,0.7386

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 = ΣXi/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(Xi - μ)2/n

For more math formulas, check out our Formula Dossier

What 20 concepts are covered in the Basic Statistics Calculator?

average deviationMean of the absolute values of the distance from the mean for each number in a number setbasic statisticscentral tendencya central or typical value for a probability distribution. Typical measures are the mode, median, meanentropyrefers to disorder or uncertaintyexpected valuepredicted value of a variable or event
E(X) = ΣxI · P(x)frequency distributionfrequency measurement of various outcomesinner fenceut-off values for upper and lower outliers in a datasetmeanA statistical measurement also known as the averagemedianthe value separating the higher half from the lower half of a data sample,modethe number that occurs the most in a number setouter fencestart 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.outlieran observation that lies an abnormal distance from other values in a random sample from a populationquartile1 of 4 equal groups in the distribution of a number setrangeDifference between the largest and smallest values in a number setrankthe 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 deviationa measure of the amount of variation or dispersion of a set of values. The square root of variancestem and leaf plota 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.varianceHow far a set of random numbers are spead out from the meanweighted averageAn 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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