Calculate Number Set Basics from 87.4,86.9,89.9,78.3,75.1,70.6

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

You entered a number set X of {87.4,86.9,89.9,78.3,75.1,70.6}


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

70.6, 75.1, 78.3, 86.9, 87.4, 89.9

Sort Descending from Highest to Lowest

89.9, 87.4, 86.9, 78.3, 75.1, 70.6

Ranked Data Calculation

Sort our number set in ascending order

and assign a ranking to each number:

Ranked Data Table

Number Set Value70.675.178.386.987.489.9
Rank123456

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 70.6,75.1,78.3,86.9,87.4,89.9

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

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 = 70.62 + 75.12 + 78.32 + 86.92 + 87.42 + 89.92

A = 4984.36 + 5640.01 + 6130.89 + 7551.61 + 7638.76 + 8082.01

A = 40027.64

Calculate Root Mean Square (RMS):

RMS  =  40027.64
  6

RMS  =  200.0690880671
  2.4494897427832

RMS = 81.677863178057

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  =  70.6 + 75.1 + 78.3 + 86.9 + 87.4 + 89.9
  6

μ  =  488.2
  6

μ = 81.366666666667

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:

(70.6,75.1,78.3,86.9,87.4,89.9)

Median = ½(78.3 + 86.9)

Median = ½(165.2)

Median = 82.6

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:

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
  1/70.6 + 1/75.1 + 1/78.3 + 1/86.9 + 1/87.4 + 1/89.9

Harmonic Mean  =  6
  0.014164305949009 + 0.013315579227696 + 0.012771392081737 + 0.01150747986191 + 0.011441647597254 + 0.011123470522803

Harmonic Mean  =  6
  0.074323875240409

Harmonic Mean = 80.727760502158

Calculate Geometric Mean:

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

Geometric Mean = (70.6 * 75.1 * 78.3 * 86.9 * 87.4 * 89.9)1/6

Geometric Mean = 283463601663.170.16666666666667

Geometric Mean = 81.049352596614

Calculate Mid-Range:

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

Mid-Range  =  89.9 + 70.6
  2

Mid-Range  =  160.5
  2

Mid-Range = 80.25

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:

{89.9,87.4,86.9,78.3,75.1,70.6}

StemLeaf
86.9,7.4,9.9
70.6,5.1,8.3

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  =  70.6 + 75.1 + 78.3 + 86.9 + 87.4 + 89.9
  6

μ  =  488.2
  6

μ = 81.366666666667

Calculate Variance denoted as σ2

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

(X1 - μ)2 = (70.6 - 81.366666666667)2 = -10.7666666666672 = 115.92111111111

(X2 - μ)2 = (75.1 - 81.366666666667)2 = -6.26666666666672 = 39.271111111111

(X3 - μ)2 = (78.3 - 81.366666666667)2 = -3.06666666666672 = 9.4044444444444

(X4 - μ)2 = (86.9 - 81.366666666667)2 = 5.53333333333332 = 30.617777777778

(X5 - μ)2 = (87.4 - 81.366666666667)2 = 6.03333333333332 = 36.401111111111

(X6 - μ)2 = (89.9 - 81.366666666667)2 = 8.53333333333332 = 72.817777777778

Adding our 6 sum of squared differences up

ΣE(Xi - μ)2 = 115.92111111111 + 39.271111111111 + 9.4044444444444 + 30.617777777778 + 36.401111111111 + 72.817777777778

ΣE(Xi - μ)2 = 304.43333333333

Use the sum of squared differences to calculate variance

PopulationSample

σ2  =  ΣE(Xi - μ)2
  n

σ2  =  ΣE(Xi - μ)2
  n - 1

σ2  =  304.43333333333
  6

σ2  =  304.43333333333
  5

Variance: σp2 = 50.738888888889Variance: σs2 = 60.886666666667
Standard Deviation: σp = √σp2 = √50.738888888889Standard Deviation: σs = √σs2 = √60.886666666667
Standard Deviation: σp = 7.1231Standard Deviation: σs = 7.803

Calculate the Standard Error of the Mean:

PopulationSample

SEM  =  σp
  n

SEM  =  σs
  n

SEM  =  7.1231
  6

SEM  =  7.803
  6

SEM  =  7.1231
  2.4494897427832

SEM  =  7.803
  2.4494897427832

SEM = 2.908SEM = 3.1856

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 = (70.6 - 81.366666666667)3 = -10.7666666666673 = -1248.083962963

(X2 - μ)3 = (75.1 - 81.366666666667)3 = -6.26666666666673 = -246.09896296296

(X3 - μ)3 = (78.3 - 81.366666666667)3 = -3.06666666666673 = -28.840296296296

(X4 - μ)3 = (86.9 - 81.366666666667)3 = 5.53333333333333 = 169.41837037037

(X5 - μ)3 = (87.4 - 81.366666666667)3 = 6.03333333333333 = 219.62003703704

(X6 - μ)3 = (89.9 - 81.366666666667)3 = 8.53333333333333 = 621.37837037037

Add our 6 sum of cubed differences up

ΣE(Xi - μ)3 = -1248.083962963 + -246.09896296296 + -28.840296296296 + 169.41837037037 + 219.62003703704 + 621.37837037037

ΣE(Xi - μ)3 = -512.60644444444

Calculate skewnes

Skewness  =  E(Xi - μ)3
  (n - 1)σ3

Skewness  =  -512.60644444444
  (6 - 1)7.12313

Skewness  =  -512.60644444444
  (5)361.41579121939

Skewness  =  -512.60644444444
  1807.078956097

Skewness = -0.2836657705049

Calculate Average Deviation (Mean Absolute Deviation) denoted below:

AD  =  Σ|Xi - μ|
  n

Evaluate the absolute value of the difference from the mean

|Xi - μ|:

|X1 - μ| = |70.6 - 81.366666666667| = |-10.766666666667| = 10.766666666667

|X2 - μ| = |75.1 - 81.366666666667| = |-6.2666666666667| = 6.2666666666667

|X3 - μ| = |78.3 - 81.366666666667| = |-3.0666666666667| = 3.0666666666667

|X4 - μ| = |86.9 - 81.366666666667| = |5.5333333333333| = 5.5333333333333

|X5 - μ| = |87.4 - 81.366666666667| = |6.0333333333333| = 6.0333333333333

|X6 - μ| = |89.9 - 81.366666666667| = |8.5333333333333| = 8.5333333333333

Average deviation numerator:

Σ|Xi - μ| = 10.766666666667 + 6.2666666666667 + 3.0666666666667 + 5.5333333333333 + 6.0333333333333 + 8.5333333333333

Σ|Xi - μ| = 40.2

Calculate average deviation (mean absolute deviation)

AD  =  Σ|Xi - μ|
  n

AD  =  40.2
  6

Average Deviation = 6.7

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:

(70.6,75.1,78.3,86.9,87.4,89.9)

Median = ½(78.3 + 86.9)

Median = ½(165.2)

Median = 82.6

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 = 89.9 - 70.6

Range = 19.3

Calculate Pearsons Skewness Coefficient 1:

PSC1  =  μ - Mode
  σ

PSC1  =  3(81.366666666667 - N/A)
  7.1231

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

Calculate Pearsons Skewness Coefficient 2:

PSC2  =  μ - Median
  σ

PSC1  =  3(81.366666666667 - 82.6)
  7.1231

PSC2  =  3 x -1.2333333333333
  7.1231

PSC2  =  -3.7
  7.1231

PSC2 = -0.5194

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  =  89.9 + 70.6
  2

Mid-Range  =  160.5
  2

Mid-Range = 80.25

Calculate the Quartile Items

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

{70.6,75.1,78.3,86.9,87.4,89.9}

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 87.4

70.6,75.1,78.3,86.9,87.4,89.9

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 75.1

70.6,75.1,78.3,86.9,87.4,89.9

Calculate Inter-Quartile Range (IQR):

IQR = UQ - LQ

IQR = 87.4 - 75.1

IQR = 12.3

Calculate Lower Inner Fence (LIF):

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

Lower Inner Fence (LIF) = 75.1 - 1.5 x 12.3

Lower Inner Fence (LIF) = 75.1 - 18.45

Lower Inner Fence (LIF) = 56.65

Calculate Upper Inner Fence (UIF):

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

Upper Inner Fence (UIF) = 87.4 + 1.5 x 12.3

Upper Inner Fence (UIF) = 87.4 + 18.45

Upper Inner Fence (UIF) = 105.85

Calculate Lower Outer Fence (LOF):

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

Lower Outer Fence (LOF) = 75.1 - 3 x 12.3

Lower Outer Fence (LOF) = 75.1 - 36.9

Lower Outer Fence (LOF) = 38.2

Calculate Upper Outer Fence (UOF):

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

Upper Outer Fence (UOF) = 87.4 + 3 x 12.3

Upper Outer Fence (UOF) = 87.4 + 36.9

Upper Outer Fence (UOF) = 124.3

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 38.2 < v < 56.65 and 105.85 < v < 124.3

70.6,75.1,78.3,86.9,87.4,89.9

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 < 38.2 or v > 124.3

70.6,75.1,78.3,86.9,87.4,89.9

Calculate weighted average

Array

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  =  
  n

Weighted Average  =  
  0

Weighted Average  =  
  0

Weighted Average  =  0
  0

Weighted Average = NAN

Frequency Distribution Table

Show the freqency distribution table for this number set

70.6, 75.1, 78.3, 86.9, 87.4, 89.9

Determine the Number of Intervals using Sturges Rule:

We need to choose the smallest integer k such that 2k ≥ n where n = 0

Therefore, we use 0 intervals

Our maximum value in our number set of 89.9 - 70.6 = 19.3

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

Interval Size  =  19.3
  0

Add 1 to this giving us INF + 1 = INF

Frequency Distribution Table

Class LimitsClass BoundariesFDCFDRFDCRFD
  0 100% 

Successive Ratio Calculation

Go through our 6 numbers

Determine the ratio of each number to the next one

Successive Ratio 1: 70.6,75.1,78.3,86.9,87.4,89.9

70.6:75.1 → 0.9401

Successive Ratio 2: 70.6,75.1,78.3,86.9,87.4,89.9

75.1:78.3 → 0.9591

Successive Ratio 3: 70.6,75.1,78.3,86.9,87.4,89.9

78.3:86.9 → 0.901

Successive Ratio 4: 70.6,75.1,78.3,86.9,87.4,89.9

86.9:87.4 → 0.9943

Successive Ratio 5: 70.6,75.1,78.3,86.9,87.4,89.9

87.4:89.9 → 0.9722

Successive Ratio Answer

Successive Ratio = 70.6:75.1,75.1:78.3,78.3:86.9,86.9:87.4,87.4:89.9 or 0.9401,0.9591,0.901,0.9943,0.9722

Final Answers

1,2,3,4,5,6
RMS = 81.677863178057
Harmonic Mean = 80.727760502158Geometric Mean = 81.049352596614
Mid-Range = 80.25
Weighted Average = NAN
Successive Ratio = Successive Ratio = 70.6:75.1,75.1:78.3,78.3:86.9,86.9:87.4,87.4:89.9 or 0.9401,0.9591,0.901,0.9943,0.9722





What is the Answer?
1,2,3,4,5,6
RMS = 81.677863178057
Harmonic Mean = 80.727760502158Geometric Mean = 81.049352596614
Mid-Range = 80.25
Weighted Average = NAN
Successive Ratio = Successive Ratio = 70.6:75.1,75.1:78.3,78.3:86.9,86.9:87.4,87.4:89.9 or 0.9401,0.9591,0.901,0.9943,0.9722
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 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
Example calculations for the Basic Statistics Calculator
Basic Statistics Calculator Video

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