Enter Number Set (Comma Separated)

You entered a number set X of {11.92,34.86,26.72,24.50,38.93,8.59,29.31,23.39,24.13,30.05,21.54,35.97,7.48,35.97}


From the 14 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

7.48, 8.59, 11.92, 21.54, 23.39, 24.13, 24.50, 26.72, 29.31, 30.05, 34.86, 35.97, 35.97, 38.93

Rank Ascending

7.48 is the 1st lowest/smallest number

8.59 is the 2nd lowest/smallest number

11.92 is the 3rd lowest/smallest number

21.54 is the 4th lowest/smallest number

23.39 is the 5th lowest/smallest number

24.13 is the 6th lowest/smallest number

24.50 is the 7th lowest/smallest number

26.72 is the 8th lowest/smallest number

29.31 is the 9th lowest/smallest number

30.05 is the 10th lowest/smallest number

34.86 is the 11th lowest/smallest number

35.97 is the 12th lowest/smallest number

35.97 is the 13th lowest/smallest number

38.93 is the 14th lowest/smallest number

Sort Descending from Highest to Lowest

38.93, 35.97, 35.97, 34.86, 30.05, 29.31, 26.72, 24.50, 24.13, 23.39, 21.54, 11.92, 8.59, 7.48

Rank Descending

38.93 is the 1st highest/largest number

35.97 is the 2nd highest/largest number

35.97 is the 3rd highest/largest number

34.86 is the 4th highest/largest number

30.05 is the 5th highest/largest number

29.31 is the 6th highest/largest number

26.72 is the 7th highest/largest number

24.50 is the 8th highest/largest number

24.13 is the 9th highest/largest number

23.39 is the 10th highest/largest number

21.54 is the 11th highest/largest number

11.92 is the 12th highest/largest number

8.59 is the 13th highest/largest number

7.48 is the 14th highest/largest number

Ranked Data Calculation

Sort our number set in ascending order

and assign a ranking to each number:

Ranked Data Table

Number Set Value7.488.5911.9221.5423.3924.1324.5026.7229.3130.0534.8635.9735.9738.93
Rank1234567891011121314

Step 2: Using original number set, assign the rank value:

Since we have 14 numbers in our original number set,
we assign ranks from lowest to highest (1 to 14)

Our original number set in unsorted order was 7.48,8.59,11.92,21.54,23.39,24.13,24.50,26.72,29.31,30.05,34.86,35.97,35.97,38.93

Our respective ranked data set is 1,2,3,4,5,6,7,8,9,10,11,13,13,14

Root Mean Square Calculation

Root Mean Square  =  √A
  √N

where A = x12 + x22 + x32 + x42 + x52 + x62 + x72 + x82 + x92 + x102 + x112 + x122 + x132 + x142 and N = 14 number set items

Calculate A

A = 7.482 + 8.592 + 11.922 + 21.542 + 23.392 + 24.132 + 24.502 + 26.722 + 29.312 + 30.052 + 34.862 + 35.972 + 35.972 + 38.932

A = 55.9504 + 73.7881 + 142.0864 + 463.9716 + 547.0921 + 582.2569 + 600.25 + 713.9584 + 859.0761 + 903.0025 + 1215.2196 + 1293.8409 + 1293.8409 + 1515.5449

A = 10259.8788

Calculate Root Mean Square (RMS):

RMS  =  √10259.8788
  √14

RMS  =  101.29105982267
  3.7416573867739

RMS = 27.071174442833

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  =  7.48 + 8.59 + 11.92 + 21.54 + 23.39 + 24.13 + 24.50 + 26.72 + 29.31 + 30.05 + 34.86 + 35.97 + 35.97 + 38.93
  14

μ  =  353.36
  14

μ = 25.24

Calculate the Median (Middle Value)

Since our number set contains 14 elements which is an even number,
our median number is determined as follows

Number Set = (n1,n2,n3,n4,n5,n6,n7,n8,n9,n10,n11,n12,n13,n14)

Median Number 1 = ½(n)

Median Number 1 = ½(14)

Median Number 1 = Number Set Entry 7

Median Number 2 = Median Number 1 + 1

Median Number 2 = Number Set Entry 7 + 1

Median Number 2 = Number Set Entry 8

For an even number set, we average the 2 median number entries:

Median = ½(n7 + n8)

Therefore, we sort our number set in ascending order

Our median is the average of entry 7 and entry 8 of our number set highlighted in red:

(7.48,8.59,11.92,21.54,23.39,24.13,24.50,26.72,29.31,30.05,34.86,35.97,35.97,38.93)

Median = ½(24.50 + 26.72)

Median = ½(51.22)

Median = 25.61

Calculate the Mode - Highest Frequency Number

The highest frequency of occurence in our number set is 2 times
by the following numbers in green:

()

Our mode is denoted as: 35.97

Mode = 35.97

Calculate Harmonic Mean:

Harmonic Mean  =  N
  1/x1 + 1/x2 + 1/x3 + 1/x4 + 1/x5 + 1/x6 + 1/x7 + 1/x8 + 1/x9 + 1/x10 + 1/x11 + 1/x12 + 1/x13 + 1/x14

With N = 14 and each xi a member of the number set you entered, we have:

Harmonic Mean  =  14
  1/7.48 + 1/8.59 + 1/11.92 + 1/21.54 + 1/23.39 + 1/24.13 + 1/24.50 + 1/26.72 + 1/29.31 + 1/30.05 + 1/34.86 + 1/35.97 + 1/35.97 + 1/38.93

Harmonic Mean  =  14
  0.13368983957219 + 0.11641443538999 + 0.083892617449664 + 0.046425255338904 + 0.042753313381787 + 0.041442188147534 + 0.040816326530612 + 0.037425149700599 + 0.034118048447629 + 0.033277870216306 + 0.028686173264487 + 0.027800945232138 + 0.027800945232138 + 0.025687130747496

Harmonic Mean  =  14
  0.72023023865147

Harmonic Mean = 19.438228567316

Calculate Geometric Mean:

Geometric Mean = (x1 * x2 * x3 * x4 * x5 * x6 * x7 * x8 * x9 * x10 * x11 * x12 * x13 * x14)1/N

Geometric Mean = (7.48 * 8.59 * 11.92 * 21.54 * 23.39 * 24.13 * 24.50 * 26.72 * 29.31 * 30.05 * 34.86 * 35.97 * 35.97 * 38.93)1/14

Geometric Mean = 9.4267014924235E+180.071428571428571

Geometric Mean = 22.662687541866

Calculate Mid-Range:

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

Mid-Range  =  38.93 + 7.48
  2

Mid-Range  =  46.41
  2

Mid-Range = 23.205

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:

{38.93,35.97,35.97,34.86,30.05,29.31,26.72,24.50,24.13,23.39,21.54,11.92,8.59,7.48}

StemLeaf
30.05,4.86,5.97,5.97,8.93
21.54,3.39,4.13,4.50,6.72,9.31
11.92
8.59
7.48

Calculate Variance denoted as σ2

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

(X1 - μ)2 = (7.48 - 25.24)2 = -17.762 = 315.4176

(X2 - μ)2 = (8.59 - 25.24)2 = -16.652 = 277.2225

(X3 - μ)2 = (11.92 - 25.24)2 = -13.322 = 177.4224

(X4 - μ)2 = (21.54 - 25.24)2 = -3.72 = 13.69

(X5 - μ)2 = (23.39 - 25.24)2 = -1.852 = 3.4225

(X6 - μ)2 = (24.13 - 25.24)2 = -1.112 = 1.2321

(X7 - μ)2 = (24.50 - 25.24)2 = -0.740000000000012 = 0.54760000000001

(X8 - μ)2 = (26.72 - 25.24)2 = 1.482 = 2.1904

(X9 - μ)2 = (29.31 - 25.24)2 = 4.072 = 16.5649

(X10 - μ)2 = (30.05 - 25.24)2 = 4.812 = 23.1361

(X11 - μ)2 = (34.86 - 25.24)2 = 9.622 = 92.5444

(X12 - μ)2 = (35.97 - 25.24)2 = 10.732 = 115.1329

(X13 - μ)2 = (35.97 - 25.24)2 = 10.732 = 115.1329

(X14 - μ)2 = (38.93 - 25.24)2 = 13.692 = 187.4161

Adding our 14 sum of squared differences up

ΣE(Xi - μ)2 = 315.4176 + 277.2225 + 177.4224 + 13.69 + 3.4225 + 1.2321 + 0.54760000000001 + 2.1904 + 16.5649 + 23.1361 + 92.5444 + 115.1329 + 115.1329 + 187.4161

ΣE(Xi - μ)2 = 1341.0724

Use the sum of squared differences to calculate variance

PopulationSample

σ2  =  ΣE(Xi - μ)2
  n

σ2  =  ΣE(Xi - μ)2
  n - 1

σ2  =  1341.0724
  14

σ2  =  1341.0724
  13

Variance: σp2 = 95.790885714286Variance: σs2 = 103.15941538462
Standard Deviation: σp = √σp2 = √95.790885714286Standard Deviation: σs = √σs2 = √103.15941538462
Standard Deviation: σp = 9.7873Standard Deviation: σs = 10.1567

Calculate the Standard Error of the Mean:

PopulationSample

SEM  =  σp
  √n

SEM  =  σs
  √n

SEM  =  9.7873
  √14

SEM  =  10.1567
  √14

SEM  =  9.7873
  3.7416573867739

SEM  =  10.1567
  3.7416573867739

SEM = 2.6158SEM = 2.7145

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 = (7.48 - 25.24)3 = -17.763 = -5601.816576

(X2 - μ)3 = (8.59 - 25.24)3 = -16.653 = -4615.754625

(X3 - μ)3 = (11.92 - 25.24)3 = -13.323 = -2363.266368

(X4 - μ)3 = (21.54 - 25.24)3 = -3.73 = -50.653

(X5 - μ)3 = (23.39 - 25.24)3 = -1.853 = -6.3316250000001

(X6 - μ)3 = (24.13 - 25.24)3 = -1.113 = -1.367631

(X7 - μ)3 = (24.50 - 25.24)3 = -0.740000000000013 = -0.40522400000001

(X8 - μ)3 = (26.72 - 25.24)3 = 1.483 = 3.241792

(X9 - μ)3 = (29.31 - 25.24)3 = 4.073 = 67.419143

(X10 - μ)3 = (30.05 - 25.24)3 = 4.813 = 111.284641

(X11 - μ)3 = (34.86 - 25.24)3 = 9.623 = 890.277128

(X12 - μ)3 = (35.97 - 25.24)3 = 10.733 = 1235.376017

(X13 - μ)3 = (35.97 - 25.24)3 = 10.733 = 1235.376017

(X14 - μ)3 = (38.93 - 25.24)3 = 13.693 = 2565.726409

Add our 14 sum of cubed differences up

ΣE(Xi - μ)3 = -5601.816576 + -4615.754625 + -2363.266368 + -50.653 + -6.3316250000001 + -1.367631 + -0.40522400000001 + 3.241792 + 67.419143 + 111.284641 + 890.277128 + 1235.376017 + 1235.376017 + 2565.726409

ΣE(Xi - μ)3 = -6530.893902

Calculate skewnes

Skewness  =  E(Xi - μ)3
  (n - 1)σ3

Skewness  =  -6530.893902
  (14 - 1)9.78733

Skewness  =  -6530.893902
  (13)937.53761587762

Skewness  =  -6530.893902
  12187.989006409

Skewness = -0.53584671749915

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| = |7.48 - 25.24| = |-17.76| = 17.76

|X2 - μ| = |8.59 - 25.24| = |-16.65| = 16.65

|X3 - μ| = |11.92 - 25.24| = |-13.32| = 13.32

|X4 - μ| = |21.54 - 25.24| = |-3.7| = 3.7

|X5 - μ| = |23.39 - 25.24| = |-1.85| = 1.85

|X6 - μ| = |24.13 - 25.24| = |-1.11| = 1.11

|X7 - μ| = |24.50 - 25.24| = |-0.74000000000001| = 0.74000000000001

|X8 - μ| = |26.72 - 25.24| = |1.48| = 1.48

|X9 - μ| = |29.31 - 25.24| = |4.07| = 4.07

|X10 - μ| = |30.05 - 25.24| = |4.81| = 4.81

|X11 - μ| = |34.86 - 25.24| = |9.62| = 9.62

|X12 - μ| = |35.97 - 25.24| = |10.73| = 10.73

|X13 - μ| = |35.97 - 25.24| = |10.73| = 10.73

|X14 - μ| = |38.93 - 25.24| = |13.69| = 13.69

Average deviation numerator:

Σ|Xi - μ| = 17.76 + 16.65 + 13.32 + 3.7 + 1.85 + 1.11 + 0.74000000000001 + 1.48 + 4.07 + 4.81 + 9.62 + 10.73 + 10.73 + 13.69

Σ|Xi - μ| = 110.26

Calculate average deviation (mean absolute deviation)

AD  =  Σ|Xi - μ|
  n

AD  =  110.26
  14

Average Deviation = 7.87571

Calculate the Range

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

Range = 38.93 - 7.48

Range = 31.45

Calculate Pearsons Skewness Coefficient 1:

PSC1  =  μ - Mode
  σ

PSC1  =  3(25.24 - 35.97)
  9.7873

PSC1  =  3 x -10.73
  9.7873

PSC1  =  -32.19
  9.7873

PSC1 = -3.289

Calculate Pearsons Skewness Coefficient 2:

PSC2  =  μ - Median
  σ

PSC1  =  3(25.24 - 25.61)
  9.7873

PSC2  =  3 x -0.36999999999999
  9.7873

PSC2  =  -1.11
  9.7873

PSC2 = -0.1134

Calculate Entropy:

Entropy = Ln(n)

Entropy = Ln(14)

Entropy = 2.6390573296153

Calculate Mid-Range:

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

Mid-Range  =  38.93 + 7.48
  2

Mid-Range  =  46.41
  2

Mid-Range = 23.205

Calculate the Quartile Items

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

{7.48,8.59,11.92,21.54,23.39,24.13,24.50,26.72,29.31,30.05,34.86,35.97,35.97,38.93}

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

V  =  y(n + 1)
  100

V  =  75(14 + 1)
  100

V  =  75(15)
  100

V  =  1125
  100

V = 11 ← Rounded down to the nearest integer

Upper quartile (UQ) point = Point # 11 in the dataset which is 34.86

7.48,8.59,11.92,21.54,23.39,24.13,24.50,26.72,29.31,30.05,34.86,35.97,35.97,38.93

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

V  =  y(n + 1)
  100

V  =  25(14 + 1)
  100

V  =  25(15)
  100

V  =  375
  100

V = 4 ← Rounded up to the nearest integer

Lower quartile (LQ) point = Point # 4 in the dataset which is 21.54

7.48,8.59,11.92,21.54,23.39,24.13,24.50,26.72,29.31,30.05,34.86,35.97,35.97,38.93

Calculate Inter-Quartile Range (IQR):

IQR = UQ - LQ

IQR = 34.86 - 21.54

IQR = 13.32

Calculate Lower Inner Fence (LIF):

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

Lower Inner Fence (LIF) = 21.54 - 1.5 x 13.32

Lower Inner Fence (LIF) = 21.54 - 19.98

Lower Inner Fence (LIF) = 1.56

Calculate Upper Inner Fence (UIF):

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

Upper Inner Fence (UIF) = 34.86 + 1.5 x 13.32

Upper Inner Fence (UIF) = 34.86 + 19.98

Upper Inner Fence (UIF) = 54.84

Calculate Lower Outer Fence (LOF):

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

Lower Outer Fence (LOF) = 21.54 - 3 x 13.32

Lower Outer Fence (LOF) = 21.54 - 39.96

Lower Outer Fence (LOF) = -18.42

Calculate Upper Outer Fence (UOF):

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

Upper Outer Fence (UOF) = 34.86 + 3 x 13.32

Upper Outer Fence (UOF) = 34.86 + 39.96

Upper Outer Fence (UOF) = 74.82

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 -18.42 < v < 1.56 and 54.84 < v < 74.82

7.48,8.59,11.92,21.54,23.39,24.13,24.50,26.72,29.31,30.05,34.86,35.97,35.97,38.93

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 < -18.42 or v > 74.82

7.48,8.59,11.92,21.54,23.39,24.13,24.50,26.72,29.31,30.05,34.86,35.97,35.97,38.93

Calculate weighted average

7.48, 8.59, 11.92, 21.54, 23.39, 24.13, 24.50, 26.72, 29.31, 30.05, 34.86, 35.97, 35.97, 38.93

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 + X7p7 + X8p8 + X9p9 + X10p10 + X11p11 + X12p12 + X13p13 + X14p14
  n

Weighted Average  =  7.48 x 0.2 + 8.59 x 0.4 + 11.92 x 0.6 + 21.54 x 0.8 + 23.39 x 0.9 + 24.13 x + 24.50 x + 26.72 x + 29.31 x + 30.05 x + 34.86 x + 35.97 x + 35.97 x + 38.93 x
  14

Weighted Average  =  1.496 + 3.436 + 7.152 + 17.232 + 21.051 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0
  14

Weighted Average  =  50.367
  14

Weighted Average = 3.5976428571429

Frequency Distribution Table

Show the freqency distribution table for this number set

7.48, 8.59, 11.92, 21.54, 23.39, 24.13, 24.50, 26.72, 29.31, 30.05, 34.86, 35.97, 35.97, 38.93

Determine the Number of Intervals using Sturges Rule:

Choose the smallest integer k such that 2k ≥ n where n = 14

For k = 1, we have 21 = 2

For k = 2, we have 22 = 4

For k = 3, we have 23 = 8

For k = 4, we have 24 = 16 ← Use this since it is greater than our n value of 14

Therefore, we use 4 intervals

Our maximum value in our number set of 38.93 - 7.48 = 31.45

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

Interval Size  =  31.45
  4

Add 1 to this giving us 7 + 1 = 8

Frequency Distribution Table

Class LimitsClass BoundariesFDCFDRFDCRFD
7.48 - 15.486.98 - 15.98333/14 = 21.43%3/14 = 21.43%
15.48 - 23.4814.98 - 23.9823 + 2 = 52/14 = 14.29%5/14 = 35.71%
23.48 - 31.4822.98 - 31.9853 + 2 + 5 = 105/14 = 35.71%10/14 = 71.43%
31.48 - 39.4830.98 - 39.9843 + 2 + 5 + 4 = 144/14 = 28.57%14/14 = 100%
  14 100% 

Successive Ratio Calculation

Go through our 14 numbers

Determine the ratio of each number to the next one

Successive Ratio 1: 7.48,8.59,11.92,21.54,23.39,24.13,24.50,26.72,29.31,30.05,34.86,35.97,35.97,38.93

7.48:8.59 → 0.8708

Successive Ratio 2: 7.48,8.59,11.92,21.54,23.39,24.13,24.50,26.72,29.31,30.05,34.86,35.97,35.97,38.93

8.59:11.92 → 0.7206

Successive Ratio 3: 7.48,8.59,11.92,21.54,23.39,24.13,24.50,26.72,29.31,30.05,34.86,35.97,35.97,38.93

11.92:21.54 → 0.5534

Successive Ratio 4: 7.48,8.59,11.92,21.54,23.39,24.13,24.50,26.72,29.31,30.05,34.86,35.97,35.97,38.93

21.54:23.39 → 0.9209

Successive Ratio 5: 7.48,8.59,11.92,21.54,23.39,24.13,24.50,26.72,29.31,30.05,34.86,35.97,35.97,38.93

23.39:24.13 → 0.9693

Successive Ratio 6: 7.48,8.59,11.92,21.54,23.39,24.13,24.50,26.72,29.31,30.05,34.86,35.97,35.97,38.93

24.13:24.50 → 0.9849

Successive Ratio 7: 7.48,8.59,11.92,21.54,23.39,24.13,24.50,26.72,29.31,30.05,34.86,35.97,35.97,38.93

24.50:26.72 → 0.9169

Successive Ratio 8: 7.48,8.59,11.92,21.54,23.39,24.13,24.50,26.72,29.31,30.05,34.86,35.97,35.97,38.93

26.72:29.31 → 0.9116

Successive Ratio 9: 7.48,8.59,11.92,21.54,23.39,24.13,24.50,26.72,29.31,30.05,34.86,35.97,35.97,38.93

29.31:30.05 → 0.9754

Successive Ratio 10: 7.48,8.59,11.92,21.54,23.39,24.13,24.50,26.72,29.31,30.05,34.86,35.97,35.97,38.93

30.05:34.86 → 0.862

Successive Ratio 11: 7.48,8.59,11.92,21.54,23.39,24.13,24.50,26.72,29.31,30.05,34.86,35.97,35.97,38.93

34.86:35.97 → 0.9691

Successive Ratio 12: 7.48,8.59,11.92,21.54,23.39,24.13,24.50,26.72,29.31,30.05,34.86,35.97,35.97,38.93

35.97:35.97 → 1

Successive Ratio 13: 7.48,8.59,11.92,21.54,23.39,24.13,24.50,26.72,29.31,30.05,34.86,35.97,35.97,38.93

35.97:38.93 → 0.924

Successive Ratio Answer

Successive Ratio = 7.48:8.59,8.59:11.92,11.92:21.54,21.54:23.39,23.39:24.13,24.13:24.50,24.50:26.72,26.72:29.31,29.31:30.05,30.05:34.86,34.86:35.97,35.97:35.97,35.97:38.93 or 0.8708,0.7206,0.5534,0.9209,0.9693,0.9849,0.9169,0.9116,0.9754,0.862,0.9691,1,0.924

Final Answers


1,2,3,4,5,6,7,8,9,10,11,13,13,14
RMS = 27.071174442833
Harmonic Mean = 19.438228567316Geometric Mean = 22.662687541866
Mid-Range = 23.205
Weighted Average = 3.5976428571429
Successive Ratio = Successive Ratio = 7.48:8.59,8.59:11.92,11.92:21.54,21.54:23.39,23.39:24.13,24.13:24.50,24.50:26.72,26.72:29.31,29.31:30.05,30.05:34.86,34.86:35.97,35.97:35.97,35.97:38.93 or 0.8708,0.7206,0.5534,0.9209,0.9693,0.9849,0.9169,0.9116,0.9754,0.862,0.9691,1,0.924

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