You entered a number set X of {69,79,54,77,98,100}
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
54, 69, 77, 79, 98, 100
Rank Ascending
54 is the 1st lowest/smallest number
69 is the 2nd lowest/smallest number
77 is the 3rd lowest/smallest number
79 is the 4th lowest/smallest number
98 is the 5th lowest/smallest number
100 is the 6th lowest/smallest number
Sort Descending from Highest to Lowest
100, 98, 79, 77, 69, 54
Rank Descending
100 is the 1st highest/largest number
98 is the 2nd highest/largest number
79 is the 3rd highest/largest number
77 is the 4th highest/largest number
69 is the 5th highest/largest number
54 is the 6th 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 Value | 54 | 69 | 77 | 79 | 98 | 100 |
| 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 54,69,77,79,98,100
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 = 542 + 692 + 772 + 792 + 982 + 1002
A = 2916 + 4761 + 5929 + 6241 + 9604 + 10000
A = 39451
Calculate Root Mean Square (RMS):
| RMS = | √39451 |
| √6 |
| RMS = | 198.62275801126 |
| 2.4494897427832 |
RMS = 81.087401405315
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 = | 54 + 69 + 77 + 79 + 98 + 100 |
| 6 |
| μ = | 477 |
| 6 |
μ = 79.5
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
Our median is the average of entry 3 and entry 4 of our number set highlighted in red:
(54,69,77,79,98,100)
Median = ½(77 + 79)
Median = ½(156)
Median = 78
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 |
| 1/54 + 1/69 + 1/77 + 1/79 + 1/98 + 1/100 |
| Harmonic Mean = | 6 |
| 0.018518518518519 + 0.014492753623188 + 0.012987012987013 + 0.012658227848101 + 0.010204081632653 + 0.01 |
| Harmonic Mean = | 6 |
| 0.078860594609474 |
Harmonic Mean = 76.083626172395
Calculate Geometric Mean:
Geometric Mean = (x1 * x2 * x3 * x4 * x5 * x6)1/N
Geometric Mean = (54 * 69 * 77 * 79 * 98 * 100)1/6
Geometric Mean = 2221195284000.16666666666667
Geometric Mean = 77.821176190776
Calculate Mid-Range:
| Mid-Range = | Maximum Value in Number Set + Minimum Value in Number Set |
| 2 |
| Mid-Range = | 100 + 54 |
| 2 |
| Mid-Range = | 154 |
| 2 |
Mid-Range = 77
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:
{100,98,79,77,69,54}
| Stem | Leaf |
|---|---|
| 1 | 00 |
| 9 | 8 |
| 7 | 7,9 |
| 6 | 9 |
| 5 | 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 = | 54 + 69 + 77 + 79 + 98 + 100 |
| 6 |
| μ = | 477 |
| 6 |
μ = 79.5
Calculate Variance denoted as σ2
Let's evaluate the square difference from the mean of each term (Xi - μ)2:
(X1 - μ)2 = (54 - 79.5)2 = -25.52 = 650.25
(X2 - μ)2 = (69 - 79.5)2 = -10.52 = 110.25
(X3 - μ)2 = (77 - 79.5)2 = -2.52 = 6.25
(X4 - μ)2 = (79 - 79.5)2 = -0.52 = 0.25
(X5 - μ)2 = (98 - 79.5)2 = 18.52 = 342.25
(X6 - μ)2 = (100 - 79.5)2 = 20.52 = 420.25
Adding our 6 sum of squared differences up
ΣE(Xi - μ)2 = 650.25 + 110.25 + 6.25 + 0.25 + 342.25 + 420.25
ΣE(Xi - μ)2 = 1529.5
Use the sum of squared differences to calculate variance
| Population | Sample | ||||||||
|---|---|---|---|---|---|---|---|---|---|
|
|
|
| ||||||
| Variance: σp2 = 254.91666666667 | Variance: σs2 = 305.9 | ||||||||
| Standard Deviation: σp = √σp2 = √254.91666666667 | Standard Deviation: σs = √σs2 = √305.9 | ||||||||
| Standard Deviation: σp = 15.9661 | Standard Deviation: σs = 17.49 |
Calculate the Standard Error of the Mean:
| Population | Sample | ||||||||
|---|---|---|---|---|---|---|---|---|---|
|
|
|
|
|
| ||||
| SEM = 6.5181 | SEM = 7.1403 |
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 = (54 - 79.5)3 = -25.53 = -16581.375
(X2 - μ)3 = (69 - 79.5)3 = -10.53 = -1157.625
(X3 - μ)3 = (77 - 79.5)3 = -2.53 = -15.625
(X4 - μ)3 = (79 - 79.5)3 = -0.53 = -0.125
(X5 - μ)3 = (98 - 79.5)3 = 18.53 = 6331.625
(X6 - μ)3 = (100 - 79.5)3 = 20.53 = 8615.125
Add our 6 sum of cubed differences up
ΣE(Xi - μ)3 = -16581.375 + -1157.625 + -15.625 + -0.125 + 6331.625 + 8615.125
ΣE(Xi - μ)3 = -2808
Calculate skewnes
| Skewness = | E(Xi - μ)3 |
| (n - 1)σ3 |
| Skewness = | -2808 |
| (6 - 1)15.96613 |
| Skewness = | -2808 |
| (5)4070.0199231218 |
| Skewness = | -2808 |
| 20350.099615609 |
Skewness = -0.13798458253473
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| = |54 - 79.5| = |-25.5| = 25.5
|X2 - μ| = |69 - 79.5| = |-10.5| = 10.5
|X3 - μ| = |77 - 79.5| = |-2.5| = 2.5
|X4 - μ| = |79 - 79.5| = |-0.5| = 0.5
|X5 - μ| = |98 - 79.5| = |18.5| = 18.5
|X6 - μ| = |100 - 79.5| = |20.5| = 20.5
Average deviation numerator:
Σ|Xi - μ| = 25.5 + 10.5 + 2.5 + 0.5 + 18.5 + 20.5
Σ|Xi - μ| = 78
Calculate average deviation (mean absolute deviation)
| AD = | Σ|Xi - μ| |
| n |
| AD = | 78 |
| 6 |
Average Deviation = 13
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:
(54,69,77,79,98,100)
Median = ½(77 + 79)
Median = ½(156)
Median = 78
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 = 100 - 54
Range = 46
Calculate Pearsons Skewness Coefficient 1:
| PSC1 = | μ - Mode |
| σ |
| PSC1 = | 3(79.5 - N/A) |
| 15.9661 |
Since no mode exists, we do not have a Pearsons Skewness Coefficient 1
Calculate Pearsons Skewness Coefficient 2:
| PSC2 = | μ - Median |
| σ |
| PSC1 = | 3(79.5 - 78) |
| 15.9661 |
| PSC2 = | 3 x 1.5 |
| 15.9661 |
| PSC2 = | 4.5 |
| 15.9661 |
PSC2 = 0.2818
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 = | 100 + 54 |
| 2 |
| Mid-Range = | 154 |
| 2 |
Mid-Range = 77
Calculate the Quartile Items
We need to sort our number set from lowest to highest shown below:
{54,69,77,79,98,100}
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 98
54,69,77,79,98,100Calculate 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 69
54,69,77,79,98,100
Calculate Inter-Quartile Range (IQR):
IQR = UQ - LQ
IQR = 98 - 69
IQR = 29
Calculate Lower Inner Fence (LIF):
Lower Inner Fence (LIF) = LQ - 1.5 x IQR
Lower Inner Fence (LIF) = 69 - 1.5 x 29
Lower Inner Fence (LIF) = 69 - 43.5
Lower Inner Fence (LIF) = 25.5
Calculate Upper Inner Fence (UIF):
Upper Inner Fence (UIF) = UQ + 1.5 x IQR
Upper Inner Fence (UIF) = 98 + 1.5 x 29
Upper Inner Fence (UIF) = 98 + 43.5
Upper Inner Fence (UIF) = 141.5
Calculate Lower Outer Fence (LOF):
Lower Outer Fence (LOF) = LQ - 3 x IQR
Lower Outer Fence (LOF) = 69 - 3 x 29
Lower Outer Fence (LOF) = 69 - 87
Lower Outer Fence (LOF) = -18
Calculate Upper Outer Fence (UOF):
Upper Outer Fence (UOF) = UQ + 3 x IQR
Upper Outer Fence (UOF) = 98 + 3 x 29
Upper Outer Fence (UOF) = 98 + 87
Upper Outer Fence (UOF) = 185
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 < v < 25.5 and 141.5 < v < 185
54,69,77,79,98,100
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 or v > 185
54,69,77,79,98,100
Calculate weighted average
54, 69, 77, 79, 98, 100
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 = | 54 x + 69 x + 77 x + 79 x + 98 x + 100 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
54, 69, 77, 79, 98, 100
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 100 - 54 = 46
Each interval size is the difference of the maximum and minimum value divided by the number of intervals
| Interval Size = | 46 |
| 3 |
Add 1 to this giving us 15 + 1 = 16
Frequency Distribution Table
| Class Limits | Class Boundaries | FD | CFD | RFD | CRFD |
|---|---|---|---|---|---|
| 54 - 70 | 53.5 - 70.5 | 2 | 2 | 2/6 = 33.33% | 2/6 = 33.33% |
| 70 - 86 | 69.5 - 86.5 | 2 | 2 + 2 = 4 | 2/6 = 33.33% | 4/6 = 66.67% |
| 86 - 102 | 85.5 - 102.5 | 2 | 2 + 2 + 2 = 6 | 2/6 = 33.33% | 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: 54,69,77,79,98,100
54:69 → 0.7826
Successive Ratio 2: 54,69,77,79,98,100
69:77 → 0.8961
Successive Ratio 3: 54,69,77,79,98,100
77:79 → 0.9747
Successive Ratio 4: 54,69,77,79,98,100
79:98 → 0.8061
Successive Ratio 5: 54,69,77,79,98,100
98:100 → 0.98
Successive Ratio Answer
Successive Ratio = 54:69,69:77,77:79,79:98,98:100 or 0.7826,0.8961,0.9747,0.8061,0.98
Final Answers
RMS = 81.087401405315
Harmonic Mean = 76.083626172395Geometric Mean = 77.821176190776
Mid-Range = 77
Weighted Average = 0
Successive Ratio = Successive Ratio = 54:69,69:77,77:79,79:98,98:100 or 0.7826,0.8961,0.9747,0.8061,0.98
Common Core State Standards In This Lesson
What is the Answer?
RMS = 81.087401405315
Harmonic Mean = 76.083626172395Geometric Mean = 77.821176190776
Mid-Range = 77
Weighted Average = 0
Successive Ratio = Successive Ratio = 54:69,69:77,77:79,79:98,98:100 or 0.7826,0.8961,0.9747,0.8061,0.98
How does the Basic Statistics Calculator work?
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?
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
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
