Step 2: Using original number set, assign the rank value:
Since we have 7 numbers in our original number set, we assign ranks from lowest to highest (1 to 7) Our original number set in unsorted order was 9,8,7,64,3,2,1 Our respective ranked data set is 6,5,4,7,3,2,1
Root Mean Square Calculation
Root Mean Square =
√A
√N
where A = x_{1}^{2} + x_{2}^{2} + x_{3}^{2} + x_{4}^{2} + x_{5}^{2} + x_{6}^{2} + x_{7}^{2} and N = 7 number set items
Calculate the Median (Middle Value) Since our number set contains 7 elements which is an odd number, our median number is determined as follows: Number Set = (n_{1},n_{2},n_{3},n_{4},n_{5},n_{6},n_{7}) Median Number = Entry ½(n + 1) Median Number = Entry ½(8) Median Number = n_{4}
Therefore, we sort our number set in ascending order and our median is entry 4 of our number set highlighted in red: (1,2,3,7,8,9,64) Median = 7
The highest frequency of occurence in our number set is 1 times by the following numbers in green: (9,8,7,64,3,2,1) Since the maximum frequency of any number is 1, no mode exists. Mode = N/A
Adding our 7 sum of cubed differences up, we have our skewness numerator: ΣE(X_{i} - μ)^{3} = -1919.833819242 + -1492.7113702624 + -1134.1603498542 + -265.67055393586 + -159.97667638484 + -86.854227405248 + 129334.88046647 ΣE(X_{i} - μ)^{3} = 124275.67346939
Now that we have the sum of cubed differences from the means, calculate skewness:
Skewness =
E(X_{i} - μ)^{3}
(n - 1)σ^{3}
Skewness =
124275.67346939
(7 - 1)20.8454^{3}
Skewness =
124275.67346939
(6)9057.9662779607
Skewness =
124275.67346939
54347.797667764
Skewness = 2.2866735875684
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|: |X_{1} - μ| = |1 - 13.428571428571| = |-12.428571428571| = 12.428571428571 |X_{2} - μ| = |2 - 13.428571428571| = |-11.428571428571| = 11.428571428571 |X_{3} - μ| = |3 - 13.428571428571| = |-10.428571428571| = 10.428571428571 |X_{4} - μ| = |7 - 13.428571428571| = |-6.4285714285714| = 6.4285714285714 |X_{5} - μ| = |8 - 13.428571428571| = |-5.4285714285714| = 5.4285714285714 |X_{6} - μ| = |9 - 13.428571428571| = |-4.4285714285714| = 4.4285714285714 |X_{7} - μ| = |64 - 13.428571428571| = |50.571428571429| = 50.571428571429
Adding our 7 absolute value of differences from the mean, we have our average deviation numerator: Σ|X_{i} - μ| = 12.428571428571 + 11.428571428571 + 10.428571428571 + 6.4285714285714 + 5.4285714285714 + 4.4285714285714 + 50.571428571429 Σ|X_{i} - μ| = 101.14285714286
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 =
101.14285714286
7
Average Deviation = 14.44898
Calculate the Median (Middle Value) Since our number set contains 7 elements which is an odd number, our median number is determined as follows: Number Set = (n_{1},n_{2},n_{3},n_{4},n_{5},n_{6},n_{7}) Median Number = Entry ½(n + 1) Median Number = Entry ½(8) Median Number = n_{4}
Therefore, we sort our number set in ascending order and our median is entry 4 of our number set highlighted in red: (1,2,3,7,8,9,64) Median = 7
The highest frequency of occurence in our number set is 1 times by the following numbers in green: (9,8,7,64,3,2,1) Since the maximum frequency of any number is 1, no mode exists. Mode = N/A
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 -19 < v < -8.5 and 19.5 < v < 30 1,2,3,7,8,9,64
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 < -19 or v > 30 1,2,3,7,8,9,64
Calculate weighted average
9,8,7,64,3,2,1
Weighted-Average Formula:
Multiply each value by each probability amount
We do this by multiplying each X_{i} x p_{i} to get a weighted score Y
9 x 0.2 + 8 x 0.4 + 7 x 0.6 + 64 x 0.8 + 3 x 0.9 + 2 x + 1 x
7
Weighted Average =
1.8 + 3.2 + 4.2 + 51.2 + 2.7 + 0 + 0
7
Weighted Average =
63.1
7
Weighted Average = 9.0142857142857
Frequency Distribution Table
Show the freqency distribution table for this number set
1, 2, 3, 7, 8, 9, 64
Determine the Number of Intervals using Sturges Rule:
We need to choose the smallest integer k such that 2^{k} ≥ n where n = 7
For k = 1, we have 2^{1} = 2
For k = 2, we have 2^{2} = 4
For k = 3, we have 2^{3} = 8 ← Use this since it is greater than our n value of 7
Therefore, we use 3 intervals
Our maximum value in our number set of 64 - 1 = 63
Each interval size is the difference of the maximum and minimum value divided by the number of intervals
Interval Size =
63
3
Add 1 to this giving us 21 + 1 = 22
Frequency Distribution Table
Class Limits
Class Boundaries
FD
CFD
RFD
CRFD
1 - 23
0.5 - 23.5
6
6
6/7 = 85.71%
6/7 = 85.71%
23 - 45
22.5 - 45.5
6 + = 6
/7 = 0%
6/7 = 85.71%
45 - 67
44.5 - 67.5
1
6 + + 1 = 7
1/7 = 14.29%
7/7 = 100%
7
100%
Successive Ratio Calculation
Go through our 7 numbers
Determine the ratio of each number to the next one
Successive Ratio 1: 1,2,3,7,8,9,64
1:2 → 0.5
Successive Ratio 2: 1,2,3,7,8,9,64
2:3 → 0.6667
Successive Ratio 3: 1,2,3,7,8,9,64
3:7 → 0.4286
Successive Ratio 4: 1,2,3,7,8,9,64
7:8 → 0.875
Successive Ratio 5: 1,2,3,7,8,9,64
8:9 → 0.8889
Successive Ratio 6: 1,2,3,7,8,9,64
9:64 → 0.1406
Successive Ratio Answer
Successive Ratio = 1:2,2:3,3:7,7:8,8:9,9:64 or 0.5,0.6667,0.4286,0.875,0.8889,0.1406
Final Answers
6,5,4,7,3,2,1 RMS = 24.796313089997 Harmonic Mean = 3.1419347656685Geometric Mean = 5.691826851234 Mid-Range = 32.5 Weighted Average = 9.0142857142857 Successive Ratio = Successive Ratio = 1:2,2:3,3:7,7:8,8:9,9:64 or 0.5,0.6667,0.4286,0.875,0.8889,0.1406
What is the Answer?
6,5,4,7,3,2,1 RMS = 24.796313089997 Harmonic Mean = 3.1419347656685Geometric Mean = 5.691826851234 Mid-Range = 32.5 Weighted Average = 9.0142857142857 Successive Ratio = Successive Ratio = 1:2,2:3,3:7,7:8,8:9,9:64 or 0.5,0.6667,0.4286,0.875,0.8889,0.1406
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 = σ^{2} 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 = n_{1}/n_{0} 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 σ^{2} = ΣE(X_{i} - 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) = 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