Step 2: Using original number set, assign the rank value:
Since we have 5 numbers in our original number set, we assign ranks from lowest to highest (1 to 5) Our original number set in unsorted order was 120,99,101,87,140 Our respective ranked data set is 4,2,3,1,5
Root Mean Square Calculation
Root Mean Square =
√A
√N
where A = x12 + x22 + x32 + x42 + x52 and N = 5 number set items
Calculate the Median (Middle Value) Since our number set contains 5 elements which is an odd number, our median number is determined as follows: Number Set = (n1,n2,n3,n4,n5) Median Number = Entry ½(n + 1) Median Number = Entry ½(6) Median Number = n3
Therefore, we sort our number set in ascending order and our median is entry 3 of our number set highlighted in red: (87,99,101,120,140) Median = 101
The highest frequency of occurence in our number set is 1 times by the following numbers in green: (120,99,101,87,140) 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
With N = 5 and each xi a member of the number set you entered, we have:
Let's evaluate the square difference from the mean of each term (Xi - μ)3: (X1 - μ)3 = (87 - 109.4)3 = -22.43 = -11239.424 (X2 - μ)3 = (99 - 109.4)3 = -10.43 = -1124.864 (X3 - μ)3 = (101 - 109.4)3 = -8.43 = -592.704 (X4 - μ)3 = (120 - 109.4)3 = 10.63 = 1191.016 (X5 - μ)3 = (140 - 109.4)3 = 30.63 = 28652.616
Adding our 5 sum of cubed differences up, we have our skewness numerator: ΣE(Xi - μ)3 = -11239.424 + -1124.864 + -592.704 + 1191.016 + 28652.616 ΣE(Xi - μ)3 = 16886.64
Now that we have the sum of cubed differences from the means, calculate skewness:
Skewness =
E(Xi - μ)3
(n - 1)σ3
Skewness =
16886.64
(5 - 1)18.59683
Skewness =
16886.64
(4)6431.5353553592
Skewness =
16886.64
25726.141421437
Skewness = 0.6564000299683
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| = |87 - 109.4| = |-22.4| = 22.4 |X2 - μ| = |99 - 109.4| = |-10.4| = 10.4 |X3 - μ| = |101 - 109.4| = |-8.4| = 8.4 |X4 - μ| = |120 - 109.4| = |10.6| = 10.6 |X5 - μ| = |140 - 109.4| = |30.6| = 30.6
Adding our 5 absolute value of differences from the mean, we have our average deviation numerator: Σ|Xi - μ| = 22.4 + 10.4 + 8.4 + 10.6 + 30.6 Σ|Xi - μ| = 82.4
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 =
82.4
5
Average Deviation = 16.48
Calculate the Median (Middle Value) Since our number set contains 5 elements which is an odd number, our median number is determined as follows: Number Set = (n1,n2,n3,n4,n5) Median Number = Entry ½(n + 1) Median Number = Entry ½(6) Median Number = n3
Therefore, we sort our number set in ascending order and our median is entry 3 of our number set highlighted in red: (87,99,101,120,140) Median = 101
The highest frequency of occurence in our number set is 1 times by the following numbers in green: (120,99,101,87,140) 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 36 < v < 67.5 and 151.5 < v < 183 87,99,101,120,140
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 < 36 or v > 183 87,99,101,120,140
Calculate weighted average
120,99,101,87,140
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
n
Weighted Average =
120 x 0.2 + 99 x 0.4 + 101 x 0.6 + 87 x 0.8 + 140 x 0.9
5
Weighted Average =
24 + 39.6 + 60.6 + 69.6 + 126
5
Weighted Average =
319.8
5
Weighted Average = 63.96
Frequency Distribution Table
Show the freqency distribution table for this number set
87, 99, 101, 120, 140
Determine the Number of Intervals using Sturges Rule:
We need to choose the smallest integer k such that 2k ≥ n where n = 5
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 5
Therefore, we use 3 intervals
Our maximum value in our number set of 140 - 87 = 53
Each interval size is the difference of the maximum and minimum value divided by the number of intervals
Interval Size =
53
3
Add 1 to this giving us 17 + 1 = 18
Frequency Distribution Table
Class Limits
Class Boundaries
FD
CFD
RFD
CRFD
87 - 105
86.5 - 105.5
3
3
3/5 = 60%
3/5 = 60%
105 - 123
104.5 - 123.5
1
3 + 1 = 4
1/5 = 20%
4/5 = 80%
123 - 141
122.5 - 141.5
1
3 + 1 + 1 = 5
1/5 = 20%
5/5 = 100%
5
100%
Successive Ratio Calculation
Go through our 5 numbers
Determine the ratio of each number to the next one
Successive Ratio 1: 87,99,101,120,140
87:99 → 0.8788
Successive Ratio 2: 87,99,101,120,140
99:101 → 0.9802
Successive Ratio 3: 87,99,101,120,140
101:120 → 0.8417
Successive Ratio 4: 87,99,101,120,140
120:140 → 0.8571
Successive Ratio Answer
Successive Ratio = 87:99,99:101,101:120,120:140 or 0.8788,0.9802,0.8417,0.8571
Final Answers
4,2,3,1,5 RMS = 110.96936514192 Harmonic Mean = 106.44538844784Geometric Mean = 107.88399507889 Mid-Range = 113.5 Weighted Average = 63.96 Successive Ratio = Successive Ratio = 87:99,99:101,101:120,120:140 or 0.8788,0.9802,0.8417,0.8571
You have 2 free calculationss remaining
What is the Answer?
4,2,3,1,5 RMS = 110.96936514192 Harmonic Mean = 106.44538844784Geometric Mean = 107.88399507889 Mid-Range = 113.5 Weighted Average = 63.96 Successive Ratio = Successive Ratio = 87:99,99:101,101:120,120:140 or 0.8788,0.9802,0.8417,0.8571
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
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