You entered a number set X of {2,4,6,8,10} Do 3 things Run statistical calculations on this Treat ths as a vector and run calculations on this Treat this as a sequence
Order This Number Set
Order lowest to highest 2 < 4 < 6 < 8 < 10
Order highest to lowest 10 > 8 > 6 > 4 > 2
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 =
ΣX_{i}
n
μ =
2 + 4 + 6 + 8 + 10
5
μ =
30
5
μ = 6
Calculate variance denoted as σ^{2}
Evaluate the square difference from the mean of each term (X_{i} - μ)^{2}:
Calculate average deviation (mean absolute deviation)
AD =
Σ|X_{i} - μ|
n
AD =
30
5
Average Deviation = 6
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 = (n_{1},n_{2},n_{3},n_{4},n_{5}) Median Number = Entry ½(n + 1) Median Number = Entry ½(6) Median Number = n_{3}
Final Median Calculation
Therefore, we sort our number set in ascending order and our median is entry 3 of our number set highlighted in red: ( 2) ,4) ,6) ,8) ,10) Median = 6
The highest frequency of occurence in our number set is 1 times by the following numbers in green: (2,4,6,8,10) Since the maximum frequency of any number is 1, no mode exists.
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 -8 < v < -2 and 14 < v < 20 2,4,6,8,10
Calculate Highly Suspect Outliers:
Highly Suspect Outliers are values outside the outer fences Mark all values in our dataset (v) in red below such that v < -8 or v > 20 2,4,6,8,10
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:
{10,8,6,4,2}
Stem and Leaf Plot
Stem
Leaf
1
0
8
6
4
2
Ranked Data Calculation
Sort our number set in ascending order and assign a ranking to each number:
Ranked Data Table
Number Set Value
2
4
6
8
10
Rank
1
2
3
4
5
Step 2: Using original number set, assign the rank value:
Assign ranks from lowest to highest (1 to 5) Our original number set in unsorted order was 2,4,6,8,10 Our respective ranked data set is 1,2,3,4,5
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} and N = 5 number set items
Calculate A
A = 2^{2} + 4^{2} + 6^{2} + 8^{2} + 10^{2} A = 4 + 16 + 36 + 64 + 100 A = 220
Calculate Root Mean Square (RMS):
RMS =
√220
√5
RMS =
14.832396974191
2.2360679774998
RMS = 6.6332495807108
Harmonic-Geometric-Mid Range Weighted Average
Calculate the harmonic mean, geometric mean, mid-range, weighted-average:
Calculate Harmonic Mean:
Harmonic Mean =
N
1/x_{1} + 1/x_{2} + 1/x_{3} + 1/x_{4} + 1/x_{5}
With N = 5 and each x_{i} a member of the number set you entered, we have:
Harmonic Mean =
5
1/2 + 1/4 + 1/6 + 1/8 + 1/10
Harmonic Mean =
5
0.5 + 0.25 + 0.16666666666667 + 0.125 + 0.1
Harmonic Mean =
5
1.1416666666667
Harmonic Mean = 4.3795620437956
Calculate Geometric Mean:
Geometric Mean = (x_{1} * x_{2} * x_{3} * x_{4} * x_{5})^{1/N} Geometric Mean = (2 * 4 * 6 * 8 * 10)^{1/5} Geometric Mean = 3840^{0.2} Geometric Mean = 5.2103421693947
Calcualte Mid-Range:
Mid-Range =
Maximum Value in Number Set + Minimum Value in Number Set
2
Mid-Range =
10 + 2
2
Mid-Range =
12
2
Mid-Range = 6
Calculate weighted average
2,4,6,8,10
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
Show the freqency distribution table for this number set 2, 4, 6, 8, 10
Determine the Number of Intervals using Sturges Rule:
We need to choose the smallest integer k such that 2^{k} ≥ n where n = 5 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 5 Therefore, we use 3 intervals Our maximum value in our number set of 10 - 2 = 8 Each interval size is the difference of the maximum and minimum value divided by the number of intervals
Interval Size =
8
3
Add 1 to this giving us 2 + 1 = 3
Frequency Distribution Table
Class Limits
Class Boundaries
FD
CFD
RFD
CRFD
2 - 5
1.5 - 5.5
2
2
2/5 = 40%
2/5 = 40%
5 - 8
4.5 - 8.5
1
2 + 1 = 3
1/5 = 20%
3/5 = 60%
8 - 11
7.5 - 11.5
2
2 + 1 + 2 = 5
2/5 = 40%
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: 2,4,6,8,10
2:4 → 0.5
Successive Ratio 2: 2,4,6,8,10
4:6 → 0.6667
Successive Ratio 3: 2,4,6,8,10
6:8 → 0.75
Successive Ratio 4: 2,4,6,8,10
8:10 → 0.8
Successive Ratio Answer
Successive Ratio = 2:4,4:6,6:8,8:10 or 0.5,0.6667,0.75,0.8
Treat this as a set
Calculate the power set and partitions For set S = {2,4,6,8,10}, show: Elements, cardinality, and power set
List the elements of S
Elements = set objects Use the ∈ symbol.
2 ∈ S
4 ∈ S
6 ∈ S
8 ∈ S
10 ∈ S
Cardinality of set S → |S|:
Cardinality = Number of set elements. Since the set S contains 5 elements |S| = 5
Determine the power set P:
Power set = Set of all subsets of S including S and ∅.
Calculate power set subsets
S contains 5 terms Power Set contains 2^{5} = 32 items
Build subsets of P
The subset A of a set B is A set where all elements of A are in B.
Find the explicit formula and terms Calculate the explicit formula Calculate term number 10 And the Sum of the first 10 terms for: 2,4,6,8,10
Explicit Formula
a_{n} = a_{1} + (n - 1)d
Define d
d = Δ between consecutive terms d = a_{n} - a_{n - 1}
We see a common difference = 2 We have a_{1} = 2 a_{n} = 2 + 2(n - 1)
Calculate Term (6)
Plug in n = 6 and d = 2 a_{6} = 2 + 2(6 - 1) a_{6} = 2 + 2(6 - 1) a_{6} = 2 + 2(5) a_{6} = 2 + 10 a_{6} = 12
Calculate Term (7)
Plug in n = 7 and d = 2 a_{7} = 2 + 2(7 - 1) a_{7} = 2 + 2(7 - 1) a_{7} = 2 + 2(6) a_{7} = 2 + 12 a_{7} = 14
Calculate Term (8)
Plug in n = 8 and d = 2 a_{8} = 2 + 2(8 - 1) a_{8} = 2 + 2(8 - 1) a_{8} = 2 + 2(7) a_{8} = 2 + 14 a_{8} = 16
Calculate Term (9)
Plug in n = 9 and d = 2 a_{9} = 2 + 2(9 - 1) a_{9} = 2 + 2(9 - 1) a_{9} = 2 + 2(8) a_{9} = 2 + 16 a_{9} = 18
Calculate Term (10)
Plug in n = 10 and d = 2 a_{10} = 2 + 2(10 - 1) a_{10} = 2 + 2(10 - 1) a_{10} = 2 + 2(9) a_{10} = 2 + 18 a_{10} = 20
Calculate S_{n}:
S_{n} = Sum of the first n terms
S_{n} =
n(a_{1} + a_{n})
2
Substituting n = 10, we get:
S_{10} =
10(a_{1} + a_{10})
2
S_{10} =
10(2 + 20)
2
S_{10} =
10(22)
2
S_{10} =
220
2
S_{10} = 110
Final Answers
Order lowest to highest = 2 < 4 < 6 < 8 < 10 Order highest to lowest = 10 > 8 > 6 > 4 > 2 μ = 6 σ^{2} = 44 σ = 6.6332 SEM = 3.3166 Skewness = 1.5418539495248 Average Deviation = 6 Median = 6 Mode = N/A Range = 8 Mid-Range = 6 PSC1 = No Pearsons Skewness Coefficient PSC2 = -2.7136 Entropy = 1.6094379124341 1,2,3,4,5 RMS = 6.6332495807108 Harmonic Mean = 4.3795620437956Geometric Mean = 5.2103421693947 Mid-Range = 6 Weighted Average = 4.2 Successive Ratio = Successive Ratio = 2:4,4:6,6:8,8:10 or 0.5,0.6667,0.75,0.8 P = {{}, {2}, {4}, {6}, {8}, {10}, {2,10}, {2,4}, {2,6}, {2,8}, {4,10}, {4,6}, {4,8}, {6,10}, {6,8}, {8,10}, {2,4,10}, {2,4,6}, {2,4,8}, {2,6,10}, {2,6,8}, {2,8,10}, {4,6,10}, {4,6,8}, {4,8,10}, {6,8,10}, {2,4,6,10}, {2,4,6,8}, {2,4,8,10}, {2,6,8,10}, {4,6,8,10}, {2,4,6,8,10}} ||A|| = 14.832396974191 S_{10} = 110
What is the Answer?
Order lowest to highest = 2 < 4 < 6 < 8 < 10 Order highest to lowest = 10 > 8 > 6 > 4 > 2 μ = 6 σ^{2} = 44 σ = 6.6332 SEM = 3.3166 Skewness = 1.5418539495248 Average Deviation = 6 Median = 6 Mode = N/A Range = 8 Mid-Range = 6 PSC1 = No Pearsons Skewness Coefficient PSC2 = -2.7136 Entropy = 1.6094379124341 1,2,3,4,5 RMS = 6.6332495807108 Harmonic Mean = 4.3795620437956Geometric Mean = 5.2103421693947 Mid-Range = 6 Weighted Average = 4.2 Successive Ratio = Successive Ratio = 2:4,4:6,6:8,8:10 or 0.5,0.6667,0.75,0.8 P = {{}, {2}, {4}, {6}, {8}, {10}, {2,10}, {2,4}, {2,6}, {2,8}, {4,10}, {4,6}, {4,8}, {6,10}, {6,8}, {8,10}, {2,4,10}, {2,4,6}, {2,4,8}, {2,6,10}, {2,6,8}, {2,8,10}, {4,6,10}, {4,6,8}, {4,8,10}, {6,8,10}, {2,4,6,10}, {2,4,6,8}, {2,4,8,10}, {2,6,8,10}, {4,6,8,10}, {2,4,6,8,10}} ||A|| = 14.832396974191 S_{10} = 110
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