# Stats

vCalc Reviewed

The Statistics Calculator provides college level statistics functionality.  This includes observation stats (min, max, mean, median, population and sample variance and standard deviation), sort, random sample, t-test, linear regression and more.  See the following:  Frequency Distribution

• Observational Stats:  This function accepts a string of numbers separated by commas and calculates observational statistics (count, min, max, sum, sum of squares (Σx²), square of the sum (Σx)², mean, median, mode, range, mid point, rand, sort up, sort down, rand, population variance, population standard deviation, the sample/experimental variance, sample/experimental standard deviation).
• Frequency Distribution:  This function lets you enter a string of numbers separated by commas, a low and high range and a number of bins.  It then computes how many of the observations are in each of the bins between the high and low values designated.
• Random Sample (k):  This generate a random sample of k items within a set.
• Percentile:  This computes the relative percentile of an observation verses a set.
• P(A) = F / T: This computes the probability of a favorable event in a total number of outcomes.   Normal Distribution and z SCORES
• P(n,S) - Binomial Probability:  Probability of S successes in n trials of a binomial distribution.
• BC: - The binomial coefficient derived from Pascal's Triangle
• zSCORE (y in X):  This computes the z SCORE of an observation in a set (X).
• zSCORE (y,μ,σ):  This computes the z SCORE of an observation based on the mean and standard deviation.
• z from P(y):  This computes the z SCORE based on a probability or percentile in a Normal Distribution table.
• P(y) left of z: This computes the percentile, probability or area under the curve of a Normal Distribution left of the z SCORE.
• P(y) right of z: This computes the percentile, probability or area under the curve of a Normal Distribution right of the z SCORE.
• Probability between z SCORE:  This computes the area under the Normal Distribution curve between z SCOREs.
• Raw Score (P,μ,σ): This computes the raw score associated with a percentile in a Normal Distribution (μ,σ).
• Paired Sample t-test:  This computes the various parameters associated with the Paired Sample t-test.
• y = a • bx:  This is linear equation used with Linear Regression to predict values of Y.
• Linear Regression:  This computes the regression line (least-squares) through a set of X and Y observations.  It also computes the regression coefficient (r).
• nCk:  This is the combinations function           Linear Regression
• nPr:  This is the permutations function
• n!:  This returns the factorial of any number.

# The Math

The formulas for the statistics are as follows:

#### sum

S = sum_1^n(x)

(Σx)² = S²

#### sum of squares (Σx²)

Σx² = sum_1^n(x^2)

#### averages

• mean:     mu = (sum(x))/n  where n is the number of observations
• median:  middle value if in an odd number of observations.  If there is an even number of observations, it's the average of the two middle values.
• mode:  the most repeated observation.
• mid-point:  mp = (min + max)/2

#### variance

• Population Variance: sigma^2 = (sum_1^n(x_n-mu)^2)/n
• Sample Variance:       sigma^2 = (sum_1^n(x_n-mu)^2)/(n-1)

#### standard deviation

• Population Standard Deviation:   sigma = sqrt((sum_1^n(x_n-mu)^2)/n)
• Sample Standard Deviation:         sigma = sqrt((sum_1^n(x_n-mu)^2)/(n-1))

#### Permutations

• Permutations:                                nPr =  (n!)/((n-r)!)
• Circular Permutations:                 Pc(n,r) =  (n!)/(r(n-r)!)
• Permutations with repetitions:   Pr(n) = nr

#### Combinations

• Combinations:                                    nCk=  (n!)/(k!(n-k)!)
• Combinations with repetitions:      Cr(n, k) =  ((n+k-1)!)/(k!(n-1)!)

#### z SCORE

The formula for the z SCORE is as follows:

z_y =   (y - μ_X)/s   where:

- y is the single observation
- μ is the mean
- s is the sample or population standard deviation.

#### Least-squares Linear Regression

The formula for the least-squared regression line is in the following form:   y = a + bx

where:  b = (  (sum(XY) - (sumX * sumY)"/n")) / (sum(X^2) - (sumX)^2"/n")

and:   a = MY - b*MX           MY is mean of Y.  MX is mean of X.

## Statistics in Culture

"There are three kinds of lies: lies, damn lies and statistics."   Mark Twain

## Thanks

Thanks to Dr. Lee Hammerstrom, professor of math stats at Eastern Nazarene College and Dr. Richard (Rich) D. Platt, Associate Professor of Psychology at St Mary's College of Maryland, for their advice and testing.  Special thanks to Ms. Caroline Robertson for getting this library of equations off the ground.

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