# Scalar Statistics

The package implements functions for computing various statistics over an array of scalar real numbers.

## Weighted sum and mean

Base.sumFunction
sum(v::AbstractArray, w::AbstractWeights{<:Real}; [dims])

Compute the weighted sum of an array v with weights w, optionally over the dimension dims.

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Base.sum!Function
sum!(R::AbstractArray, A::AbstractArray,
w::AbstractWeights{<:Real}, dim::Int;
init::Bool=true)

Compute the weighted sum of A with weights w over the dimension dim and store the result in R. If init=false, the sum is added to R rather than starting from zero.

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StatsBase.wsumFunction
wsum(v, w::AbstractVector, [dim])

Compute the weighted sum of an array v with weights w, optionally over the dimension dim.

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StatsBase.wsum!Function
wsum!(R::AbstractArray, A::AbstractArray,
w::AbstractVector, dim::Int;
init::Bool=true)

Compute the weighted sum of A with weights w over the dimension dim and store the result in R. If init=false, the sum is added to R rather than starting from zero.

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Statistics.meanFunction
mean(A::AbstractArray, w::AbstractWeights[, dims::Int])

Compute the weighted mean of array A with weight vector w (of type AbstractWeights). If dim is provided, compute the weighted mean along dimension dims.

Examples

n = 20
x = rand(n)
w = rand(n)
mean(x, weights(w))
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Statistics.mean!Function
mean!(R::AbstractArray, A::AbstractArray, w::AbstractWeights[; dims=nothing])

Compute the weighted mean of array A with weight vector w (of type AbstractWeights) along dimension dims, and write results to R.

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## Means

The package provides functions to compute means of different kinds.

StatsBase.genmeanFunction
genmean(a, p)

Return the generalized/power mean with exponent p of a real-valued array, i.e. $\left( \frac{1}{n} \sum_{i=1}^n a_i^p \right)^{\frac{1}{p}}$, where n = length(a). It is taken to be the geometric mean when p == 0.

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## Moments

Statistics.varFunction
var(x::AbstractArray, w::AbstractWeights, [dim]; mean=nothing, corrected=false)

Compute the variance of a real-valued array x, optionally over a dimension dim. Observations in x are weighted using weight vector w. The uncorrected (when corrected=false) sample variance is defined as:

$\frac{1}{\sum{w}} \sum_{i=1}^n {w_i\left({x_i - μ}\right)^2 }$

where $n$ is the length of the input and $μ$ is the mean. The unbiased estimate (when corrected=true) of the population variance is computed by replacing $\frac{1}{\sum{w}}$ with a factor dependent on the type of weights used:

• AnalyticWeights: $\frac{1}{\sum w - \sum {w^2} / \sum w}$
• FrequencyWeights: $\frac{1}{\sum{w} - 1}$
• ProbabilityWeights: $\frac{n}{(n - 1) \sum w}$ where $n$ equals count(!iszero, w)
• Weights: ArgumentError (bias correction not supported)
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Statistics.stdFunction
std(x::AbstractArray, w::AbstractWeights, [dim]; mean=nothing, corrected=false)

Compute the standard deviation of a real-valued array x, optionally over a dimension dim. Observations in x are weighted using weight vector w. The uncorrected (when corrected=false) sample standard deviation is defined as:

$\sqrt{\frac{1}{\sum{w}} \sum_{i=1}^n {w_i\left({x_i - μ}\right)^2 }}$

where $n$ is the length of the input and $μ$ is the mean. The unbiased estimate (when corrected=true) of the population standard deviation is computed by replacing $\frac{1}{\sum{w}}$ with a factor dependent on the type of weights used:

• AnalyticWeights: $\frac{1}{\sum w - \sum {w^2} / \sum w}$
• FrequencyWeights: $\frac{1}{\sum{w} - 1}$
• ProbabilityWeights: $\frac{n}{(n - 1) \sum w}$ where $n$ equals count(!iszero, w)
• Weights: ArgumentError (bias correction not supported)
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StatsBase.mean_and_varFunction
mean_and_var(x, [w::AbstractWeights], [dim]; corrected=false) -> (mean, var)

Return the mean and variance of collection x. If x is an AbstractArray, dim can be specified as a tuple to compute statistics over these dimensions. A weighting vector w can be specified to weight the estimates. Finally, bias correction is be applied to the variance calculation if corrected=true. See var documentation for more details.

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StatsBase.mean_and_stdFunction
mean_and_std(x, [w::AbstractWeights], [dim]; corrected=false) -> (mean, std)

Return the mean and standard deviation of collection x. If x is an AbstractArray, dim can be specified as a tuple to compute statistics over these dimensions. A weighting vector w can be specified to weight the estimates. Finally, bias correction is applied to the standard deviation calculation if corrected=true. See std documentation for more details.

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StatsBase.skewnessFunction
skewness(v, [wv::AbstractWeights], m=mean(v))

Compute the standardized skewness of a real-valued array v, optionally specifying a weighting vector wv and a center m.

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StatsBase.kurtosisFunction
kurtosis(v, [wv::AbstractWeights], m=mean(v))

Compute the excess kurtosis of a real-valued array v, optionally specifying a weighting vector wv and a center m.

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StatsBase.momentFunction
moment(v, k, [wv::AbstractWeights], m=mean(v))

Return the kth order central moment of a real-valued array v, optionally specifying a weighting vector wv and a center m.

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## Measurements of Variation

StatsBase.spanFunction
span(x)

Return the span of a collection, i.e. the range minimum(x):maximum(x). The minimum and maximum of x are computed in one pass using extrema.

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StatsBase.variationFunction
variation(x, m=mean(x))

Return the coefficient of variation of collection x, optionally specifying a precomputed mean m. The coefficient of variation is the ratio of the standard deviation to the mean.

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StatsBase.semFunction
sem(x)

Return the standard error of the mean of collection x, i.e. sqrt(var(x, corrected=true) / length(x)).

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StatsBase.madFunction
mad(x; center=median(x), normalize=true)

Compute the median absolute deviation (MAD) of collection x around center (by default, around the median).

If normalize is set to true, the MAD is multiplied by 1 / quantile(Normal(), 3/4) ≈ 1.4826, in order to obtain a consistent estimator of the standard deviation under the assumption that the data is normally distributed.

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StatsBase.mad!Function
StatsBase.mad!(x; center=median!(x), normalize=true)

Compute the median absolute deviation (MAD) of array x around center (by default, around the median), overwriting x in the process. x must be able to hold values of generated by calling middle on its elements (for example an integer vector is not appropriate since middle can produce non-integer values).

If normalize is set to true, the MAD is multiplied by 1 / quantile(Normal(), 3/4) ≈ 1.4826, in order to obtain a consistent estimator of the standard deviation under the assumption that the data is normally distributed.

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## Z-scores

StatsBase.zscoreFunction
zscore(X, [μ, σ])

Compute the z-scores of X, optionally specifying a precomputed mean μ and standard deviation σ. z-scores are the signed number of standard deviations above the mean that an observation lies, i.e. $(x - μ) / σ$.

μ and σ should be both scalars or both arrays. The computation is broadcasting. In particular, when μ and σ are arrays, they should have the same size, and size(μ, i) == 1 || size(μ, i) == size(X, i) for each dimension.

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StatsBase.zscore!Function
zscore!([Z], X, μ, σ)

Compute the z-scores of an array X with mean μ and standard deviation σ. z-scores are the signed number of standard deviations above the mean that an observation lies, i.e. $(x - μ) / σ$.

If a destination array Z is provided, the scores are stored in Z and it must have the same shape as X. Otherwise X is overwritten.

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StatsBase.entropyFunction
entropy(p, [b])

Compute the entropy of a collection of probabilities p, optionally specifying a real number b such that the entropy is scaled by 1/log(b). Elements with probability 0 or 1 add 0 to the entropy.

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StatsBase.crossentropyFunction
crossentropy(p, q, [b])

Compute the cross entropy between p and q, optionally specifying a real number b such that the result is scaled by 1/log(b).

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StatsBase.kldivergenceFunction
kldivergence(p, q, [b])

Compute the Kullback-Leibler divergence from q to p, also called the relative entropy of p with respect to q, that is the sum pᵢ * log(pᵢ / qᵢ). Optionally a real number b can be specified such that the divergence is scaled by 1/log(b).

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StatsBase.percentileFunction
percentile(x, p)

Return the pth percentile of a collection x, i.e. quantile(x, p / 100).

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StatsBase.iqrFunction
iqr(x)

Compute the interquartile range (IQR) of collection x, i.e. the 75th percentile minus the 25th percentile.

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StatsBase.nquantileFunction
nquantile(x, n::Integer)

Return the n-quantiles of collection x, i.e. the values which partition v into n subsets of nearly equal size.

Equivalent to quantile(x, [0:n]/n). For example, nquantiles(x, 5) returns a vector of quantiles, respectively at [0.0, 0.2, 0.4, 0.6, 0.8, 1.0].

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Statistics.quantileFunction
quantile(v, w::AbstractWeights, p)

Compute the weighted quantiles of a vector v at a specified set of probability values p, using weights given by a weight vector w (of type AbstractWeights). Weights must not be negative. The weights and data vectors must have the same length. NaN is returned if x contains any NaN values. An error is raised if w contains any NaN values.

With FrequencyWeights, the function returns the same result as quantile for a vector with repeated values. Weights must be integers.

With non FrequencyWeights, denote $N$ the length of the vector, $w$ the vector of weights, $h = p (\sum_{i<= N} w_i - w_1) + w_1$ the cumulative weight corresponding to the probability $p$ and $S_k = \sum_{i<=k} w_i$ the cumulative weight for each observation, define $v_{k+1}$ the smallest element of v such that $S_{k+1}$ is strictly superior to $h$. The weighted $p$ quantile is given by $v_k + \gamma (v_{k+1} - v_k)$ with $\gamma = (h - S_k)/(S_{k+1} - S_k)$. In particular, when all weights are equal, the function returns the same result as the unweighted quantile.

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Statistics.medianMethod
median(v::RealVector, w::AbstractWeights)

Compute the weighted median of v with weights w (of type AbstractWeights). See the documentation for quantile for more details.

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## Mode and Modes

StatsBase.modeFunction
mode(a, [r])
mode(a::AbstractArray, wv::AbstractWeights)

Return the mode (most common number) of an array, optionally over a specified range r or weighted via a vector wv. If several modes exist, the first one (in order of appearance) is returned.

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StatsBase.modesFunction
modes(a, [r])::Vector
mode(a::AbstractArray, wv::AbstractWeights)::Vector

Return all modes (most common numbers) of an array, optionally over a specified range r or weighted via vector wv.

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## Summary Statistics

StatsBase.summarystatsFunction
summarystats(a)

Compute summary statistics for a real-valued array a. Returns a SummaryStats object containing the mean, minimum, 25th percentile, median, 75th percentile, and maxmimum.

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## Reliability Measures

StatsBase.cronbachalphaFunction
cronbachalpha(covmatrix::AbstractMatrix{<:Real})

Calculate Cronbach's alpha (1951) from a covariance matrix covmatrix according to the formula:

$\rho = \frac{k}{k-1} (1 - \frac{\sum^k_{i=1} \sigma^2_i}{\sum_{i=1}^k \sum_{j=1}^k \sigma_{ij}})$

where $k$ is the number of items, i.e. columns, $\sigma_i^2$ the item variance, and $\sigma_{ij}$ the inter-item covariance.

Returns a CronbachAlpha object that holds:

• alpha: the Cronbach's alpha score for all items, i.e. columns, in covmatrix; and
• dropped: a vector giving Cronbach's alpha scores if a specific item, i.e. column, is dropped from covmatrix.

Example

julia> using StatsBase

julia> cov_X = [10 6 6 6;
6 11 6 6;
6 6 12 6;
6 6 6 13];

julia> cronbachalpha(cov_X)
Cronbach's alpha for all items: 0.8136

Cronbach's alpha if an item is dropped:
item 1: 0.7500
item 2: 0.7606
item 3: 0.7714
item 4: 0.7826
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