# zscore

Standardized *z*-scores

## Syntax

## Description

returns
the `Z`

= zscore(`X`

)*z*-score for
each element of `X`

such that columns of `X`

are
centered to have mean 0 and scaled to have standard deviation 1. `Z`

is
the same size as `X`

.

If

`X`

is a vector, then`Z`

is a vector of*z*-scores.If

`X`

is a matrix, then`Z`

is a matrix of the same size as`X`

, and each column of`Z`

has mean 0 and standard deviation 1.For multidimensional arrays,

*z*-scores in`Z`

are computed along the first nonsingleton dimension of`X`

.

scales `Z`

= zscore(`X`

,`flag`

)`X`

using
the standard deviation indicated by `flag`

.

If

`flag`

is 0 (default), then`zscore`

scales`X`

using the sample standard deviation, with*n*- 1 in the denominator of the standard deviation formula.`zscore(X,0)`

is the same as`zscore(X)`

.If

`flag`

is 1, then`zscore`

scales`X`

using the population standard deviation, with*n*in the denominator of standard deviation formula.

## Examples

### Z-Scores of Two Data Vectors

Compute and plot the $$z$$-scores of two data vectors, and then compare the results.

Load the sample data.

`load lawdata`

Two variables load into the workspace: `gpa`

and `lsat`

.

Plot both variables on the same axes.

plot([gpa,lsat]) legend('gpa','lsat','Location','East')

It is difficult to compare these two measures because they are on a very different scale.

Plot the $$z$$-scores of `gpa`

and `lsat`

on the same axes.

Zgpa = zscore(gpa); Zlsat = zscore(lsat); plot([Zgpa, Zlsat]) legend('gpa z-scores','lsat z-scores','Location','Northeast')

Now, you can see the relative performance of individuals with respect to both their `gpa`

and `lsat`

results. For example, the third individual’s `gpa`

and `lsat`

results are both one standard deviation below the sample mean. The eleventh individual’s `gpa`

is around the sample mean but has an `lsat`

score almost 1.25 standard deviations above the sample average.

Check the mean and standard deviation of the $$z$$-scores you created.

mean([Zgpa,Zlsat])

ans =1×210^{-14}× -0.1088 0.0357

std([Zgpa,Zlsat])

`ans = `*1×2*
1 1

By definition, $$z$$-scores of `gpa`

and `lsat`

have mean 0 and standard deviation 1.

### Z-Scores for a Population vs. Sample

Load the sample data.

`load lawdata`

Two variables load into the workspace: `gpa`

and `lsat`

.

Compute the $$z$$-scores of `gpa`

using the population formula for standard deviation.

Z1 = zscore(gpa,1); % population formula Z0 = zscore(gpa,0); % sample formula disp([Z1 Z0])

1.2554 1.2128 0.8728 0.8432 -1.2100 -1.1690 -0.2749 -0.2656 1.4679 1.4181 -0.1049 -0.1013 -0.4024 -0.3888 1.4254 1.3771 1.1279 1.0896 0.1502 0.1451 0.1077 0.1040 -1.5076 -1.4565 -1.4226 -1.3743 -0.9125 -0.8815 -0.5724 -0.5530

For a sample from a population, the population standard deviation formula with $$n$$ in the denominator corresponds to the maximum likelihood estimate of the population standard deviation, and might be biased. The sample standard deviation formula, on the other hand, is the unbiased estimator of the population standard deviation for a sample.

### Z-Scores of a Data Matrix

Compute $$z$$-scores using the mean and standard deviation computed along the columns or rows of a data matrix.

Load the sample data.

`load flu`

The dataset array `flu`

is loaded in the workplace. `flu`

has 52 observations on 11 variables. The first variable contains dates (in weeks). The other variables contain the flu estimates for different regions in the U.S.

Convert the dataset array to a data matrix.

flu2 = double(flu(:,2:end));

The new data matrix, `flu2`

, is a 52-by-10 double data matrix. The rows correspond to the weeks and the columns correspond to the U.S. regions in the data set array `flu`

.

Standardize the flu estimate for each region (the *columns* of `flu2`

).

Z1 = zscore(flu2,[ ],1);

You can see the $$z$$-scores in the variable editor by double-clicking on the matrix `Z1`

created in the workspace.

Standardize the flu estimate for each week (the *rows* of `flu2`

).

Z2 = zscore(flu2,[ ],2);

### Z-Scores of Multidimensional Array

Find the z-scores of a multidimensional array by specifying to standardize the data along different dimensions. Compare the results when using the `'all'`

, `dim`

, and `vecdim`

input arguments.

Create a 3-by-4-by-2 array.

X = reshape(1:24,[3 4 2])

X = X(:,:,1) = 1 4 7 10 2 5 8 11 3 6 9 12 X(:,:,2) = 13 16 19 22 14 17 20 23 15 18 21 24

Standardize `X`

by using the mean and standard deviation of all the values in `X`

.

`Zall = zscore(X,0,'all')`

Zall = Zall(:,:,1) = -1.6263 -1.2021 -0.7778 -0.3536 -1.4849 -1.0607 -0.6364 -0.2121 -1.3435 -0.9192 -0.4950 -0.0707 Zall(:,:,2) = 0.0707 0.4950 0.9192 1.3435 0.2121 0.6364 1.0607 1.4849 0.3536 0.7778 1.2021 1.6263

The resulting multidimensional array of z-scores has mean 0 and standard deviation 1. For example, compute the mean and standard deviation of `Zall`

.

`mZall = mean(Zall(:,:,:),'all')`

mZall = -9.2519e-18

`sZall = std(Zall(:,:,:),0,'all')`

sZall = 1.0000

Now standardize `X`

along the second dimension.

Zdim = zscore(X,0,2)

Zdim = Zdim(:,:,1) = -1.1619 -0.3873 0.3873 1.1619 -1.1619 -0.3873 0.3873 1.1619 -1.1619 -0.3873 0.3873 1.1619 Zdim(:,:,2) = -1.1619 -0.3873 0.3873 1.1619 -1.1619 -0.3873 0.3873 1.1619 -1.1619 -0.3873 0.3873 1.1619

The elements in each row of each page of `Zdim`

have mean 0 and standard deviation 1. For example, compute the mean and standard deviation of the first row of the second page of `Zdim`

.

`mZdim = mean(Zdim(1,:,2),'all')`

mZdim = 0

`sZdim = std(Zdim(1,:,2),0,'all')`

sZdim = 1

Finally, standardize `X`

based on the second and third dimensions.

Zvecdim = zscore(X,0,[2 3])

Zvecdim = Zvecdim(:,:,1) = -1.4289 -1.0206 -0.6124 -0.2041 -1.4289 -1.0206 -0.6124 -0.2041 -1.4289 -1.0206 -0.6124 -0.2041 Zvecdim(:,:,2) = 0.2041 0.6124 1.0206 1.4289 0.2041 0.6124 1.0206 1.4289 0.2041 0.6124 1.0206 1.4289

The elements in each `Zvecdim(i,:,:)`

slice have mean 0 and standard deviation 1. For example, compute the mean and standard deviation of the elements in `Zvecdim(1,:,:)`

.

`mZvecdim = mean(Zvecdim(1,:,:),'all')`

mZvecdim = 2.7756e-17

`sZvecdim = std(Zvecdim(1,:,:),0,'all')`

sZvecdim = 1

### Z-Scores, Mean, and Standard Deviation

Return the mean and standard deviation used to compute the $$z$$-scores.

Load the sample data.

`load lawdata`

Two variables load into the workspace: `gpa`

and `lsat`

.

Return the $$z$$-scores, mean, and standard deviation of `gpa`

.

[Z,gpamean,gpastdev] = zscore(gpa)

`Z = `*15×1*
1.2128
0.8432
-1.1690
-0.2656
1.4181
-0.1013
-0.3888
1.3771
1.0896
0.1451
⋮

gpamean = 3.0947

gpastdev = 0.2435

## Input Arguments

`X`

— Input data

vector | matrix | multidimensional array

Input data, specified as a vector, matrix, or multidimensional array.

**Data Types: **`double`

| `single`

`flag`

— Indicator for the standard deviation

0 (default) | 1

Indicator for the standard deviation used to compute the *z*-scores,
specified as 0 or 1.

If

`flag`

is 0 (default), then`zscore`

scales`X`

using the sample standard deviation.`zscore(X,0)`

is the same as`zscore(X)`

.If

`flag`

is 1, then`zscore`

scales`X`

using the population standard deviation.

`dim`

— Dimension

positive integer scalar

Dimension along which to calculate the *z*-scores of `X`

,
specified as a positive integer scalar. If you do not specify a value, then the default
value is the first array dimension whose size does not equal 1.

For example, for a matrix `X`

, if `dim`

= 1, then
`zscore`

uses the means and standard deviations along the columns of
`X`

, and if `dim`

= 2, then
`zscore`

uses the means and standard deviations along the rows of
`X`

.

`vecdim`

— Vector of dimensions

positive integer vector

Vector of dimensions along which to calculate the *z*-scores of
`X`

, specified as a positive integer vector. Each element of
`vecdim`

represents a dimension of the input array
`X`

. The output `Z`

has the same dimensions as
`X`

, but the mean `mu`

and standard deviation
`sigma`

each have length 1 in the operating dimensions. The other
dimension lengths are the same for `X`

, `mu`

, and
`sigma`

.

For example, if `X`

is a 2-by-3-by-3 array, then
`zscore(X,0,[1 2])`

uses the means and standard deviations along the
pages of `X`

to standardize the values of
`X`

.

**Data Types: **`single`

| `double`

## Output Arguments

`Z`

— *z-*scores

vector | matrix | multidimensional array

*z-*scores, returned as a vector, matrix, or multidimensional array.
`Z`

has the same dimensions as `X`

.

The values of `Z`

depend on whether you specify
`'all'`

, `dim`

, or `vecdim`

. If
you do not specify any of these input arguments, then the following conditions apply:

If

`X`

is a vector, then`Z`

is a vector of*z*-scores with mean 0 and variance 1.If

`X`

is an array, then`zscore`

standardizes along the first nonsingleton dimension of`X`

.

For an example that demonstrates the differences in `Z`

when you
use `'all'`

, `dim`

, and `vecdim`

,
see Z-Scores of Multidimensional Array.

`mu`

— Mean

scalar | vector | matrix | multidimensional array

Mean of `X`

used to compute the *z*-scores, returned as a
scalar, vector, matrix, or multidimensional array. `mu`

has length 1
in the specified operating dimensions. The other dimension lengths are the same for
`X`

and `mu`

.

For example, if `X`

is a 2-by-3-by-3 array and
`vecdim`

is `[1 2]`

, then `mu`

is a 1-by-1-by-3 array of means. Each value in `mu`

corresponds to
the mean of a page in `X`

.

`sigma`

— Standard deviation

scalar | vector | matrix | multidimensional array

Standard deviation of `X`

used to compute the *z*-scores,
returned as a scalar, vector, matrix, or multidimensional array.
`sigma`

has length 1 in the specified operating dimensions. The
other dimension lengths are the same for `X`

and
`sigma`

.

For example, if `X`

is a 2-by-3-by-3 array and
`vecdim`

is `[1 2]`

, then
`sigma`

is a 1-by-1-by-3 array of standard deviations. Each value
in `sigma`

corresponds to the standard deviation of a page in
`X`

.

## More About

### Z-Score

For a random variable *X* with
mean μ and standard deviation σ, the *z*-score
of a value *x* is

$$z=\frac{\left(x-\mu \right)}{\sigma}.$$

For sample data with mean $$\overline{X}$$ and standard deviation *S*,
the *z*-score of a data point *x* is

$$z=\frac{\left(x-\overline{X}\right)}{S}.$$

*z*-scores measure the distance of a data point
from the mean in terms of the standard deviation. This is also called *standardization* of
data. The standardized data set has mean 0 and standard deviation
1, and retains the shape properties of the original data set (same
skewness and kurtosis).

You can use *z*-scores to put data on the same scale before further analysis.
This lets you compare two or more data sets with different units.

### Multidimensional Array

A *multidimensional array* is
an array with more than two dimensions. For example, if X is a 1-by-3-by-4
array, then `X`

is a three-dimensional array.

### First Nonsingleton Dimension

A *first nonsingleton dimension*
is the first dimension of an array whose size is not equal to 1. For
example, if `X`

is a 1-by-2-by-3-by-4 array, then
the second dimension is the first nonsingleton dimension of `X`

.

### Sample Standard Deviation

The *sample standard deviation*
*S* is given by

$$S=\sqrt{\frac{{\displaystyle {\sum}_{i=1}^{n}{\left({x}_{i}-\overline{X}\right)}^{2}}}{n-1}}.$$

*S* is the square root of an unbiased estimator of the
variance of the population from which `X`

is drawn, as long as
`X`

consists of independent, identically distributed samples. $$\overline{X}$$ is the sample mean.

Notice that the denominator in this variance formula is *n* – 1.

### Population Standard Deviation

If the data is the entire population of values, then you can use the
*population standard deviation*,

$$\sigma =\sqrt{\frac{{\displaystyle {\sum}_{i=1}^{n}{\left({x}_{i}-\mu \right)}^{2}}}{n}}.$$

If `X`

is a random sample from a population, then the
mean *μ* is estimated by the sample mean, and *σ* is the
biased maximum likelihood estimator of the population standard deviation.

Notice that the denominator in this variance formula is *n*.

## Algorithms

`zscore`

returns `NaN`

s for
any sample containing `NaN`

s.

`zscore`

returns `0`

s for any sample that is constant (all
values are the same). For example, if `X`

is a vector of the same numeric
value, then `Z`

is a vector of `0`

s.

## Extended Capabilities

### Tall Arrays

Calculate with arrays that have more rows than fit in memory.

This function fully supports tall arrays. For more information, see Tall Arrays.

### C/C++ Code Generation

Generate C and C++ code using MATLAB® Coder™.

Usage notes and limitations:

The

`'all'`

and`vecdim`

input arguments are not supported.The

`dim`

input argument must be a compile-time constant.If you do not specify the

`dim`

input argument, the working (or operating) dimension can be different in the generated code. As a result, run-time errors can occur. For more details, see Automatic dimension restriction (MATLAB Coder).

For more information on code generation, see Introduction to Code Generation and General Code Generation Workflow.

### Thread-Based Environment

Run code in the background using MATLAB® `backgroundPool`

or accelerate code with Parallel Computing Toolbox™ `ThreadPool`

.

This function fully supports thread-based environments. For more information, see Run MATLAB Functions in Thread-Based Environment.

### GPU Arrays

Accelerate code by running on a graphics processing unit (GPU) using Parallel Computing Toolbox™.

Usage notes and limitations:

The

`'all'`

and`vecdim`

input arguments are not supported.

For more information, see Run MATLAB Functions on a GPU (Parallel Computing Toolbox).

## Version History

**Introduced before R2006a**

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