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Histogram

Histogram plot

Description

Histograms are a type of bar plot for numeric data that group the data into bins. After you create a Histogram object, you can modify aspects of the histogram by changing its property values. This is particularly useful for quickly modifying the properties of the bins or changing the display.

Creation

Syntax

histogram(X)
histogram(X,nbins)
histogram(X,edges)
histogram('BinEdges',edges,'BinCounts',counts)
histogram(C)
histogram(C,Categories)
histogram('Categories',Categories,'BinCounts',counts)
histogram(___,Name,Value)
histogram(ax,___)
h = histogram(___)

Description

example

histogram(X) creates a histogram plot of X. The histogram function uses an automatic binning algorithm that returns bins with a uniform width, chosen to cover the range of elements in X and reveal the underlying shape of the distribution. histogram displays the bins as rectangles such that the height of each rectangle indicates the number of elements in the bin.

example

histogram(X,nbins) uses a number of bins specified by the scalar, nbins.

example

histogram(X,edges) sorts X into bins with the bin edges specified by the vector, edges. Each bin includes the left edge, but does not include the right edge, except for the last bin which includes both edges.

histogram('BinEdges',edges,'BinCounts',counts) manually specifies bin edges and associated bin counts. histogram plots the specified bin counts and does not do any data binning.

example

histogram(C), where C is a categorical array, plots a histogram with a bar for each category in C.

histogram(C,Categories) plots only the subset of categories specified by Categories.

histogram('Categories',Categories,'BinCounts',counts) manually specifies categories and associated bin counts. histogram plots the specified bin counts and does not do any data binning.

example

histogram(___,Name,Value) specifies additional options with one or more Name,Value pair arguments using any of the previous syntaxes. For example, you can specify 'BinWidth' and a scalar to adjust the width of the bins, or 'Normalization' with a valid option ('count', 'probability', 'countdensity', 'pdf', 'cumcount', or 'cdf') to use a different type of normalization. For a list of properties, see Histogram Properties.

histogram(ax,___) plots into the axes specified by ax instead of into the current axes (gca). The option ax can precede any of the input argument combinations in the previous syntaxes.

example

h = histogram(___) returns a Histogram object. Use this to inspect and adjust the properties of the histogram. For a list of properties, see Histogram Properties.

Input Arguments

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Data to distribute among bins, specified as a vector, matrix, or multidimensional array. If X is not a vector, then histogram treats it as a single column vector, X(:), and plots a single histogram.

histogram ignores all NaN values. Similarly, histogram ignores Inf and -Inf values, unless the bin edges explicitly specify Inf or -Inf as a bin edge.

Note

If X contains integers of type int64 or uint64 that are larger than flintmax, then it is recommended that you explicitly specify the histogram bin edges. histogram automatically bins the input data using double precision, which lacks integer precision for numbers greater than flintmax.

Data Types: single | double | int8 | int16 | int32 | int64 | uint8 | uint16 | uint32 | uint64 | logical | datetime | duration

Categorical data, specified as a categorical array. histogram ignores undefined categorical values.

Data Types: categorical

Number of bins, specified as a positive integer. If you do not specify nbins, then histogram automatically calculates how many bins to use based on the values in X.

Example: histogram(X,15) creates a histogram with 15 bins.

Bin edges, specified as a vector. edges(1) is the left edge of the first bin, and edges(end) is the right edge of the last bin.

The value X(i) is in the kth bin if edges(k)X(i) < edges(k+1). The last bin also includes the right bin edge, so that it contains X(i) if edges(end-1)X(i)edges(end).

For datetime and duration data, edges must be a datetime or duration vector in monotonically increasing order.

Data Types: single | double | int8 | int16 | int32 | int64 | uint8 | uint16 | uint32 | uint64 | logical | datetime | duration

Note

This option only applies to categorical histograms.

Categories included in histogram, specified as a cell array of character vectors or categorical vector.

  • If you specify an input categorical array C, then by default, histogram plots a bar for each category in C. In that case, use Categories to specify a unique subset of the categories instead.

  • If you specify bin counts, then Categories specifies the associated category names for the histogram.

Example: h = histogram(C,{'Large','Small'}) plots only the categorical data in the categories 'Large' and 'Small'.

Example: histogram('Categories',{'Yes','No','Maybe'},'BinCounts',[22 18 3]) plots a histogram that has three categories with the associated bin counts.

Example: h.Categories queries the categories that are in histogram object h.

Data Types: cell | categorical

Bin counts, specified as a vector. Use this input to pass bin counts to histogram when the bin counts calculation is performed separately and you do not want histogram to do any data binning.

The length of counts must be equal to the number of bins.

  • For numeric histograms, the number of bins is length(edges)-1.

  • For categorical histograms, the number of bins is equal to the number of categories.

Example: histogram('BinEdges',-2:2,'BinCounts',[5 8 15 9])

Example: histogram('Categories',{'Yes','No','Maybe'},'BinCounts',[22 18 3])

Target axes, specified as an Axes object or a PolarAxes object. If you do not specify the axes and if the current axes are Cartesian axes, then the histogram function uses the current axes (gca). To plot into polar axes, specify the PolarAxes object as the first input argument or use the polarhistogram function.

Name-Value Pair Arguments

Specify optional comma-separated pairs of Name,Value arguments. Name is the argument name and Value is the corresponding value. Name must appear inside single quotes (' '). You can specify several name and value pair arguments in any order as Name1,Value1,...,NameN,ValueN.

Example: histogram(X,'BinWidth',5)

The histogram properties listed here are only a subset. For a complete list, see Histogram Properties.

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Note

This option only applies to histograms of categorical data.

Relative width of categorical bars, specified as a scalar value in the range [0,1]. Use this property to control the separation of categorical bars within the histogram. The default value is 0.9, which means that the bar width is 90% of the space from the previous bar to the next bar, with 5% of that space on each side.

If you set this property to 1, then adjacent bars touch.

Example: 0.5

Data Types: single | double | int8 | int16 | int32 | int64 | uint8 | uint16 | uint32 | uint64

Bin limits, specified as a two-element vector, [bmin,bmax]. This option plots a histogram using the values in the input array, X, that fall between bmin and bmax inclusive. That is, X(X>=bmin & X<=bmax).

This option does not apply to histograms of categorical data.

Example: histogram(X,'BinLimits',[1,10]) plots a histogram using only the values in X that are between 1 and 10 inclusive.

Selection mode for bin limits, specified as 'auto' or 'manual'. The default value is 'auto', so that the bin limits automatically adjust to the data.

If you explicitly specify either BinLimits or BinEdges, then BinLimitsMode is automatically set to 'manual'. In that case, specify BinLimitsMode as 'auto' to rescale the bin limits to the data.

This option does not apply to histograms of categorical data.

Binning algorithm, specified as one of the values in this table.

Value

Description

'auto'

The default 'auto' algorithm chooses a bin width to cover the data range and reveal the shape of the underlying distribution.

'scott'

Scott’s rule is optimal if the data is close to being normally distributed. This rule is appropriate for most other distributions, as well. It uses a bin width of 3.5*std(X(:))*numel(X)^(-1/3).

'fd'

The Freedman-Diaconis rule is less sensitive to outliers in the data, and might be more suitable for data with heavy-tailed distributions. It uses a bin width of 2*IQR(X(:))*numel(X)^(-1/3), where IQR is the interquartile range of X.

'integers'

The integer rule is useful with integer data, as it creates a bin for each integer. It uses a bin width of 1 and places bin edges halfway between integers. To avoid accidentally creating too many bins, you can use this rule to create a limit of 65536 bins (216). If the data range is greater than 65536, then the integer rule uses wider bins instead.

Note

'integers' does not support datetime or duration data.

'sturges'

Sturges’ rule is popular due to its simplicity. It chooses the number of bins to be ceil(1 + log2(numel(X))).

'sqrt'

The Square Root rule is widely used in other software packages. It chooses the number of bins to be ceil(sqrt(numel(X))).

For datetime data, the bin method can be one of these units of time:

'second''month'
'minute''quarter'
'hour''year'
'day''decade'
'week''century'

For duration data, the bin method can be one of these units of time:

'second''day'
'minute''year'
'hour' 

If you specify BinMethod with datetime or duration data, then histogram can use a maximum of 65,536 bins (or 216). If the specified bin duration requires more bins, then histogram uses a larger bin width corresponding to the maximum number of bins.

This option does not apply to histograms of categorical data.

Note

If you set the BinLimits, NumBins, BinEdges, or BinWidth property, then the BinMethod property is set to 'manual'.

Example: histogram(X,'BinMethod','integers') creates a histogram with the bins centered on integers.

Width of bins, specified as a scalar. When you specify BinWidth, then histogram can use a maximum of 65,536 bins (or 216). If instead the specified bin width requires more bins, then histogram uses a larger bin width corresponding to the maximum number of bins.

For datetime and duration data, the value of 'BinWidth' can be a scalar duration or calendar duration.

This option does not apply to histograms of categorical data.

Example: histogram(X,'BinWidth',5) uses bins with a width of 5.

Category display order, specified as 'ascend', 'descend', or 'data'. With 'ascend' or 'descend', the histogram displays with increasing or decreasing bar heights. The default 'data' value uses the category order in the input data, C.

This option only works with categorical data.

Histogram display style, specified as either 'bar' or 'stairs'. Specify 'stairs' to display a stairstep plot, which displays the outline of the histogram without filling the interior.

The default value of 'bar' displays a histogram bar plot.

Example: histogram(X,'DisplayStyle','stairs') plots the outline of the histogram.

Transparency of histogram bar edges, specified as a scalar value between 0 and 1 inclusive. A value of 1 means fully opaque and 0 means completely transparent (invisible).

Example: histogram(X,'EdgeAlpha',0.5) creates a histogram plot with semi-transparent bar edges.

Histogram edge color, specified as one of these values:

  • 'none' — Edges are not drawn.

  • 'auto' — Color of each edge is chosen automatically.

  • RGB triplet or a color name — Edges use the specified color.

    An RGB triplet is a three-element row vector whose elements specify the intensities of the red, green, and blue components of the color. The intensities must be in the range [0,1]; for example, [0.4 0.6 0.7]. Alternatively, you can specify some common colors by name. This table lists the long and short color name options and the equivalent RGB triplet values.

    OptionDescriptionEquivalent RGB Triplet
    'red' or 'r'Red[1 0 0]
    'green' or 'g'Green[0 1 0]
    'blue' or 'b'Blue[0 0 1]
    'yellow' or 'y'Yellow[1 1 0]
    'magenta' or 'm'Magenta[1 0 1]
    'cyan' or 'c'Cyan[0 1 1]
    'white' or 'w'White[1 1 1]
    'black' or 'k'Black[0 0 0]

Example: histogram(X,'EdgeColor','r') creates a histogram plot with red bar edges.

Transparency of histogram bars, specified as a scalar value between 0 and 1 inclusive. histogram uses the same transparency for all the bars of the histogram. A value of 1 means fully opaque and 0 means completely transparent (invisible).

Example: histogram(X,'FaceAlpha',1) creates a histogram plot with fully opaque bars.

Histogram bar color, specified as one of these values:

  • 'none' — Bars are not filled.

  • 'auto' — Histogram bar color is chosen automatically (default).

  • RGB triplet or a color name — Bars are filled with the specified color.

    An RGB triplet is a three-element row vector whose elements specify the intensities of the red, green, and blue components of the color. The intensities must be in the range [0,1]; for example, [0.4 0.6 0.7]. Alternatively, you can specify some common colors by name. This table lists the long and short color name options and the equivalent RGB triplet values.

    OptionDescriptionEquivalent RGB Triplet
    'red' or 'r'Red[1 0 0]
    'green' or 'g'Green[0 1 0]
    'blue' or 'b'Blue[0 0 1]
    'yellow' or 'y'Yellow[1 1 0]
    'magenta' or 'm'Magenta[1 0 1]
    'cyan' or 'c'Cyan[0 1 1]
    'white' or 'w'White[1 1 1]
    'black' or 'k'Black[0 0 0]

If you specify DisplayStyle as 'stairs', then histogram does not use the FaceColor property.

Example: histogram(X,'FaceColor','g') creates a histogram plot with green bars.

Line style, specified as one of the line styles listed in this table.

Line StyleDescriptionResulting Line
'-'Solid line

'--'Dashed line

':'Dotted line

'-.'Dash-dotted line

'none'No lineNo line

Width of bar outlines, specified as a positive value in point units. One point equals 1/72 inch.

Example: 1.5

Data Types: single | double | int8 | int16 | int32 | int64 | uint8 | uint16 | uint32 | uint64

Type of normalization, specified as one of the values in this table. For each bin i:

  • vi is the bin value.

  • ci is the number of elements in the bin.

  • wi is the width of the bin.

  • N is the number of elements in the input data. This value can be greater than the binned data if the data contains NaN, NaT, or <undefined> values, or if some of the data lies outside the bin limits.

ValueBin ValuesNotes
'count' (default)

vi=ci

  • Count or frequency of observations.

  • Sum of bin values is less than or equal to numel(X). The sum is less than numel(X) only when some of the input data is not included in the bins.

  • For categorical data, sum of bin values is less than or equal to either numel(X) or sum(ismember(X(:),Categories)).

'countdensity'

vi=ciwi

  • Count or frequency scaled by width of bin.

  • The area (height * width) of each bar is the number of observations in the bin. The sum of the bar areas is less than or equal to numel(X).

  • For categorical histograms, this is the same as 'count'.

Note

'countdensity' does not support datetime or duration data.

'cumcount'

vi=j=1icj

  • Cumulative count. Each bin value is the cumulative number of observations in that bin and all previous bins.

  • The height of the last bar is less than or equal to numel(X).

  • For categorical histograms, the height of the last bar is less than or equal to numel(X) or sum(ismember(X(:),Categories)).

'probability'

vi=ciN

  • Relative probability.

  • The sum of the bar heights is less than or equal to 1.

'pdf'

vi=ciNwi

  • Probability density function estimate.

  • The area of each bar is the relative number of observations. The sum of the bar areas is less than or equal to 1.

  • For categorical histograms, this is the same as 'probability'.

Note

'pdf' does not support datetime or duration data.

'cdf'

vi=j=1icjN

  • Cumulative density function estimate.

  • The height of each bar is equal to the cumulative relative number of observations in the bin and all previous bins. The height of the last bar is less than or equal to 1.

  • For categorical data, the height of each bar is equal to the cumulative relative number of observations in each category and all previous categories.

Example: histogram(X,'Normalization','pdf') plots an estimate of the probability density function for X.

Number of categories to display, specified as a scalar. You can change the ordering of categories displayed in the histogram using the 'DisplayOrder' option.

This option only works with categorical data.

Orientation of bars, specified as 'vertical' or 'horizontal'.

Example: histogram(X,'Orientation','horizontal') creates a histogram plot with horizontal bars.

Toggle summary display of data belonging to undisplayed categories, specified as 'off' or 'on'. Set this option to 'on' to display an additional bar in the histogram with the name 'Others'. This extra bar counts all elements that do not belong to categories displayed in the histogram.

You can change the number of categories displayed in the histogram, as well as their order, using the 'NumDisplayBins' and 'DisplayOrder' options.

This option only works with categorical data.

Output Arguments

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Histogram, returned as an object. For more information, see histogram .

Properties

Histogram PropertiesControl histogram appearance and behavior

Object Functions

morebinsIncrease number of histogram bins
fewerbinsDecrease number of histogram bins

Examples

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Generate 10,000 random numbers and create a histogram. The histogram function automatically chooses an appropriate number of bins to cover the range of values in x and show the shape of the underlying distribution.

x = randn(10000,1);
h = histogram(x)
h = 
  Histogram with properties:

             Data: [10000x1 double]
           Values: [1x37 double]
          NumBins: 37
         BinEdges: [1x38 double]
         BinWidth: 0.2000
        BinLimits: [-3.8000 3.6000]
    Normalization: 'count'
        FaceColor: 'auto'
        EdgeColor: [0 0 0]

  Show all properties

When you specify an output argument to the histogram function, it returns a histogram object. You can use this object to inspect the properties of the histogram, such as the number of bins or the width of the bins.

Find the number of histogram bins.

nbins = h.NumBins
nbins = 37

Plot a histogram of 1,000 random numbers sorted into 25 equally spaced bins.

x = randn(1000,1);
nbins = 25;
h = histogram(x,nbins)
h = 
  Histogram with properties:

             Data: [1000x1 double]
           Values: [1x25 double]
          NumBins: 25
         BinEdges: [1x26 double]
         BinWidth: 0.2800
        BinLimits: [-3.4000 3.6000]
    Normalization: 'count'
        FaceColor: 'auto'
        EdgeColor: [0 0 0]

  Show all properties

Find the bin counts.

counts = h.Values
counts = 

  Columns 1 through 13

     1     3     0     6    14    19    31    54    74    80    92   122   104

  Columns 14 through 25

   115    88    80    38    32    21     9     5     5     5     0     2

Generate 1,000 random numbers and create a histogram.

X = randn(1000,1);
h = histogram(X)
h = 
  Histogram with properties:

             Data: [1000x1 double]
           Values: [1x23 double]
          NumBins: 23
         BinEdges: [1x24 double]
         BinWidth: 0.3000
        BinLimits: [-3.3000 3.6000]
    Normalization: 'count'
        FaceColor: 'auto'
        EdgeColor: [0 0 0]

  Show all properties

Use the morebins function to coarsely adjust the number of bins.

Nbins = morebins(h);
Nbins = morebins(h)
Nbins = 29

Adjust the bins at a fine grain level by explicitly setting the number of bins.

h.NumBins = 31;

Generate 1,000 random numbers and create a histogram. Specify the bin edges as a vector with wide bins on the edges of the histogram to capture the outliers that do not satisfy . The first vector element is the left edge of the first bin, and the last vector element is the right edge of the last bin.

x = randn(1000,1);
edges = [-10 -2:0.25:2 10];
h = histogram(x,edges);

Specify the Normalization property as 'countdensity' to flatten out the bins containing the outliers. Now, the area of each bin (rather than the height) represents the frequency of observations in that interval.

h.Normalization = 'countdensity';

Create a categorical vector that represents votes. The categories in the vector are 'yes', 'no', or 'undecided'.

A = [0 0 1 1 1 0 0 0 0 NaN NaN 1 0 0 0 1 0 1 0 1 0 0 0 1 1 1 1];
C = categorical(A,[1 0 NaN],{'yes','no','undecided'})
C = 1x27 categorical array
  Columns 1 through 9

     no      no      yes      yes      yes      no      no      no      no 

  Columns 10 through 16

     undecided      undecided      yes      no      no      no      yes 

  Columns 17 through 25

     no      yes      no      yes      no      no      no      yes      yes 

  Columns 26 through 27

     yes      yes 

Plot a categorical histogram of the votes, using a relative bar width of 0.5.

h = histogram(C,'BarWidth',0.5)
h = 
  Histogram with properties:

              Data: [1x27 categorical]
            Values: [11 14 2]
    NumDisplayBins: 3
        Categories: {'yes'  'no'  'undecided'}
      DisplayOrder: 'data'
     Normalization: 'count'
      DisplayStyle: 'bar'
         FaceColor: 'auto'
         EdgeColor: [0 0 0]

  Show all properties

Generate 1,000 random numbers and create a histogram using the 'probability' normalization.

x = randn(1000,1);
h = histogram(x,'Normalization','probability')
h = 
  Histogram with properties:

             Data: [1000x1 double]
           Values: [1x23 double]
          NumBins: 23
         BinEdges: [1x24 double]
         BinWidth: 0.3000
        BinLimits: [-3.3000 3.6000]
    Normalization: 'probability'
        FaceColor: 'auto'
        EdgeColor: [0 0 0]

  Show all properties

Compute the sum of the bar heights. With this normalization, the height of each bar is equal to the probability of selecting an observation within that bin interval, and the height of all of the bars sums to 1.

S = sum(h.Values)
S = 1

Generate two vectors of random numbers and plot a histogram for each vector in the same figure.

x = randn(2000,1);
y = 1 + randn(5000,1);
h1 = histogram(x);
hold on
h2 = histogram(y);

Since the sample size and bin width of the histograms are different, it is difficult to compare them. Normalize the histograms so that all of the bar heights add to 1, and use a uniform bin width.

h1.Normalization = 'probability';
h1.BinWidth = 0.25;
h2.Normalization = 'probability';
h2.BinWidth = 0.25;

Generate 1,000 random numbers and create a histogram. Return the histogram object to adjust the properties of the histogram without recreating the entire plot.

x = randn(1000,1);
h = histogram(x)
h = 
  Histogram with properties:

             Data: [1000x1 double]
           Values: [1x23 double]
          NumBins: 23
         BinEdges: [1x24 double]
         BinWidth: 0.3000
        BinLimits: [-3.3000 3.6000]
    Normalization: 'count'
        FaceColor: 'auto'
        EdgeColor: [0 0 0]

  Show all properties

Specify exactly how many bins to use.

h.NumBins = 15;

Specify the edges of the bins with a vector. The first value in the vector is the left edge of the first bin. The last value is the right edge of the last bin.

h.BinEdges = [-3:3];

Change the color of the histogram bars.

h.FaceColor = [0 0.5 0.5];
h.EdgeColor = 'r';

Generate 5,000 normally distributed random numbers with a mean of 5 and a standard deviation of 2. Plot a histogram with Normalization set to 'pdf' to produce an estimation of the probability density function.

x = 2*randn(5000,1) + 5;
histogram(x,'Normalization','pdf')

In this example, the underlying distribution for the normally distributed data is known. You can, however, use the 'pdf' histogram plot to determine the underlying probability distribution of the data by comparing it against a known probability density function.

The probability density function for a normal distribution with mean , standard deviation , and variance is

Overlay a plot of the probability density function for a normal distribution with a mean of 5 and a standard deviation of 2.

hold on
y = -5:0.1:15;
mu = 5;
sigma = 2;
f = exp(-(y-mu).^2./(2*sigma^2))./(sigma*sqrt(2*pi));
plot(y,f,'LineWidth',1.5)

Use the savefig function to save a histogram figure.

y = histogram(randn(10));
savefig('histogram.fig');
clear all
close all

Use openfig to load the histogram figure back into MATLAB. openfig also returns a handle to the figure, h.

h = openfig('histogram.fig');

Use the findobj function to locate the correct object handle from the figure handle. This allows you to continue manipulating the original histogram object used to generate the figure.

y = findobj(h, 'type', 'histogram')
y = 

  Histogram with properties:

             Data: [10x10 double]
           Values: [2 17 28 32 16 3 2]
          NumBins: 7
         BinEdges: [-3 -2 -1 0 1 2 3 4]
         BinWidth: 1
        BinLimits: [-3 4]
    Normalization: 'count'
        FaceColor: 'auto'
        EdgeColor: [0 0 0]

  Use GET to show all properties

Tips

  • Histogram plots created using histogram have a context menu in plot edit mode that enables interactive manipulations in the figure window. For example, you can use the context menu to interactively change the number of bins, align multiple histograms, or change the display order.

Extended Capabilities

Introduced in R2014b

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