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Median

Status: Stable

documented, exercised by the test suite and/or worked examples, with no known limitations recorded.

Description

Median[data]
    gives the median estimate of the elements in data.
Median[dist]
    gives the median of the distribution dist.

Examples

All examples below are verified against the current Mathilda build.

In[1]:= Median[{1, 2, 3, 4, 5, 6, 7}]
Out[1]= 4

In[2]:= Median[{1, 2, 3, 4, 5, 6, 7, 8}]
Out[2]= 9/2

In[3]:= Median[{1, 2, 3, 4}]
Out[3]= 5/2

In[4]:= Median[{Pi, E, 2}]
Out[4]= E

In[5]:= Median[{1., 2., 3., 4.}]
Out[5]= 2.5

In[6]:= Median[{{1, 11, 3}, {4, 6, 7}}]
Out[6]= {5/2, 17/2, 5}

In[7]:= Median[{{{3, 7}, {2, 1}}, {{5, 19}, {12, 4}}}]
Out[7]= {{4, 13}, {7, 5/2}}

In[8]:= Median[{a, b, c}]
Out[8]= Median[{a, b, c}]

Implementation notes

Algorithm. builtin_median requires a List. If the first element is itself a List it treats the input as a matrix/tensor and reduces column-wise through apply_columnwise (Map[Median, Transpose[...]]). For a 1-D vector it first verifies every element is a real numeric via the helper is_real_numeric (which checks NumericQ and FreeQ[#, I]); non-real data prints Median::rectn and leaves the call unevaluated. It then evaluates Sort[data]: for odd n it returns the middle element (sorted[n/2]); for even n it returns (sorted[n/2-1] + sorted[n/2]) / 2, built as Plus then Divide and re-evaluated so the result stays exact (rational) when the inputs are exact. ATTR_PROTECTED.

  • Protected.
  • Median is a robust location estimator, which means it not very sensitive to outliers.
  • For VectorQ data ${x_1, \dots, x_n}$, the median can be thought of as the "middle value". Formally, when data is sorted as ${x_{(1)}, \dots, x_{(n)}}$, the median is given by the center element $x_{((n+1)/2)}$ if $n$ is odd and the mean of the two center elements $(x_{(n/2)} + x_{(n/2+1)})/2$ if $n$ is even.
  • For MatrixQ data, the median is computed for each column vector. Median for a tensor gives columnwise medians at the first level.
  • Median requires numeric values.

Attributes: Protected.

Implementation status

Stable — documented, exercised by the test suite and/or worked examples, with no known limitations recorded.

References

Notes & additional examples

Worked examples

In[1]:= Median[{5, 1, 3, 2, 4}]
Out[1]= 3

In[1]:= Median[{1, 2, 3, 4}]
Out[1]= 5/2

In[1]:= Median[Table[k^2, {k, 1, 10}]]
Out[1]= 61/2

Notes

Median[data] returns the middle value of the sorted data. For an odd number of elements it is the single central element (the first example sorts to {1, 2, 3, 4, 5}, giving 3); for an even number it is the exact average of the two central elements, so Median[{1, 2, 3, 4}] is 5/2. The result is kept in exact arithmetic — the median of the first ten squares is 61/2, the average of the 5th and 6th sorted values 25 and 36. Median expects numeric data; an even-length list of unresolved symbols cannot be averaged and is left unevaluated with a Median::rectn message.