[[search-aggregations-metrics-median-absolute-deviation-aggregation]]
=== Median absolute deviation aggregation
++++
<titleabbrev>Median absolute deviation</titleabbrev>
++++

This `single-value` aggregation approximates the {wikipedia}/Median_absolute_deviation[median absolute deviation]
of its search results.

Median absolute deviation is a measure of variability. It is a robust
statistic, meaning that it is useful for describing data that may have
outliers, or may not be normally distributed. For such data it can be more
descriptive than standard deviation.

It is calculated as the median of each data point's deviation from the median
of the entire sample. That is, for a random variable X, the median absolute
deviation is median(|median(X) - X~i~|).

==== Example

Assume our data represents product reviews on a one to five star scale.
Such reviews are usually summarized as a mean, which is easily understandable
but doesn't describe the reviews' variability. Estimating the median absolute
deviation can provide insight into how much reviews vary from one another.

In this example we have a product which has an average rating of
3 stars. Let's look at its ratings' median absolute deviation to determine
how much they vary

[source,console]
---------------------------------------------------------
GET reviews/_search
{
  "size": 0,
  "aggs": {
    "review_average": {
      "avg": {
        "field": "rating"
      }
    },
    "review_variability": {
      "median_absolute_deviation": {
        "field": "rating" <1>
      }
    }
  }
}
---------------------------------------------------------
// TEST[setup:reviews]
<1> `rating` must be a numeric field

The resulting median absolute deviation of `2` tells us that there is a fair
amount of variability in the ratings. Reviewers must have diverse opinions about
this product.

[source,console-result]
---------------------------------------------------------
{
  ...
  "aggregations": {
    "review_average": {
      "value": 3.0
    },
    "review_variability": {
      "value": 2.0
    }
  }
}
---------------------------------------------------------
// TESTRESPONSE[s/\.\.\./"took": $body.took,"timed_out": false,"_shards": $body._shards,"hits": $body.hits,/]

==== Approximation

The naive implementation of calculating median absolute deviation stores the
entire sample in memory, so this aggregation instead calculates an
approximation. It uses the https://github.com/tdunning/t-digest[TDigest data structure]
to approximate the sample median and the median of deviations from the sample
median. For more about the approximation characteristics of TDigests, see
<<search-aggregations-metrics-percentile-aggregation-approximation>>.

The tradeoff between resource usage and accuracy of a TDigest's quantile
approximation, and therefore the accuracy of this aggregation's approximation
of median absolute deviation, is controlled by the `compression` parameter. A
higher `compression` setting provides a more accurate approximation at the
cost of higher memory usage. For more about the characteristics of the TDigest
`compression` parameter see
<<search-aggregations-metrics-percentile-aggregation-compression>>.

[source,console]
---------------------------------------------------------
GET reviews/_search
{
  "size": 0,
  "aggs": {
    "review_variability": {
      "median_absolute_deviation": {
        "field": "rating",
        "compression": 100
      }
    }
  }
}
---------------------------------------------------------
// TEST[setup:reviews]

The default `compression` value for this aggregation is `1000`. At this
compression level this aggregation is usually within 5% of the exact result,
but observed performance will depend on the sample data.

==== Script

In the example above, product reviews are on a scale of one to five. If you
want to modify them to a scale of one to ten, use a <<runtime,runtime field>>.

[source,console]
----
GET reviews/_search?filter_path=aggregations
{
  "size": 0,
  "runtime_mappings": {
    "rating.out_of_ten": {
      "type": "long",
      "script": {
        "source": "emit(doc['rating'].value * params.scaleFactor)",
        "params": {
          "scaleFactor": 2
        }
      }
    }
  },
  "aggs": {
    "review_average": {
      "avg": {
        "field": "rating.out_of_ten"
      }
    },
    "review_variability": {
      "median_absolute_deviation": {
        "field": "rating.out_of_ten"
      }
    }
  }
}
----
// TEST[setup:reviews]

Which will result in:

[source,console-result]
---------------------------------------------------------
{
  "aggregations": {
    "review_average": {
      "value": 6.0
    },
    "review_variability": {
      "value": 4.0
    }
  }
}
---------------------------------------------------------

==== Missing value

The `missing` parameter defines how documents that are missing a value should be
treated. By default they will be ignored but it is also possible to treat them
as if they had a value.

Let's be optimistic and assume some reviewers loved the product so much that
they forgot to give it a rating. We'll assign them five stars

[source,console]
---------------------------------------------------------
GET reviews/_search
{
  "size": 0,
  "aggs": {
    "review_variability": {
      "median_absolute_deviation": {
        "field": "rating",
        "missing": 5
      }
    }
  }
}
---------------------------------------------------------
// TEST[setup:reviews]
