# Median

**averagesample medianmedian-unbiased estimatormedian averagespatial medianHalf of affected patientsmedian debtmedian vectormedian-median linemedian-unbiasedness**

The median is the value separating the higher half from the lower half of a data sample (a population or a probability distribution).wikipedia

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### Mode (statistics)

**modemodalmodes**

The median is 2 in this case, (as is the mode), and it might be seen as a better indication of central tendency (less susceptible to the exceptionally large value in data) than the arithmetic mean of 4.

Like the statistical mean and median, the mode is a way of expressing, in a (usually) single number, important information about a random variable or a population.

### Skewness

**skewedskewskewed distribution**

The basic advantage of the median in describing data compared to the mean (often simply described as the "average") is that it is not skewed so much by extremely large or small values, and so it may give a better idea of a "typical" value. The median can be used as a measure of location when a distribution is skewed, when end-values are not known, or when one requires reduced importance to be attached to outliers, e.g., because they may be measurement errors.

In the older notion of nonparametric skew, defined as where \mu is the mean, \nu is the median, and \sigma is the standard deviation, the skewness is defined in terms of this relationship: positive/right nonparametric skew means the mean is greater than (to the right of) the median, while negative/left nonparametric skew means the mean is less than (to the left of) the median.

### Central tendency

**Localitycentral locationcentral point**

The median is 2 in this case, (as is the mode), and it might be seen as a better indication of central tendency (less susceptible to the exceptionally large value in data) than the arithmetic mean of 4.

The most common measures of central tendency are the arithmetic mean, the median and the mode.

### Geometric median

**minimizing the sum of distances**

A geometric median, on the other hand, is defined in any number of dimensions.

This generalizes the median, which has the property of minimizing the sum of distances for one-dimensional data, and provides a central tendency in higher dimensions.

### Probability distribution

**distributioncontinuous probability distributiondiscrete probability distribution**

The median is the value separating the higher half from the lower half of a data sample (a population or a probability distribution).

Median: the value such that the set of values less than the median, and the set greater than the median, each have probabilities no greater than one-half.

### Quartile

**quartileslower quartilelower and upper quartiles**

The median is the 2nd quartile, 5th decile, and 50th percentile.

The first quartile (Q 1 ) is defined as the middle number between the smallest number and the median of the data set.

### Trimmed estimator

**trimmedtrimming**

(In more technical terms, this interprets the median as the fully trimmed mid-range).

The median is the most trimmed statistic (nominally 50%), as it discards all but the most central data, and equals the fully trimmed mean – or indeed fully trimmed mid-range, or (for odd-size data sets) the fully trimmed maximum or minimum.

### Arithmetic mean

**meanaveragearithmetic**

The median is 2 in this case, (as is the mode), and it might be seen as a better indication of central tendency (less susceptible to the exceptionally large value in data) than the arithmetic mean of 4. The basic advantage of the median in describing data compared to the mean (often simply described as the "average") is that it is not skewed so much by extremely large or small values, and so it may give a better idea of a "typical" value.

Notably, for skewed distributions, such as the distribution of income for which a few people's incomes are substantially greater than most people's, the arithmetic mean may not coincide with one's notion of "middle", and robust statistics, such as the median, may be a better description of central tendency.

### Outlier

**outliersconservative estimateirregularities**

The median is a popular summary statistic used in descriptive statistics, since it is simple to understand and easy to calculate, while also giving a measure that is more robust in the presence of outlier values than is the mean. The median can be used as a measure of location when a distribution is skewed, when end-values are not known, or when one requires reduced importance to be attached to outliers, e.g., because they may be measurement errors.

For example, if one is calculating the average temperature of 10 objects in a room, and nine of them are between 20 and 25 degrees Celsius, but an oven is at 175 °C, the median of the data will be between 20 and 25 °C but the mean temperature will be between 35.5 and 40 °C. In this case, the median better reflects the temperature of a randomly sampled object (but not the temperature in the room) than the mean; naively interpreting the mean as "a typical sample", equivalent to the median, is incorrect.

### Mean

**mean valuepopulation meanaverage**

The median is a popular summary statistic used in descriptive statistics, since it is simple to understand and easy to calculate, while also giving a measure that is more robust in the presence of outlier values than is the mean.

In descriptive statistics, the mean may be confused with the median, mode or mid-range, as any of these may be called an "average" (more formally, a measure of central tendency).

### Mid-range

**midsummarymidrangehalf-range**

(In more technical terms, this interprets the median as the fully trimmed mid-range).

The median can be interpreted as the fully trimmed (50%) mid-range; this accords with the convention that the median of an even number of points is the mean of the two middle points.

### Percentile

**percentiles50th percentile85th percentile speed**

The median is the 2nd quartile, 5th decile, and 50th percentile.

The 25th percentile is also known as the first quartile (Q 1 ), the 50th percentile as the median or second quartile (Q 2 ), and the 75th percentile as the third quartile (Q 3 ).

### Average absolute deviation

**mean absolute deviationmean deviationMAD**

When the median is used as a location parameter in descriptive statistics, there are several choices for a measure of variability: the range, the interquartile range, the mean absolute deviation, and the median absolute deviation.

In this general form, the central point can be the mean, median, mode, or the result of another measure of central tendency.

### Normal distribution

**normally distributednormalGaussian**

The median of a normal distribution with mean μ and variance σ 2 is μ. In fact, for a normal distribution, mean = median = mode.

\mu is the mean or expectation of the distribution (and also its median and mode),

### Descriptive statistics

**descriptivedescriptive statisticstatistics**

The median is a popular summary statistic used in descriptive statistics, since it is simple to understand and easy to calculate, while also giving a measure that is more robust in the presence of outlier values than is the mean.

Measures of central tendency include the mean, median and mode, while measures of variability include the standard deviation (or variance), the minimum and maximum values of the variables, kurtosis and skewness.

### Median absolute deviation

**MAD**

When the median is used as a location parameter in descriptive statistics, there are several choices for a measure of variability: the range, the interquartile range, the mean absolute deviation, and the median absolute deviation.

For a univariate data set X 1, X 2, ..., X n, the MAD is defined as the median of the absolute deviations from the data's median :

### Location parameter

**locationlocation modelshift parameter**

The median can be used as a measure of location when a distribution is skewed, when end-values are not known, or when one requires reduced importance to be attached to outliers, e.g., because they may be measurement errors.

Here, x_0 is called the location parameter. Examples of location parameters include the mean, the median, and the mode.

### Summary statistics

**summary statisticSummarizationdata summarization**

The median is a popular summary statistic used in descriptive statistics, since it is simple to understand and easy to calculate, while also giving a measure that is more robust in the presence of outlier values than is the mean.

Common measures of location, or central tendency, are the arithmetic mean, median, mode, and interquartile mean.

### Symmetric probability distribution

**symmetricsymmetric distributionSymmetry**

The median of a symmetric distribution which possesses a mean μ also takes the value μ.

The median and the mean (if it exists) of a symmetric distribution both occur at the point x_0 about which the symmetry occurs.

### Exponential distribution

**exponentialexponentially distributedexponentially**

The median of an exponential distribution with rate parameter λ is the natural logarithm of 2 divided by the rate parameter: λ −1 ln 2.

The median of X is given by

### Order statistic

**order statisticsorderedth-smallest of items**

operations, selection algorithms can compute the 'th-smallest of items with only

Important special cases of the order statistics are the minimum and maximum value of a sample, and (with some qualifications discussed below) the sample median and other sample quantiles.

### Resampling (statistics)

**resamplingstatistical supportstrongly supported**

The standard "delete one" jackknife method produces inconsistent results.

1) Estimating the precision of sample statistics (medians, variances, percentiles) by using subsets of available data (jackknifing) or drawing randomly with replacement from a set of data points (bootstrapping)

### Interquartile range

**inter-quartile rangebelowinterquartile**

When the median is used as a location parameter in descriptive statistics, there are several choices for a measure of variability: the range, the interquartile range, the mean absolute deviation, and the median absolute deviation.

The median is the corresponding measure of central tendency.

### Efficiency (statistics)

**efficientefficiencyinefficient**

For example, a comparison of the efficiency of candidate estimators shows that the sample mean is more statistically efficient than the sample median when data are uncontaminated by data from heavy-tailed distributions or from mixtures of distributions, but less efficient otherwise, and that the efficiency of the sample median is higher than that for a wide range of distributions.

Now consider the sample median,.

### K-medians clustering

**k''-medians clusteringk-Mediansk-median problem**

This optimization-based definition of the median is useful in statistical data-analysis, for example, in k-medians clustering.

It is a variation of k-means clustering where instead of calculating the mean for each cluster to determine its centroid, one instead calculates the median.