The interquartile mean (IQM) (or midmean) is a

statistical Statistics (from German: ''Statistik'', "description of a state, a country") is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of data. In applying statistics to a scientific, industria ...

measure of

central tendency In statistics, a central tendency (or measure of central tendency) is a central or typical value for a probability distribution.Weisberg H.F (1992) ''Central Tendency and Variability'', Sage University Paper Series on Quantitative Applications ...

based on the

truncated mean A truncated mean or trimmed mean is a statistical measure of central tendency, much like the mean and median. It involves the calculation of the mean after discarding given parts of a probability distribution or sample at the high and low end, an ...

of the

interquartile range In descriptive statistics, the interquartile range (IQR) is a measure of statistical dispersion, which is the spread of the data. The IQR may also be called the midspread, middle 50%, fourth spread, or H‑spread. It is defined as the difference ...

. The IQM is very similar to the scoring method used in sports that are evaluated by a panel of judges: ''discard the lowest and the highest scores; calculate the mean value of the remaining scores''.

Calculation

In calculation of the IQM, only the data between the first and third

quartiles In statistics, a quartile is a type of quantile which divides the number of data points into four parts, or ''quarters'', of more-or-less equal size. The data must be ordered from smallest to largest to compute quartiles; as such, quartiles are ...

is used, and the lowest 25% and the highest 25% of the data are discarded. :

x_\mathrm =  \sum_^

assuming the values have been ordered.

Examples

Dataset size divisible by four

The method is best explained with an example. Consider the following dataset: :5, 8, 4, 38, 8, 6, 9, 7, 7, 3, 1, 6 First sort the list from lowest-to-highest: :1, 3, 4, 5, 6, 6, 7, 7, 8, 8, 9, 38 There are 12 observations (datapoints) in the dataset, thus we have 4 quartiles of 3 numbers. Discard the lowest and the highest 3 values: :~~1, 3, 4~~, 5, 6, 6, 7, 7, 8, ~~8, 9, 38~~ We now have 6 of the 12 observations remaining; next, we calculate the arithmetic

mean There are several kinds of mean in mathematics, especially in statistics. Each mean serves to summarize a given group of data, often to better understand the overall value (magnitude and sign) of a given data set. For a data set, the ''arithme ...

of these numbers: :''x''_IQM = (5 + 6 + 6 + 7 + 7 + 8) / 6 = 6.5 This is the interquartile mean. For comparison, the arithmetic mean of the original dataset is :(5 + 8 + 4 + 38 + 8 + 6 + 9 + 7 + 7 + 3 + 1 + 6) / 12 = 8.5 due to the strong influence of the outlier, 38.

Dataset size not divisible by four

The above example consisted of 12 observations in the dataset, which made the determination of the quartiles very easy. Of course, not all datasets have a number of observations that is divisible by 4. We can adjust the method of calculating the IQM to accommodate this. So ideally we want to have the IQM equal to the

for symmetric distributions, e.g.: :1, 2, 3, 4, 5 has a mean value ''x''_mean = 3, and since it is a symmetric distribution, ''x''_IQM = 3 would be desired. We can solve this by using a

weighted average The weighted arithmetic mean is similar to an ordinary arithmetic mean (the most common type of average), except that instead of each of the data points contributing equally to the final average, some data points contribute more than others. The ...

of the quartiles and the interquartile dataset: Consider the following dataset of 9 observations: :1, 3, 5, 7, 9, 11, 13, 15, 17 There are 9/4 = 2.25 observations in each quartile, and 4.5 observations in the interquartile range. Truncate the fractional quartile size, and remove this number from the 1st and 4th quartiles (2.25 observations in each quartile, thus the lowest 2 and the highest 2 are removed). :~~1, 3~~, (5), 7, 9, 11, (13), ~~15, 17~~ Thus, there are 3 ''full'' observations in the interquartile range, and 2 fractional observations. Since we have a total of 4.5 observations in the interquartile range, the two fractional observations each count for 0.75 (and thus 3×1 + 2×0.75 = 4.5 observations). The IQM is now calculated as follows: :''x''_IQM = {(7 + 9 + 11) + 0.75 × (5 + 13)} / 4.5 = 9 In the above example, the mean has a value x_mean = 9. The same as the IQM, as was expected. The method of calculating the IQM for any number of observations is analogous; the fractional contributions to the IQM can be either 0, 0.25, 0.50, or 0.75.

Comparison with mean and median

The interquartile mean shares some properties of both the

and the

median In statistics and probability theory, the median is the value separating the higher half from the lower half of a data sample, a population, or a probability distribution. For a data set, it may be thought of as "the middle" value. The basic fe ...

: *Like the

, the IQM is insensitive to

outlier In statistics, an outlier is a data point that differs significantly from other observations. An outlier may be due to a variability in the measurement, an indication of novel data, or it may be the result of experimental error; the latter are ...

s; in the example given, the highest value (38) was an obvious outlier of the dataset, but its value is not used in the calculation of the IQM. On the other hand, the common average (the

arithmetic mean In mathematics and statistics, the arithmetic mean ( ) or arithmetic average, or just the ''mean'' or the ''average'' (when the context is clear), is the sum of a collection of numbers divided by the count of numbers in the collection. The colle ...

) is sensitive to these outliers: ''x''_mean = 8.5. *Like the

, the IQM is a distinct parameter, based on a large number of observations from the dataset. The

is always equal to ''one'' of the observations in the dataset (assuming an odd number of observations). The mean can be equal to ''any'' value between the lowest and highest observation, depending on the value of ''all'' the other observations. The IQM can be equal to ''any'' value between the first and third quartiles, depending on ''all'' the observations in the interquartile range.

Calculation

Examples

Dataset size divisible by four

Dataset size not divisible by four

Comparison with mean and median

See also

Related statistics

Applications