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Trimean
In statistics the trimean (TM), or Tukey's trimean, is a measure of a probability distribution's location defined as a weighted average of the distribution's median and its two quartiles: : TM= \frac This is equivalent to the arithmetic mean of the median and the midhinge: : TM= \frac\left(Q_2 + \frac\right) The foundations of the trimean were part of Arthur Bowley's teachings, and later popularized by statistician John Tukey in his 1977 book which has given its name to a set of techniques called exploratory data analysis. Like the median and the midhinge, but unlike the sample mean, it is a statistically resistant L-estimator with a breakdown point of 25%. This beneficial property has been described as follows: Efficiency Despite its simplicity, the trimean is a remarkably efficient estimator of population mean. More precisely, for a large data set (over 100 points) from a symmetric population, the average of the 18th, 50th, and 82nd percentile is the most efficient 3-p ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
L-estimator
In statistics, an L-estimator (or L-statistic) is an estimator which is a linear combination of order statistics of the measurements. This can be as little as a single point, as in the median (of an odd number of values), or as many as all points, as in the mean. The main benefits of L-estimators are that they are often extremely simple, and often robust statistics: assuming sorted data, they are very easy to calculate and interpret, and are often resistant to outliers. They thus are useful in robust statistics, as descriptive statistics, in statistics education, and when computation is difficult. However, they are inefficient, and in modern times robust statistics M-estimators are preferred, although these are much more difficult computationally. In many circumstances L-estimators are reasonably efficient, and thus adequate for initial estimation. Examples A basic example is the median. Given ''n'' values x_1, \ldots, x_n, if n=2k+1 is odd, the median equals x_, the (n+1)/2 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
John Tukey
John Wilder Tukey (; June 16, 1915 – July 26, 2000) was an American mathematician and statistician, best known for the development of the fast Fourier Transform (FFT) algorithm and box plot. The Tukey range test, the Tukey lambda distribution, the Tukey test of additivity, and the Teichmüller–Tukey lemma all bear his name. He is also credited with coining the term '' bit'' and the first published use of the word ''software''. Biography Tukey was born in New Bedford, Massachusetts, in 1915, to a Latin teacher father and a private tutor. He was mainly taught by his mother and attended regular classes only for certain subjects like French. Tukey obtained a B.A. in 1936 and M.S. in 1937 in chemistry, from Brown University, before moving to Princeton University, where in 1939 he received a PhD in mathematics after completing a doctoral dissertation titled "On denumerability in topology". During World War II, Tukey worked at the Fire Control Research Office and coll ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Average
In colloquial, ordinary language, an average is a single number or value that best represents a set of data. The type of average taken as most typically representative of a list of numbers is the arithmetic mean the sum of the numbers divided by how many numbers are in the list. For example, the mean or average of the numbers 2, 3, 4, 7, and 9 (summing to 25) is 5. Depending on the context, the most representative statistics, statistic to be taken as the average might be another measure of central tendency, such as the mid-range, median, Mode (statistics), mode or geometric mean. For example, the average income, personal income is often given as the median the number below which are 50% of personal incomes and above which are 50% of personal incomes because the mean would be higher by including personal incomes from a few billionaires. General properties If all numbers in a list are the same number, then their average is also equal to this number. This property is shared by each o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Midhinge
In statistics, the midhinge () is the average of the first and third quartiles and is thus a measure of location. Equivalently, it is the 25% trimmed mid-range or 25% midsummary; it is an L-estimator. The midhinge is defined as \begin \operatorname(X) &= \overline \\ &= \frac \\ &= \frac \\ &= M_(X). \end The midhinge is related to the interquartile range (), the difference of the third and first quartiles (i.e. ), which is a measure of statistical dispersion. The two are complementary in sense that if one knows the midhinge and the , one can find the first and third quartiles. The use of the term ''hinge'' for the lower or upper quartiles derives from John Tukey's work on exploratory data analysis in the late 1970s,Tukey, J. W. (1977) ''Exploratory Data Analysis'', Addison-Wesley. and ''midhinge'' is a fairly modern term dating from around that time. The midhinge is slightly simpler to calculate than the trimean (), which originated in the same context and equals the avera ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Arthur Bowley
Sir Arthur Lyon Bowley, FBA (6 November 1869 – 21 January 1957) was an English statistician and economist who worked on economic statistics and pioneered the use of sampling techniques in social surveys. Early life Bowley's father, James William Lyon Bowley, was a minister in the Church of England. He died at the age of 40 when Arthur was one, leaving Arthur's mother as mother or stepmother to seven children. Arthur was educated at Christ's Hospital, and won a scholarship to Trinity College, Cambridge to study mathematics. He graduated as Tenth Wrangler. At Cambridge Bowley had a short course of study with the economist Alfred Marshall who had also been a Cambridge wrangler. Under Marshall's influence Bowley became an economic statistician. His ''Account of England's Foreign Trade'' won the Cobden Essay Prize and was published as a book. Marshall watched over Bowley's career, recommending him for jobs and offering him advice. Most notoriously Marshall told him the ''Elemen ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Midsummary
In statistics, the mid-range or mid-extreme is a measure of central tendency of a sample defined as the arithmetic mean of the maximum and minimum values of the data set: :M=\frac. The mid-range is closely related to the range, a measure of statistical dispersion defined as the difference between maximum and minimum values. The two measures are complementary in sense that if one knows the mid-range and the range, one can find the sample maximum and minimum values. The mid-range is rarely used in practical statistical analysis, as it lacks efficiency as an estimator for most distributions of interest, because it ignores all intermediate points, and lacks robustness, as outliers change it significantly. Indeed, for many distributions it is one of the least efficient and least robust statistics. However, it finds some use in special cases: it is the maximally efficient estimator for the center of a uniform distribution, trimmed mid-ranges address robustness, and as an L-estimator ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Statistics
Statistics (from German language, German: ', "description of a State (polity), state, a country") is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of data. In applying statistics to a scientific, industrial, or social problem, it is conventional to begin with a statistical population or a statistical model to be studied. Populations can be diverse groups of people or objects such as "all people living in a country" or "every atom composing a crystal". Statistics deals with every aspect of data, including the planning of data collection in terms of the design of statistical survey, surveys and experimental design, experiments. When census data (comprising every member of the target population) cannot be collected, statisticians collect data by developing specific experiment designs and survey sample (statistics), samples. Representative sampling assures that inferences and conclusions can reasonably extend from the sample ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Summary Statistics
In descriptive statistics, summary statistics are used to summarize a set of observations, in order to communicate the largest amount of information as simply as possible. Statisticians commonly try to describe the observations in * a measure of location, or central tendency, such as the arithmetic mean * a measure of statistical dispersion like the standard mean absolute deviation * a measure of the shape of the distribution like skewness or kurtosis * if more than one variable is measured, a measure of statistical dependence such as a correlation coefficient A common collection of order statistics used as summary statistics are the five-number summary, sometimes extended to a seven-number summary, and the associated box plot. Entries in an analysis of variance table can also be regarded as summary statistics. Examples Location Common measures of location, or central tendency, are the arithmetic mean, median, mode, and interquartile mean. Spread Common measures ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
MathWorld
''MathWorld'' is an online mathematics reference work, created and largely written by Eric W. Weisstein. It is sponsored by and licensed to Wolfram Research, Inc. and was partially funded by the National Science Foundation's National Science Digital Library grant to the University of Illinois at Urbana–Champaign. History Eric W. Weisstein, the creator of the site, was a physics and astronomy student who got into the habit of writing notes on his mathematical readings. In 1995 he put his notes online and called it "Eric's Treasure Trove of Mathematics." It contained hundreds of pages/articles, covering a wide range of mathematical topics. The site became popular as an extensive single resource on mathematics on the web. In 1998, he made a contract with CRC Press and the contents of the site were published in print and CD-ROM form, titled ''CRC Concise Encyclopedia of Mathematics''. The free online version became only partially accessible to the public. In 1999 Weisstein we ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Interquartile Mean
The interquartile mean (IQM) (or midmean) is a statistical measure of central tendency based on the truncated mean of the interquartile range. 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 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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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, and typically discarding an equal amount of both. This number of points to be discarded is usually given as a percentage of the total number of points, but may also be given as a fixed number of points. For most statistical applications, 5 to 25 percent of the ends are discarded. For example, given a set of 8 points, trimming by 12.5% would discard the minimum and maximum value in the sample: the smallest and largest values, and would compute the mean of the remaining 6 points. The 25% trimmed mean (when the lowest 25% and the highest 25% are discarded) is known as the interquartile mean. The median can be regarded as a fully truncated mean and is most robust. As with other trimmed estimators, the main advantage of the trimmed mean is ro ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Efficiency (statistics)
In statistics, efficiency is a measure of quality of an estimator, of an experimental design, or of a hypothesis testing procedure. Essentially, a more efficient estimator needs fewer input data or observations than a less efficient one to achieve the Cramér–Rao bound. An ''efficient estimator'' is characterized by having the smallest possible variance, indicating that there is a small deviance between the estimated value and the "true" value in the L2 norm sense. The relative efficiency of two procedures is the ratio of their efficiencies, although often this concept is used where the comparison is made between a given procedure and a notional "best possible" procedure. The efficiencies and the relative efficiency of two procedures theoretically depend on the sample size available for the given procedure, but it is often possible to use the asymptotic relative efficiency (defined as the limit of the relative efficiencies as the sample size grows) as the principal comparison ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
