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The **five-number summary** is a set of descriptive statistics that provides information about a dataset. It consists of the five most important sample percentiles:

- Use and representation
- Example
- Example in R
- Example in Python
- Example in SAS
- Example in Stata
- See also
- References

- the sample minimum
*(smallest observation)* - the lower quartile or
*first quartile* - the median (the middle value)
- the upper quartile or
*third quartile* - the sample maximum (largest observation)

In addition to the median of a single set of data there are two related statistics called the upper and lower quartiles. If data are placed in order, then the lower quartile is central to the lower half of the data and the upper quartile is central to the upper half of the data. These quartiles are used to calculate the interquartile range, which helps to describe the spread of the data, and determine whether or not any data points are outliers.

In order for these statistics to exist the observations must be from a univariate variable that can be measured on an ordinal, interval or ratio scale.

*The five-number summary provides a concise summary of the distribution of the observations. Reporting five numbers avoids the need to decide on the most appropriate summary statistic. The five-number summary gives information about the location (from the median), spread (from the quartiles) and range (from the sample minimum and maximum) of the observations. Since it reports order statistics (rather than, say, the mean) the five-number summary is appropriate for ordinal measurements, as well as interval and ratio measurements.*

*It is possible to quickly compare several sets of observations by comparing their five-number summaries, which can be represented graphically using a boxplot.*

*In addition to the points themselves, many L-estimators can be computed from the five-number summary, including interquartile range, midhinge, range, mid-range, and trimean.*

*The five-number summary is sometimes represented as in the following table: *

median | |

1st quartile | 3rd quartile |

Minimum | Maximum |

*This example calculates the five-number summary for the following set of observations: 0, 0, 1, 2, 63, 61, 27, 13. These are the number of moons of each planet in the Solar System.*

*It helps to put the observations in ascending order: 0, 0, 1, 2, 13, 27, 61, 63. There are eight observations, so the median is the mean of the two middle numbers, (2 + 13)/2 = 7.5. Splitting the observations either side of the median gives two groups of four observations. The median of the first group is the lower or first quartile, and is equal to (0 + 1)/2 = 0.5. The median of the second group is the upper or third quartile, and is equal to (27 + 61)/2 = 44. The smallest and largest observations are 0 and 63.*

*So the five-number summary would be 0, 0.5, 7.5, 44, 63.*

*It is possible to calculate the five-number summary in the R programming language using the fivenum function. The summary function, when applied to a vector, displays the five-number summary together with the mean (which is not itself a part of the five-number summary).*

> moons <- c(0, 0, 1, 2, 63, 61, 27, 13) > fivenum(moons)[1] 0.0 0.5 7.5 44.0 63.0> summary(moons) Min. 1st Qu. Median Mean 3rd Qu. Max. 0.00 0.75 7.50 20.88 35.50 63.00

*This python example uses the percentile function from the numerical library numpy and works in Python 2 and 3. *

*importnumpyasnpdeffivenum(data):"""Five-number summary."""returnnp.percentile(data,[0,25,50,75,100],interpolation='midpoint')moons=[0,0,1,2,63,61,27,13]print(fivenum(moons))[0.0.57.544.63.]*

*You can use PROC UNIVARIATE in SAS (software) to get the five number summary:*

data fivenum; input x @@; datalines; 1 2 3 4 20 202 392 4 38 20 ; run; ods select Quantiles; proc univariate data = fivenum; output out = fivenums min = min Q1 = Q1 Q2 = median Q3 = Q3 max = max; run; proc print data = fivenums; run;

input byte y 0 0 1 2 63 61 27 13 end list tabstat y, statistics (min q max)

*A five number summary of a distribution of data.*

*In statistics, a central tendency is a central or typical value for a probability distribution. It may also be called a center or location of the distribution. Colloquially, measures of central tendency are often called averages. The term central tendency dates from the late 1920s.*

*A descriptive statistic is a summary statistic that quantitatively describes or summarizes features from a collection of information, while descriptive statistics is the process of using and analysing those statistics. Descriptive statistics is distinguished from inferential statistics by its aim to summarize a sample, rather than use the data to learn about the population that the sample of data is thought to represent. This generally means that descriptive statistics, unlike inferential statistics, is not developed on the basis of probability theory, and are frequently non-parametric statistics. Even when a data analysis draws its main conclusions using inferential statistics, descriptive statistics are generally also presented. For example, in papers reporting on human subjects, typically a table is included giving the overall sample size, sample sizes in important subgroups, and demographic or clinical characteristics such as the average age, the proportion of subjects of each sex, the proportion of subjects with related co-morbidities, etc.*

In descriptive statistics, the **interquartile range** (**IQR**), also called the **midspread**, **middle 50%**, or **H‑spread**, is a measure of statistical dispersion, being equal to the difference between 75th and 25th percentiles, or between upper and lower quartiles, IQR = *Q*_{3} − *Q*_{1}. In other words, the IQR is the first quartile subtracted from the third quartile; these quartiles can be clearly seen on a box plot on the data. It is a trimmed estimator, defined as the 25% trimmed range, and is a commonly used robust measure of scale.

In statistics and probability theory, a **median** is a 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. For example, the basic advantage of the median in describing data compared to the mean is that it is not skewed so much by a small proportion of extremely large or small values, and so it may give a better idea of a "typical" value. For example, in understanding statistics like household income or assets, which vary greatly, the mean may be skewed by a small number of extremely high or low values. Median income, for example, may be a better way to suggest what a "typical" income is. Because of this, the median is of central importance in robust statistics, as it is the most resistant statistic, having a breakdown point of 50%: so long as no more than half the data are contaminated, the median will not give an arbitrarily large or small result.

* A quartile is a type of quantile which divides the number of data points into four more or less equal parts, or quarters. The first quartile (Q_{1}) is defined as the middle number between the smallest number and the median of the data set. It is also known as the lower quartile or the 25th empirical quartile and it marks where 25% of the data is below or to the left of it. The second quartile (Q_{2}) is the median of a data set and 50% of the data lies below this point. The third quartile (Q_{3}) is the middle value between the median and the highest value of the data set. It is also known as the upper quartile or the 75th empirical quartile and 75% of the data lies below this point. Due to the fact that the data needs to be ordered from smallest to largest to compute quartiles, quartiles are a form of Order statistic.*

In statistics and probability, **quantiles** are cut points dividing the range of a probability distribution into continuous intervals with equal probabilities, or dividing the observations in a sample in the same way. There is one fewer quantile than the number of groups created. Thus quartiles are the three cut points that will divide a dataset into four equal-sized groups. Common quantiles have special names: for instance quartile, decile. The groups created are termed halves, thirds, quarters, etc., though sometimes the terms for the quantile are used for the groups created, rather than for the cut points.

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

*The interquartile mean (IQM) 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.*

In descriptive statistics, a **box plot** or **boxplot** is a method for graphically depicting groups of numerical data through their quartiles. Box plots may also have lines extending from the boxes (*whiskers*) indicating variability outside the upper and lower quartiles, hence the terms **box-and-whisker plot** and **box-and-whisker diagram**. Outliers may be plotted as individual points. Box plots are non-parametric: they display variation in samples of a statistical population without making any assumptions of the underlying statistical distribution. The spacings between the different parts of the box indicate the degree of dispersion (spread) and skewness in the data, and show outliers. In addition to the points themselves, they allow one to visually estimate various L-estimators, notably the interquartile range, midhinge, range, mid-range, and trimean. Box plots can be drawn either horizontally or vertically. Box plots received their name from the box in the middle.

*A percentile is a measure used in statistics indicating the value below which a given percentage of observations in a group of observations falls. For example, the 20th percentile is the value below which 20% of the observations may be found. Equivalently, 80% of the observations are found above the 20th percentile.*

*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.*

*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:*

*In statistics, the median absolute deviation (MAD) is a robust measure of the variability of a univariate sample of quantitative data. It can also refer to the population parameter that is estimated by the MAD calculated from a sample.*

*In statistics, the quartile coefficient of dispersion is a descriptive statistic which measures dispersion and which is used to make comparisons within and between data sets.*

*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.*

In statistics, an **L-estimator** is an estimator which is an L-statistic – a linear combination of order statistics of the measurements. This can be as little as a single point, as in the median, or as many as all points, as in the mean.

*In descriptive statistics, the seven-number summary is a collection of seven summary statistics, and is an extension of the five-number summary. There are two similar, common forms.*

*In statistics, a trimmed estimator is an estimator derived from another estimator by excluding some of the extreme values, a process called truncation. This is generally done to obtain a more robust statistic, and the extreme values are considered outliers. Trimmed estimators also often have higher efficiency for mixture distributions and heavy-tailed distributions than the corresponding untrimmed estimator, at the cost of lower efficiency for other distributions, such as the normal distribution.*

*In statistics, the interdecile range is the difference between the first and the ninth deciles. The interdecile range is a measure of statistical dispersion of the values in a set of data, similar to the range and the interquartile range, and can be computed from the (non-parametric) seven-number summary.*

*In statistics, a robust measure of scale is a robust statistic that quantifies the statistical dispersion in a set of numerical data. The most common such statistics are the interquartile range (IQR) and the median absolute deviation (MAD). These are contrasted with conventional measures of scale, such as sample variance or sample standard deviation, which are non-robust, meaning greatly influenced by outliers.*

*David C. Hoaglin, Frederick Mosteller and John W. Tukey. "**Understanding robust and exploratory data analysis*". Wiley, 1983. ISBN 0-471-09777-2.

*Cambridge university(2016). "Chapter 9 statistics". Retrieved 11 August 2016, from http://cambridge.edu.au/go/resource/?pid=1145*

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