Quartile

Last updated March 22, 2024

In statistics, quartiles are a type of quantiles which divide 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 a form of order statistic. The three quartiles, resulting in four data divisions, are as follows:

The first quartile (Q₁) is defined as the 25th percentile where lowest 25% data is below this point. It is also known as the lower quartile.
The second quartile (Q₂) is the median of a data set; thus 50% of the data lies below this point.
The third quartile (Q₃) is the 75th percentile where lowest 75% data is below this point. It is known as the upper quartile, as 75% of the data lies below this point.^[1]

Along with the minimum and maximum of the data (which are also quartiles), the three quartiles described above provide a five-number summary of the data. This summary is important in statistics because it provides information about both the center and the spread of the data. Knowing the lower and upper quartile provides information on how big the spread is and if the dataset is skewed toward one side. Since quartiles divide the number of data points evenly, the range is generally not the same between adjacent quartiles (i.e. usually (Q₃ - Q₂) ≠ (Q₂ - Q₁)). Interquartile range (IQR) is defined as the difference between the 75th and 25th percentiles or Q₃ - Q₁. While the maximum and minimum also show the spread of the data, the upper and lower quartiles can provide more detailed information on the location of specific data points, the presence of outliers in the data, and the difference in spread between the middle 50% of the data and the outer data points.^[2]

Definitions

Symbol	Names	Definition
Q₁	First quartile Lower quartile 25th percentile	Splits off the lowest 25% of data from the highest 75%
Q₂	Second quartile Median 50th percentile	Cuts data set in half
Q₃	Third quartile Upper quartile 75th percentile	Splits off the highest 25% of data from the lowest 75%

Computing methods

Discrete distributions

For discrete distributions, there is no universal agreement on selecting the quartile values.^[3]

Method 1

Use the median to divide the ordered data set into two-halves. The median becomes the second quartiles.
- If there are an odd number of data points in the original ordered data set, do not include the median (the central value in the ordered list) in either half.
- If there are an even number of data points in the original ordered data set, split this data set exactly in half.
The lower quartile value is the median of the lower half of the data. The upper quartile value is the median of the upper half of the data.

This rule is employed by the TI-83 calculator boxplot and "1-Var Stats" functions.

Method 2

Use the median to divide the ordered data set into two-halves. The median becomes the second quartiles.
- If there are an odd number of data points in the original ordered data set, include the median (the central value in the ordered list) in both halves.
- If there are an even number of data points in the original ordered data set, split this data set exactly in half.
The lower quartile value is the median of the lower half of the data. The upper quartile value is the median of the upper half of the data.

The values found by this method are also known as "Tukey's hinges";^[4] see also midhinge.

Method 3

Use the median to divide the ordered data set into two-halves. The median becomes the second quartiles.
1. If there are odd numbers of data points, then go to the next step.
2. If there are even numbers of data points, then the Method 3 starts off the same as the Method 1 or the Method 2 above and you can choose to include or not include the median as a new datapoint. If you choose to include the median as the new datapoint, then proceed to the step 2 or 3 below because you now have an odd number of datapoints. If you do not choose the median as the new data point, then continue the Method 1 or 2 where you have started.
If there are (4n+1) data points, then the lower quartile is 25% of the nth data value plus 75% of the (n+1)th data value; the upper quartile is 75% of the (3n+1)th data point plus 25% of the (3n+2)th data point.
If there are (4n+3) data points, then the lower quartile is 75% of the (n+1)th data value plus 25% of the (n+2)th data value; the upper quartile is 25% of the (3n+2)th data point plus 75% of the (3n+3)th data point.

Method 4

If we have an ordered dataset $x_{1},x_{2},...,x_{n}$ , then we can interpolate between data points to find the $p$ th empirical quantile if $x_{i}$ is in the $i/(n+1)$ quantile. If we denote the integer part of a number $a$ by $\lfloor a\rfloor$ , then the empirical quantile function is given by,

$q(p/4)=x_{k}+\alpha (x_{k+1}-x_{k})$ ,

where $k=\lfloor p(n+1)/4\rfloor$ and $\alpha =p(n+1)/4-\lfloor p(n+1)/4\rfloor$ .^[1]

To find the first, second, and third quartiles of the dataset we would evaluate $q(0.25)$ , $q(0.5)$ , and $q(0.75)$ respectively.

Example 1

Ordered Data Set (of an odd number of data points): 6, 7, 15, 36, 39, 40, 41, 42, 43, 47, 49.

The bold number (40) is the median splitting the data set into two halves with equal number of data points.

	Method 1	Method 2	Method 3	Method 4
Q₁	15	25.5	20.25	15
Q₂	40	40	40	40
Q₃	43	42.5	42.75	43

Example 2

Ordered Data Set (of an even number of data points): 7, 15, 36, 39, 40, 41.

The bold numbers (36, 39) are used to calculate the median as their average. As there are an even number of data points, the first three methods all give the same results. (The Method 3 is executed such that the median is not chosen as a new data point and the Method 1 started.)

	Method 1	Method 2	Method 3	Method 4
Q₁	15	15	15	13
Q₂	37.5	37.5	37.5	37.5
Q₃	40	40	40	40.25

Continuous probability distributions

Quartiles on a cumulative distribution function of a normal distribution NormalCDFQuartile3.svg — Quartiles on a cumulative distribution function of a normal distribution

If we define a continuous probability distributions as $P(X)$ where $X$ is a real valued random variable, its cumulative distribution function (CDF) is given by

$F_{X}(x)=P(X\leq x)$ .^[1]

The CDF gives the probability that the random variable $X$ is less than or equal to the value $x$ . Therefore, the first quartile is the value of $x$ when $F_{X}(x)=0.25$ , the second quartile is $x$ when $F_{X}(x)=0.5$ , and the third quartile is $x$ when $F_{X}(x)=0.75$ .^[5] The values of $x$ can be found with the quantile function $Q(p)$ where $p=0.25$ for the first quartile, $p=0.5$ for the second quartile, and $p=0.75$ for the third quartile. The quantile function is the inverse of the cumulative distribution function if the cumulative distribution function is monotonically increasing because the one-to-one correspondence between the input and output of the cumulative distribution function holds.

Outliers

There are methods by which to check for outliers in the discipline of statistics and statistical analysis. Outliers could be a result from a shift in the location (mean) or in the scale (variability) of the process of interest.^[6] Outliers could also be evidence of a sample population that has a non-normal distribution or of a contaminated population data set. Consequently, as is the basic idea of descriptive statistics, when encountering an outlier, we have to explain this value by further analysis of the cause or origin of the outlier. In cases of extreme observations, which are not an infrequent occurrence, the typical values must be analyzed. The Interquartile Range (IQR), defined as the difference between the upper and lower quartiles ( ${\textstyle Q_{3}-Q_{1}}$ ), may be used to characterize the data when there may be extremities that skew the data; the interquartile range is a relatively robust statistic (also sometimes called "resistance") compared to the range and standard deviation. There is also a mathematical method to check for outliers and determining "fences", upper and lower limits from which to check for outliers.

After determining the first (lower) and third (upper) quartiles ( ${\textstyle Q_{1}}$ and ${\textstyle Q_{3}}$ respectively) and the interquartile range ( ${\textstyle {\textrm {IQR}}=Q_{3}-Q_{1}}$ ) as outlined above, then fences are calculated using the following formula:

{\text{Lower fence}}=Q_{1}-(1.5\times \mathrm {IQR} )

{\text{Upper fence}}=Q_{3}+(1.5\times \mathrm {IQR} )

The lower fence is the "lower limit" and the upper fence is the "upper limit" of data, and any data lying outside these defined bounds can be considered an outlier. The fences provide a guideline by which to define an outlier, which may be defined in other ways. The fences define a "range" outside which an outlier exists; a way to picture this is a boundary of a fence. It is common for the lower and upper fences along with the outliers to be represented by a boxplot. For the boxplot shown on the right, only the vertical heights correspond to the visualized data set while horizontal width of the box is irrelevant. Outliers located outside the fences in a boxplot can be marked as any choice of symbol, such as an "x" or "o". The fences are sometimes also referred to as "whiskers" while the entire plot visual is called a "box-and-whisker" plot.

When spotting an outlier in the data set by calculating the interquartile ranges and boxplot features, it might be easy to mistakenly view it as evidence that the population is non-normal or that the sample is contaminated. However, this method should not take place of a hypothesis test for determining normality of the population. The significance of the outliers varies depending on the sample size. If the sample is small, then it is more probable to get interquartile ranges that are unrepresentatively small, leading to narrower fences. Therefore, it would be more likely to find data that are marked as outliers.^[7]

Computer software for quartiles


Environment	Function	Quartile Method
Microsoft Excel	QUARTILE.EXC	Method 4
Microsoft Excel	QUARTILE.INC	Method 3
TI-8X series calculators	1-Var Stats	Method 1
R	fivenum	Method 2
Python	numpy.percentile	Method 3
Python	pandas.DataFrame.describe	Method 3

Excel

The Excel function QUARTILE(array, quart) provides the desired quartile value for a given array of data, using Method 3 from above. In the QUARTILE function (a legacy function from Excel 2007 or earlier, giving the same output of the function QUARTILE.INC), array is the dataset of numbers that is being analyzed and quart is any of the following 5 values depending on which quartile is being calculated. ^[8]


Quart	Output QUARTILE Value
0	Minimum value
1	Lower Quartile (25th percentile)
2	Median
3	Upper Quartile (75th percentile)
4	Maximum value

MATLAB

In order to calculate quartiles in Matlab, the function quantile(A,p) can be used. Where A is the vector of data being analyzed and p is the percentage that relates to the quartiles as stated below. ^[9]


p	Output QUARTILE Value
0	Minimum value
0.25	Lower Quartile (25th percentile)
0.5	Median
0.75	Upper Quartile (75th percentile)
1	Maximum value

Related Research Articles

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 between the 75th and 25th percentiles of the data. To calculate the IQR, the data set is divided into quartiles, or four rank-ordered even parts via linear interpolation. These quartiles are denoted by Q₁ (also called the lower quartile), Q₂ (the median), and Q₃ (also called the upper quartile). The lower quartile corresponds with the 25th percentile and the upper quartile corresponds with the 75th percentile, so IQR = Q₃ − Q₁_.

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. Common quantiles have special names, such as quartiles, deciles, and percentiles. 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.

<span class="mw-page-title-main">Skewness</span> Measure of the asymmetry of random variables

In probability theory and statistics, skewness is a measure of the asymmetry of the probability distribution of a real-valued random variable about its mean. The skewness value can be positive, zero, negative, or undefined.

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.

<span class="mw-page-title-main">Box plot</span> Data visualization

In descriptive statistics, a box plot or boxplot is a method for graphically demonstrating the locality, spread and skewness groups of numerical data through their quartiles. In addition to the box on a box plot, there can be lines extending from the box indicating variability outside the upper and lower quartiles, thus, the plot is also called the box-and-whisker plot and the box-and-whisker diagram. Outliers that differ significantly from the rest of the dataset may be plotted as individual points beyond the whiskers on the box-plot. 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 in each subsection of the box-plot indicate the degree of dispersion (spread) and skewness of the data, which are usually described using the five-number summary. In addition, the box-plot allows 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.

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:

the sample minimum (smallest observation)
the lower quartile or first quartile
the median
the upper quartile or third quartile
the sample maximum

In statistics, a k-thpercentile, also known as percentile score or centile, is a score below which a given percentage k of scores in its frequency distribution falls or a score at or below which a given percentage falls. Percentiles are expressed in the same unit of measurement as the input scores, not in percent; for example, if the scores refer to human weight, the corresponding percentiles will be expressed in kilograms or pounds. In the limit of an infinite sample size, the percentile approximates the percentile function, the inverse of the cumulative distribution function.

This glossary of statistics and probability is a list of definitions of terms and concepts used in the mathematical sciences of statistics and probability, their sub-disciplines, and related fields. For additional related terms, see Glossary of mathematics and Glossary of experimental design.

In statistics, a Q–Q plot (quantile–quantile plot) is a probability plot, a graphical method for comparing two probability distributions by plotting their quantiles against each other. A point $(x, y)$ on the plot corresponds to one of the quantiles of the second distribution ( $y$ -coordinate) plotted against the same quantile of the first distribution ( $x$ -coordinate). This defines a parametric curve where the parameter is the index of the quantile interval.

The root-mean-square deviation (RMSD) or root-mean-square error (RMSE) is either one of two closely related and frequently used measures of the differences between true or predicted values on the one hand and observed values or an estimator on the other.

In probability and statistics, the quantile function outputs the value of a random variable such that its probability is less than or equal to an input probability value. Intuitively, the quantile function associates with a range at and below a probability input the likelihood that a random variable is realized in that range for some probability distribution. It is also called the percentile function, percent-point function, inverse cumulative distribution function or inverse distribution function.

In statistics, the quartile coefficient of dispersion is a descriptive statistic which measures dispersion and is used to make comparisons within and between data sets. Since it is based on quantile information, it is less sensitive to outliers than measures such as the coefficient of variation. As such, it is one of several robust measures of scale.

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.

<span class="mw-page-title-main">L-estimator</span>

In statistics, an L-estimator 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, or as many as all points, as in the mean.

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.

<span class="mw-page-title-main">Plot (graphics)</span> Graphical technique for data sets

A plot is a graphical technique for representing a data set, usually as a graph showing the relationship between two or more variables. The plot can be drawn by hand or by a computer. In the past, sometimes mechanical or electronic plotters were used. Graphs are a visual representation of the relationship between variables, which are very useful for humans who can then quickly derive an understanding which may not have come from lists of values. Given a scale or ruler, graphs can also be used to read off the value of an unknown variable plotted as a function of a known one, but this can also be done with data presented in tabular form. Graphs of functions are used in mathematics, sciences, engineering, technology, finance, and other areas.

In statistics, robust measures of scale are methods that quantify the statistical dispersion in a sample of numerical data while resisting outliers. The most common such robust statistics are the interquartile range (IQR) and the median absolute deviation (MAD). These are contrasted with conventional or non-robust measures of scale, such as sample standard deviation, which are greatly influenced by outliers.

In statistics, dispersion is the extent to which a distribution is stretched or squeezed. Common examples of measures of statistical dispersion are the variance, standard deviation, and interquartile range. For instance, when the variance of data in a set is large, the data is widely scattered. On the other hand, when the variance is small, the data in the set is clustered.

In statistical graphics, the functional boxplot is an informative exploratory tool that has been proposed for visualizing functional data. Analogous to the classical boxplot, the descriptive statistics of a functional boxplot are: the envelope of the 50% central region, the median curve and the maximum non-outlying envelope.

In statistical graphics and scientific visualization, the contour boxplot is an exploratory tool that has been proposed for visualizing ensembles of feature-sets determined by a threshold on some scalar function. Analogous to the classical boxplot and considered an expansion of the concepts defining functional boxplot, the descriptive statistics of a contour boxplot are: the envelope of the 50% central region, the median curve and the maximum non-outlying envelope.

References

1 2 3 Dekking, Michel (2005). A modern introduction to probability and statistics: understanding why and how . London: Springer. pp. 236-238. ISBN 978-1-85233-896-1. OCLC 262680588.
↑ Knoch, Jessica (February 23, 2018). "How are Quartiles Used in Statistics?". Magoosh . Archived from the original on December 10, 2019. Retrieved February 24, 2023.
↑ Hyndman, Rob J; Fan, Yanan (November 1996). "Sample quantiles in statistical packages". American Statistician. 50 (4): 361–365. doi:10.2307/2684934. JSTOR 2684934.
↑ Tukey, John Wilder (1977). Exploratory Data Analysis . ISBN 978-0-201-07616-5.
↑ "6. Distribution and Quantile Functions" (PDF). math.bme.hu.
↑ Walfish, Steven (November 2006). "A Review of Statistical Outlier Method". Pharmaceutical Technology.
↑ Dawson, Robert (July 1, 2011). "How Significant is a Boxplot Outlier?". Journal of Statistics Education. 19 (2). doi: 10.1080/10691898.2011.11889610 .
↑ "How to use the Excel QUARTILE function | Exceljet". exceljet.net. Retrieved December 11, 2019.
↑ "Quantiles of a data set – MATLAB quantile". www.mathworks.com. Retrieved December 11, 2019.

External links

Quartile – from MathWorld Includes references and compares various methods to compute quartiles
Quartiles – From MathForum.org
Quartiles calculator – simple quartiles calculator
Quartiles – An example how to calculate it

This page is based on this Wikipedia article
Text is available under the CC BY-SA 4.0 license; additional terms may apply.
Images, videos and audio are available under their respective licenses.

[:0-1] 1 2 3 Dekking, Michel (2005). A modern introduction to probability and statistics: understanding why and how . London: Springer. pp. 236-238. ISBN 978-1-85233-896-1. OCLC 262680588.

[2] Knoch, Jessica (February 23, 2018). "How are Quartiles Used in Statistics?". Magoosh . Archived from the original on December 10, 2019. Retrieved February 24, 2023.

[3] Hyndman, Rob J; Fan, Yanan (November 1996). "Sample quantiles in statistical packages". American Statistician. 50 (4): 361–365. doi:10.2307/2684934. JSTOR 2684934.

[4] Tukey, John Wilder (1977). Exploratory Data Analysis . ISBN 978-0-201-07616-5.

[5] "6. Distribution and Quantile Functions" (PDF). math.bme.hu.

[6] Walfish, Steven (November 2006). "A Review of Statistical Outlier Method". Pharmaceutical Technology.

[7] Dawson, Robert (July 1, 2011). "How Significant is a Boxplot Outlier?". Journal of Statistics Education. 19 (2). doi: 10.1080/10691898.2011.11889610 .

[8] "How to use the Excel QUARTILE function | Exceljet". exceljet.net. Retrieved December 11, 2019.

[9] "Quantiles of a data set – MATLAB quantile". www.mathworks.com. Retrieved December 11, 2019.

[1]

[2]

[3]

[4]

[5]

[6]

[7]

[8]

[9]