Quartile

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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 (Q1) 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 (if data is ordered on a timeline from smallest to largest). The second quartile (Q2) is the median of a data set and 50% of the data lies below this point. The third quartile (Q3) 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. [1] 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.

Contents

Along with the minimum and the 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 not the same between quartiles (i.e., Q3-Q2Q2-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

SymbolNamesDefinition
Q1
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

1. Use the median to divide the ordered data set into two-halves.
• If there is 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 is an even number of data points in the original ordered data set, split this data set exactly in half.
2. 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

1. Use the median to divide the ordered data set into two-halves.
• 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.
2. 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

1. If there are even numbers of data points, then Method 3 is the same as either method above.
2. 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.
3. 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 ${\displaystyle x_{1},x_{2},...,x_{n}}$, we can interpolate between data points to find the ${\displaystyle p}$th empirical quantile if ${\displaystyle x_{i}}$ is in the ${\displaystyle i/(n+1)}$ quantile. If we denote the integer part of a number ${\displaystyle a}$ by ${\displaystyle [a]}$, then the empirical quantile function is given by,

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

where ${\displaystyle k=[p(n+1)]}$ and ${\displaystyle \alpha =p(n+1)-[p(n+1)]}$. [1]

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

Example 1

Ordered Data Set: 6, 7, 15, 36, 39, 40, 41, 42, 43, 47, 49

Method 1Method 2Method 3Method 4
Q11525.520.2515
Q240404040
Q34342.542.7543

Example 2

Ordered Data Set: 7, 15, 36, 39, 40, 41

As there are an even number of data points, the first three methods all give the same results.

Method 1Method 2Method 3Method 4
Q115151513
Q237.537.537.537.5
Q340404040.25

Continuous probability distributions

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

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

The CDF gives the probability that the random variable ${\displaystyle X}$ is less than the value ${\displaystyle x}$. Therefore, the first quartile is the value of ${\displaystyle x}$ when ${\displaystyle F_{X}(x)=0.25}$, the second quartile is ${\displaystyle x}$ when ${\displaystyle F_{X}(x)=0.5}$, and the third quartile is ${\displaystyle x}$ when ${\displaystyle F_{X}(x)=0.75}$. [5] The values of ${\displaystyle x}$ can be found with the quantile function ${\displaystyle Q(p)}$where ${\displaystyle p=0.25}$ for the first quartile, ${\displaystyle p=0.5}$ for the second quartile, and ${\displaystyle p=0.75}$ for the third quartile. The quantile function is the inverse of the cumulative distribution function if the cumulative distribution function is monotonically increasing.

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 may 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. In the case of quartiles, the Interquartile Range (IQR) 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 and third quartiles and the interquartile range as outlined above, then fences are calculated using the following formula:

${\displaystyle {\text{Lower fence}}=Q_{1}-1.5(\mathrm {IQR} )\,}$
${\displaystyle {\text{Upper fence}}=Q_{3}+1.5(\mathrm {IQR} ),\,}$

where Q1 and Q3 are the first and third quartiles, respectively. 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. Anything below the Lower fence or above the Upper fence can be considered such a case. 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, outside which are "outsiders" as opposed to outliers. It is common for the lower and upper fences along with the outliers to be represented by a boxplot. For a boxplot, 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 simple 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 vary 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

Excel:

The Excel function QUARTILE(array, quart) provides the desired quartile value for a given array of data. In the Quartile function, 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]

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

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

Related Research Articles

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.

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 = Q3 − Q1. 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, 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 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.

In statistics, an outlier is a data point that differs significantly from other observations. An outlier may be due to variability in the measurement or it may indicate experimental error; the latter are sometimes excluded from the data set. An outlier can cause serious problems in statistical analyses.

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.

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:

1. the sample minimum (smallest observation)
2. the lower quartile or first quartile
3. the median
4. the upper quartile or third quartile
5. the sample maximum

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.

In statistics, a Q–Q (quantile-quantile) plot is a probability plot, which is a graphical method for comparing two probability distributions by plotting their quantiles against each other. First, the set of intervals for the quantiles is chosen. A point (x, y) on the plot corresponds to one of the quantiles of the second distribution plotted against the same quantile of the first distribution. Thus the line is a parametric curve with the parameter which is the number of the interval for the quantile.

The root-mean-square deviation (RMSD) or root-mean-square error (RMSE) is a frequently used measure of the differences between values predicted by a model or an estimator and the values observed. The RMSD represents the square root of the second sample moment of the differences between predicted values and observed values or the quadratic mean of these differences. These deviations are called residuals when the calculations are performed over the data sample that was used for estimation and are called errors when computed out-of-sample. The RMSD serves to aggregate the magnitudes of the errors in predictions for various times into a single measure of predictive power. RMSD is a measure of accuracy, to compare forecasting errors of different models for a particular dataset and not between datasets, as it is scale-dependent.

It is also called the percent-point function or inverse cumulative distribution function.

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 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.

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, 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.

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.

In statistics, the medcouple is a robust statistic that measures the skewness of a univariate distribution. It is defined as a scaled median difference of the left and right half of a distribution. Its robustness makes it suitable for identifying outliers in adjusted boxplots. Ordinary box plots do not fare well with skew distributions, since they label the longer unsymmetrical tails as outliers. Using the medcouple, the whiskers of a boxplot can be adjusted for skew distributions and thus have a more accurate identification of outliers for non-symmetrical distributions.

References

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