Quartiles Deciles And Percentiles
Ogier Dudley
Put another way, 1st Quartile contains the part of data, 2nd Quartile contains of the data and 3rd Quartile contains the part of data. Repeating, it may be noted that the data should be arranged in
ascending or descending order of magnitude.
Quartiles and deciles are statistical concepts that are commonly used to analyze datasets. Quartiles divide a dataset into four equal parts, with the first quartile (Q1) representing the 25th percentile, the second quartile (Q2) representing the median or 50th percentile, and the third quartile (Q3) representing the 75th percentile. On the other hand, deciles divide a dataset into ten equal parts, with the first decile (D1) representing the 10th percentile, the second decile (D2) representing the 20th percentile, and so on, until the tenth decile (D10) which represents the maximum value of the data.
Quartiles Q1, Q2 (median), and Q3 are statistical measures that split a dataset into four equal parts. Specifically, Q1 represents the value below which 25% of the data falls, Q2 represents the midpoint value that splits the data into the lower and upper halves, and Q3 represents the value below which 75% of the data falls. Percentile is another statistical measure that divides a dataset into 100 equal parts, with each percentile representing the percentage of data below that point.
The decile formula is used to calculate the value of a specific decile for a dataset. The formula involves finding the position of the value in the ordered dataset and then calculating the decile value based on that position.
To sum up, quartiles, deciles, and percentiles serve as important tools to measure the spread of data and provide insights into its distribution. They are frequently utilized in data analysis and offer valuable information on the relative positions of individual values within a given dataset.
This document discusses different methods for organizing data, including percentiles, quartiles, and deciles. It provides the definitions and formulas for calculating each. Percentiles indicate the value below which a given percentage of observations fall. Quartiles divide a data set into four equal parts, with the median (Q2) separating the lower and upper halves. Deciles divide a data set into ten equal parts. The document gives examples of calculating percentiles, quartiles, and deciles for sample data sets.Read less
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 (four groups), deciles (ten groups), and percentiles (100 groups). 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.
As in the computation of, for example, standard deviation, the estimation of a quantile depends upon whether one is operating with a statistical population or with a sample drawn from it. For a population, of discrete values or for a continuous population density, the k-th q-quantile is the data value where the cumulative distribution function crosses k/q. That is, x is a k-th q-quantile for a variable X if
If, instead of using integers k and q, the "p-quantile" is based on a real number p with 0 < p < 1 then p replaces k/q in the above formulas. This broader terminology is used when quantiles are used to parameterize continuous probability distributions. Moreover, some software programs (including Microsoft Excel) regard the minimum and maximum as the 0th and 100th percentile, respectively. However, this broader terminology is an extension beyond traditional statistics definitions.
The first three are piecewise constant, changing abruptly at each data point, while the last six use linear interpolation between data points, and differ only in how the index h used to choose the point along the piecewise linear interpolation curve, is chosen.
Mathematica,[3] Matlab,[4] R[5] and GNU Octave[6] programming languages support all nine sample quantile methods. SAS includes five sample quantile methods, SciPy[7] and Maple[8] both include eight, EViews[9] and Julia[10] include the six piecewise linear functions, Stata[11] includes two, Python[12] includes two, and Microsoft Excel includes two. Mathematica, SciPy and Julia support arbitrary parameters for methods which allow for other, non-standard, methods.
The sample median is the most examined one amongst quantiles, being an alternative to estimate a location parameter, when the expected value of the distribution does not exist, and hence the sample mean is not a meaningful estimator of a population characteristic. Moreover, the sample median is a more robust estimator than the sample mean.
Computing approximate quantiles from data arriving from a stream can be done efficiently using compressed data structures. The most popular methods are t-digest[16] and KLL.[17] These methods read a stream of values in a continuous fashion and can, at any time, be queried about the approximate value of a specified quantile.
Both algorithms are based on a similar idea: compressing the stream of values by summarizing identical or similar values with a weight. If the stream is made of a repetition of 100 times v1 and 100 times v2, there is no reason to keep a sorted list of 200 elements, it is enough to keep two elements and two counts to be able to recover the quantiles. With more values, these algorithms maintain a trade-off between the number of unique values stored and the precision of the resulting quantiles. Some values may be discarded from the stream and contribute to the weight of a nearby value without changing the quantile results too much. The t-digest maintains a data structure of bounded size using an approach motivated by k-means clustering to group similar values. The KLL algorithm uses a more sophisticated "compactor" method that leads to better control of the error bounds at the cost of requiring an unbounded size if errors must be bounded relative to p.
Both methods belong to the family of data sketches that are subsets of Streaming Algorithms with useful properties: t-digest or KLL sketches can be combined. Computing the sketch for a very large vector of values can be split into trivially parallel processes where sketches are computed for partitions of the vector in parallel and merged later.
Standardized test results are commonly reported as a student scoring "in the 80th percentile", for example. This uses an alternative meaning of the word percentile as the interval between (in this case) the 80th and the 81st scalar percentile.[18] This separate meaning of percentile is also used in peer-reviewed scientific research articles.[19] The meaning used can be derived from its context.
If a distribution is symmetric, then the median is the mean (so long as the latter exists). But, in general, the median and the mean can differ. For instance, with a random variable that has an exponential distribution, any particular sample of this random variable will have roughly a 63% chance of being less than the mean. This is because the exponential distribution has a long tail for positive values but is zero for negative numbers.
Quantiles are useful measures because they are less susceptible than means to long-tailed distributions and outliers. Empirically, if the data being analyzed are not actually distributed according to an assumed distribution, or if there are other potential sources for outliers that are far removed from the mean, then quantiles may be more useful descriptive statistics than means and other moment-related statistics.
Closely related is the subject of least absolute deviations, a method of regression that is more robust to outliers than is least squares, in which the sum of the absolute value of the observed errors is used in place of the squared error. The connection is that the mean is the single estimate of a distribution that minimizes expected squared error while the median minimizes expected absolute error. Least absolute deviations shares the ability to be relatively insensitive to large deviations in outlying observations, although even better methods of robust regression are available.
The quantiles of a random variable are preserved under increasing transformations, in the sense that, for example, if m is the median of a random variable X, then 2m is the median of 2X, unless an arbitrary choice has been made from a range of values to specify a particular quantile. (See quantile estimation, above, for examples of such interpolation.) Quantiles can also be used in cases where only ordinal data are available.
In Statistics, we commonly encounter the concept of the median, which represents the middle value or mean of the two middle values in a dataset. However, there are other essential values that divide data into equal parts for more comprehensive analysis.
Quartiles divide data into four parts, deciles into ten parts, and percentiles into one hundred parts. These measures provide valuable insights into the distribution and spread of data, allowing us to analyze specific segments of a dataset with precision. By comprehending quartiles, deciles, and percentiles, we gain a deeper understanding of the patterns and characteristics of our data.
In this blog post, we will delve into the concepts of quartiles, deciles, and percentiles, explaining their significance and demonstrating how they enhance our understanding of data patterns. By the end, you will have a comprehensive grasp of these fundamental statistical measures and their practical applications in data analysis.
The values which divide an array (a set of data arranged in ascending or descending order) into four equal parts are called Quartiles. The first, second and third quartiles are denoted by Q1, Q2,Q3 respectively. The first and third quartiles are also called the lower and upper quartiles respectively. The second quartile represents the median, the middle value.
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