What Does It Mean to Be in the 90th Percentile

Statistic which divides a data set into 100 parts and analyzes it equally a percentage

In statistics, a k-th percentile (percentile score or centile) is a score below which a given percent k of scores in its frequency distribution falls (exclusive definition) or a score at or below which a given percentage falls (inclusive definition). For example, the 50th percentile (the median) is the score below which (sectional) or at or beneath which (inclusive) 50% of the scores in the distribution may be found. Percentiles are expressed in the same unit of measurement as the input scores; for instance, if the scores refer to homo weight, the respective percentiles will exist expressed in kilograms or pounds.

The percentile score and the percentile rank are related terms. The percentile rank of a score is the per centum of scores in its distribution that are less than it, an exclusive definition, and ane that can be expressed with a single, uncomplicated formula. Percentile scores and percentile ranks are often used in the reporting of test scores from norm-referenced tests, but, as just noted, they are not the same. For percentile rank, a score is given and a percentage is computed. Percentile ranks are exclusive. If the percentile rank for a specified score is 90%, and so xc% of the scores were lower. In contrast, for percentiles a pct is given and a corresponding score is determined, which tin can be either exclusive or inclusive. The score for a specified percent (e.g., 90th) indicates a score beneath which (exclusive definition) or at or beneath which (inclusive definition) other scores in the distribution fall.

The 25th percentile is also known as the first quartile (Q _ane), the 50th percentile every bit the median or second quartile (Q ₂), and the 75th percentile every bit the tertiary quartile (Q ₃).

Applications [edit]

When ISPs bill "burstable" internet bandwidth, the 95th or 98th percentile ordinarily cuts off the height 5% or ii% of bandwidth peaks in each month, and and so bills at the nearest rate. In this way, infrequent peaks are ignored, and the customer is charged in a fairer way. The reason this statistic is so useful in measuring information throughput is that it gives a very authentic picture show of the price of the bandwidth. The 95th percentile says that 95% of the fourth dimension, the usage is below this corporeality: so, the remaining 5% of the time, the usage is above that amount.

Physicians will often utilize babe and children'due south weight and height to assess their growth in comparison to national averages and percentiles which are found in growth charts.

The 85th percentile speed of traffic on a road is oft used as a guideline in setting speed limits and assessing whether such a limit is likewise loftier or low.^{[1] [2]}

In finance, value at adventure is a standard measure to appraise (in a model-dependent way) the quantity under which the value of the portfolio is not expected to sink within a given period of fourth dimension and given a conviction value.

The normal distribution and percentiles [edit]

Representation of the three-sigma dominion. The dark blue zone represents observations within ane standard deviation (σ) to either side of the mean (μ), which accounts for about 68.3% of the population. Two standard deviations from the hateful (dark and medium blueish) business relationship for nearly 95.four%, and 3 standard deviations (dark, medium, and lite blue) for about 99.7%.

The methods given in the definitions section (beneath) are approximations for use in small-sample statistics. In general terms, for very big populations following a normal distribution, percentiles may ofttimes be represented by reference to a normal curve plot. The normal distribution is plotted along an axis scaled to standard deviations, or sigma ( ${\displaystyle \sigma }$ ) units. Mathematically, the normal distribution extends to negative infinity on the left and positive infinity on the right. Note, however, that only a very small proportion of individuals in a population will fall exterior the −iiiσ to +3σ range. For example, with human heights very few people are to a higher place the +3σ height level.

Percentiles represent the surface area nether the normal curve, increasing from left to right. Each standard divergence represents a fixed percentile. Thus, rounding to two decimal places, −iiiσ is the 0.13th percentile, −twoσ the 2.28th percentile, −1σ the 15.87th percentile, 0σ the 50th percentile (both the mean and median of the distribution), +oneσ the 84.13th percentile, +twoσ the 97.72nd percentile, and +3σ the 99.87th percentile. This is related to the 68–95–99.7 dominion or the three-sigma rule. Note that in theory the 0th percentile falls at negative infinity and the 100th percentile at positive infinity, although in many practical applications, such every bit test results, natural lower and/or upper limits are enforced.

Definitions [edit]

There is no standard definition of percentile,^{[3] [4] [5]} however all definitions yield similar results when the number of observations is very large and the probability distribution is continuous.^[6] In the limit, as the sample size approaches infinity, the 100p ^thursday percentile (0<p<ane) approximates the inverse of the cumulative distribution function (CDF) thus formed, evaluated at p, equally p approximates the CDF. This tin be seen equally a issue of the Glivenko–Cantelli theorem. Some methods for calculating the percentiles are given below.

Calculation methods [edit]

Interpolated and nearest-rank, exclusive and inclusive, percentiles for x-score distribution.

At that place are many formulas or algorithms^[vii] for a percentile score. Hyndman and Fan ^[iii] identified nine and most statistical and spreadsheet software employ one of the methods they draw.^[8] Algorithms either render the value of a score that exists in the fix of scores (nearest-rank methods) or interpolate betwixt existing scores and are either exclusive or inclusive.

Nearest-rank methods (exclusive/inclusive)

PC: percentile specified	0.ten	0.25	0.fifty	0.75	0.90
N: Number of scores	x	10	10	10	10
OR: ordinal rank = PC × N	1	two.5	v	7.five	nine
Rank: >OR / ≥OR	two/1	iii/three	half dozen/5	eight/8	10/ix
Score at rank (exc/inc)	ii/i	iii/iii	4/3	5/five	7/five

The figure shows a 10-score distribution, illustrates the percentile scores that event from these different algorithms, and serves as an introduction to the examples given subsequently. The simplest are nearest-rank methods that return a score from the distribution, although compared to interpolation methods, results can exist a bit crude. The Nearest-Rank Methods table shows the computational steps for exclusive and inclusive methods.

Interpolated methods (sectional/inclusive)

PC: percentile specified	0.10	0.25	0.50	0.75	0.90
N: number of scores	10	10	10	10	10
OR: PC×(Northward+1) / PC×(N−1)+1	1.1/ane.ix	2.75/iii.25	v.5/5.five	8.25/7.75	9.ix/9.1
LoRank: OR truncated	ane/1	2/3	v/5	eight/7	9/9
HIRank: OR rounded up	2/2	iii/iv	half dozen/vi	9/viii	x/10
LoScore: score at LoRank	1/1	2/3	3/3	5/4	5/5
HiScore: score at HiRank	2/two	iii/3	4/4	5/five	seven/vii
Difference: HiScore − LoScore	one/1	i/0	1/1	0/one	ii/2
Modernistic: fractional part of OR	0.ane/0.9	0.75/0.25	0.5/0.5	0.25/0.75	0.9/0.1
Interpolated score (exc/inc) = LoScore + Mod × Difference	1.ane/i.9	2.75/3	3.5/3.v	v/4.75	six.8/5.ii

Interpolation methods, every bit the name implies, tin can return a score that is between scores in the distribution. Algorithms used by statistical programs typically use interpolation methods, for case, the percentile.exc and percentile.inc functions in Microsoft Excel. The Interpolated Methods table shows the computational steps.

The nearest-rank method [edit]

The percentile values for the ordered list {15, 20, 35, 40, 50}

One definition of percentile, often given in texts, is that the P-thursday percentile ${\displaystyle (0<P\leq 100)}$ of a list of N ordered values (sorted from least to greatest) is the smallest value in the list such that no more than P percent of the data is strictly less than the value and at least P per centum of the data is less than or equal to that value. This is obtained by start calculating the ordinal rank and and then taking the value from the ordered list that corresponds to that rank. The ordinal rank n is calculated using this formula

${\displaystyle n=\left\lceil {\frac {P}{100}}\times Northward\right\rceil .}$

Note the post-obit:

• Using the nearest-rank method on lists with fewer than 100 singled-out values tin can upshot in the aforementioned value being used for more than than one percentile.
• A percentile calculated using the nearest-rank method volition ever be a fellow member of the original ordered listing.
• The 100th percentile is defined to be the largest value in the ordered list.

Worked examples of the nearest-rank method [edit]

Example i

Consider the ordered listing {15, twenty, 35, 40, 50}, which contains five data values. What are the 5th, 30th, 40th, 50th and 100th percentiles of this list using the nearest-rank method?

Percentile P	Number in list N	Ordinal rank due north	Number from the ordered list that has that rank	Percentile value	Notes
5th	5	${\displaystyle \left\lceil {\frac {5}{100}}\times 5\right\rceil =\lceil 0.25\rceil =1}$	the first number in the ordered listing, which is 15	15	15 is the smallest element of the list; 0% of the information is strictly less than 15, and xx% of the data is less than or equal to 15.
30th	5	${\displaystyle \left\lceil {\frac {xxx}{100}}\times five\correct\rceil =\lceil 1.5\rceil =2}$	the 2nd number in the ordered list, which is twenty	20	20 is an element of the ordered list.
40th	5	${\displaystyle \left\lceil {\frac {40}{100}}\times 5\correct\rceil =\lceil 2.0\rceil =ii}$	the 2d number in the ordered list, which is xx	20	In this example, information technology is the aforementioned equally the 30th percentile.
50th	5	${\displaystyle \left\lceil {\frac {50}{100}}\times five\right\rceil =\lceil ii.5\rceil =3}$	the 3rd number in the ordered list, which is 35	35	35 is an chemical element of the ordered list.
100th	five	${\displaystyle \left\lceil {\frac {100}{100}}\times 5\correct\rceil =\lceil five\rceil =5}$	the final number in the ordered list, which is 50	l	The 100th percentile is defined to be the largest value in the list, which is l.

So the 5th, 30th, 40th, 50th and 100th percentiles of the ordered list {xv, 20, 35, 40, 50} using the nearest-rank method are {xv, 20, twenty, 35, fifty}.

Example 2

Consider an ordered population of 10 data values {3, 6, 7, 8, 8, ten, 13, fifteen, 16, 20}. What are the 25th, 50th, 75th and 100th percentiles of this list using the nearest-rank method?

Percentile P	Number in list North	Ordinal rank n	Number from the ordered list that has that rank	Percentile value	Notes
25th	x	${\displaystyle \left\lceil {\frac {25}{100}}\times 10\right\rceil =\lceil 2.5\rceil =3}$	the 3rd number in the ordered list, which is 7	7	7 is an element of the list.
50th	10	${\displaystyle \left\lceil {\frac {50}{100}}\times 10\right\rceil =\lceil v.0\rceil =5}$	the 5th number in the ordered list, which is viii	viii	8 is an element of the list.
75th	ten	${\displaystyle \left\lceil {\frac {75}{100}}\times ten\right\rceil =\lceil seven.5\rceil =viii}$	the eighth number in the ordered list, which is 15	15	15 is an chemical element of the listing.
100th	10	Final	20, which is the last number in the ordered list	20	The 100th percentile is defined to be the largest value in the list, which is 20.

Then the 25th, 50th, 75th and 100th percentiles of the ordered list {iii, 6, 7, 8, 8, 10, 13, 15, xvi, 20} using the nearest-rank method are {vii, 8, 15, 20}.

Example 3

Consider an ordered population of eleven information values {three, 6, 7, 8, viii, 9, 10, 13, 15, sixteen, 20}. What are the 25th, 50th, 75th and 100th percentiles of this list using the nearest-rank method?

Percentile P	Number in list Due north	Ordinal rank north	Number from the ordered list that has that rank	Percentile value	Notes
25th	11	${\displaystyle \left\lceil {\frac {25}{100}}\times 11\right\rceil =\lceil 2.75\rceil =3}$	the third number in the ordered list, which is 7	7	7 is an element of the list.
50th	eleven	${\displaystyle \left\lceil {\frac {50}{100}}\times 11\right\rceil =\lceil five.50\rceil =half dozen}$	the 6th number in the ordered list, which is 9	9	9 is an element of the list.
75th	11	${\displaystyle \left\lceil {\frac {75}{100}}\times 11\correct\rceil =\lceil 8.25\rceil =9}$	the 9th number in the ordered list, which is 15	xv	xv is an element of the list.
100th	11	Final	twenty, which is the last number in the ordered listing	20	The 100th percentile is divers to be the largest value in the list, which is 20.

So the 25th, 50th, 75th and 100th percentiles of the ordered list {3, half-dozen, 7, 8, 8, ix, 10, 13, 15, 16, 20} using the nearest-rank method are {seven, nine, 15, twenty}.

The linear interpolation between closest ranks method [edit]

An alternative to rounding used in many applications is to employ linear interpolation between adjacent ranks.

Commonalities between the variants of this method [edit]

All of the following variants accept the following in mutual. Given the order statistics

${\displaystyle \{v_{i},i=one,2,\ldots ,N:v_{i+1}\geq v_{i},\forall i=1,two,\ldots ,N-1\},}$

we seek a linear interpolation office that passes through the points ${\displaystyle (v_{i},i)}$ . This is simply accomplished past

${\displaystyle v(x)=v_{\lfloor x\rfloor }+(10{\bmod {1}})(v_{\lfloor x\rfloor +1}-v_{\lfloor ten\rfloor }),\forall 10\in [1,N]:v(i)=v_{i}{\text{, for }}i=ane,2,\ldots ,N,}$

where ${\displaystyle \lfloor x\rfloor }$ uses the floor office to correspond the integral role of positive x, whereas ${\displaystyle x{\bmod {1}}}$ uses the mod function to represent its fractional office (the balance later division by 1). (Note that, though at the endpoint ${\displaystyle 10=Northward}$ , ${\displaystyle v_{\lfloor x\rfloor +1}}$ is undefined, information technology does not need to exist because it is multiplied by ${\displaystyle ten{\bmod {1}}=0}$ .) As nosotros can encounter, x is the continuous version of the subscript i, linearly interpolating v between adjacent nodes.

There are ii ways in which the variant approaches differ. The first is in the linear human relationship between the rank x, the percentage rank ${\displaystyle P=100p}$ , and a constant that is a function of the sample size N:

${\displaystyle x=f(p,N)=(N+c_{1})p+c_{2}.}$

There is the additional requirement that the midpoint of the range ${\displaystyle (1,N)}$ , corresponding to the median, occur at ${\displaystyle p=0.5}$ :

${\displaystyle {\begin{aligned}f(0.five,Northward)&={\frac {Northward+c_{ane}}{2}}+c_{2}={\frac {N+ane}{2}}\\\therefore 2c_{two}+c_{ane}&=1\end{aligned}},}$

and our revised function now has just one degree of liberty, looking like this:

${\displaystyle 10=f(p,N)=(N+ane-2C)p+C.}$

The second fashion in which the variants differ is in the definition of the function near the margins of the ${\displaystyle [0,1]}$ range of p: ${\displaystyle f(p,North)}$ should produce, or be forced to produce, a issue in the range ${\displaystyle [1,North]}$ , which may mean the absence of a 1-to-one correspondence in the wider region. One author has suggested a selection of ${\displaystyle C={\tfrac {i}{two}}(i+\xi )}$ where ξ is the shape of the Generalized extreme value distribution which is the extreme value limit of the sampled distribution.

First variant, C = i/2 [edit]

The result of using each of the 3 variants on the ordered list {15, twenty, 35, 40, 50}

(Sources: Matlab "prctile" function,^{[9] [ten]})

${\displaystyle ten=f(p)={\brainstorm{cases}Np+{\frac {1}{2}},\forall p\in \left[p_{one},p_{N}\correct],\\1,\forall p\in \left[0,p_{1}\right],\\N,\forall p\in \left[p_{Northward},1\right].\finish{cases}}}$

where

${\displaystyle p_{i}={\frac {ane}{N}}\left(i-{\frac {one}{ii}}\right),i\in [i,N]\cap \mathbb {Northward} }$

${\displaystyle \therefore p_{1}={\frac {ane}{2N}},p_{Northward}={\frac {2N-1}{2N}}.}$

Furthermore, permit

${\displaystyle P_{i}=100p_{i}.}$

The inverse relationship is restricted to a narrower region:

${\displaystyle p={\frac {1}{North}}\left(ten-{\frac {i}{two}}\right),x\in (one,N)\cap \mathbb {R} .}$

Worked case of the kickoff variant [edit]

Consider the ordered listing {15, 20, 35, xl, 50}, which contains five data values. What are the fifth, 30th, 40th and 95th percentiles of this list using the Linear Interpolation Between Closest Ranks method? Starting time, we calculate the percentage rank for each listing value.

List value ${\displaystyle v_{i}}$	Position of that value in the ordered listing i	Number of values N	Calculation of percent rank	Pct rank, ${\displaystyle P_{i}}$
xv	ane	5	${\displaystyle {\frac {100}{5}}\left(1-{\frac {i}{2}}\correct)=10.}$	10
20	2	5	${\displaystyle {\frac {100}{v}}\left(two-{\frac {1}{2}}\right)=30.}$	30
35	3	five	${\displaystyle {\frac {100}{5}}\left(3-{\frac {i}{ii}}\right)=50.}$	l
xl	iv	5	${\displaystyle {\frac {100}{5}}\left(4-{\frac {1}{two}}\right)=70.}$	70
50	5	5	${\displaystyle {\frac {100}{5}}\left(5-{\frac {1}{two}}\correct)=ninety.}$	90

Then we accept those per centum ranks and calculate the percentile values as follows:

Per centum rank P	Number of values North	Is ${\displaystyle P<P_{i}}$ ?	Is ${\displaystyle P>P_{n}}$ ?	Is there a percent rank equal to P?	What practise we utilize for percentile value?	Percentile value ${\displaystyle v(f(p))}$	Notes
v	5	Yes	No	No	Nosotros encounter that ${\displaystyle P=5}$ , which is less than the starting time per centum rank ${\displaystyle p_{1}=10}$ , so use the first list value ${\displaystyle v_{1}}$ , which is fifteen	15	xv is a member of the ordered listing
30	5	No	No	Yes	Nosotros see that ${\displaystyle P=30}$ is the same as the second pct rank ${\displaystyle p_{ii}=30}$ , and so use the second listing value ${\displaystyle v_{two}}$ , which is 20	20	20 is a member of the ordered list
40	five	No	No	No	Nosotros run across that ${\displaystyle P=40}$ is between percent rank ${\displaystyle p_{ii}=xxx}$ and ${\displaystyle p_{iii}=fifty}$ , then nosotros take ${\displaystyle yard=two,k+1=3,P=40,p_{k}=p_{two}=30,v_{k}=v_{two}=20,v_{k+i}=v_{3}=35,Northward=v}$ . Given those values nosotros tin so calculate v as follows: ${\displaystyle five=20+v\times {\frac {40-30}{100}}(35-xx)=27.v}$	27.five	27.v is not a member of the ordered list
95	five	No	Yep	No	We see that ${\displaystyle P=95}$ , which is greater than the terminal percentage rank ${\displaystyle p_{Northward}=90}$ , then employ the final list value, which is 50	50	l is a member of the ordered list

And then the 5th, 30th, 40th and 95th percentiles of the ordered list {fifteen, 20, 35, 40, 50} using the Linear Interpolation Between Closest Ranks method are {xv, twenty, 27.5, 50}

Second variant, C = 1 [edit]

(Source: Some software packages, including NumPy^[11] and Microsoft Excel^[five] (up to and including version 2013 by means of the PERCENTILE.INC office). Noted as an alternative by NIST^[8])

${\displaystyle x=f(p,North)=p(N-1)+1{\text{, }}p\in [0,1]}$

${\displaystyle \therefore p={\frac {x-one}{N-one}}{\text{, }}x\in [1,N].}$

Note that the ${\displaystyle 10\leftrightarrow p}$ human relationship is one-to-one for ${\displaystyle p\in [0,1]}$ , the merely one of the three variants with this holding; hence the "INC" suffix, for inclusive, on the Excel function.

Worked examples of the second variant [edit]

Case ane [edit]

Consider the ordered list {15, twenty, 35, xl, fifty}, which contains five data values. What is the 40th percentile of this list using this variant method?

Get-go we calculate the rank of the 40th percentile:

${\displaystyle ten={\frac {xl}{100}}(5-ane)+1=2.half-dozen}$

Then, x=2.six, which gives usa ${\displaystyle \lfloor x\rfloor =ii}$ and ${\displaystyle x{\bmod {one}}=0.6}$ . So, the value of the 40th percentile is

${\displaystyle 5(2.6)=v_{two}+0.6(v_{three}-v_{ii})=20+0.six(35-20)=29.}$

Example two [edit]

Consider the ordered listing {1,2,3,4} which contains four data values. What is the 75th percentile of this list using the Microsoft Excel method?

First we summate the rank of the 75th percentile equally follows:

${\displaystyle x={\frac {75}{100}}(4-ane)+i=three.25}$

Then, ten=3.25, which gives us an integral part of 3 and a fractional role of 0.25. So, the value of the 75th percentile is

${\displaystyle v(3.25)=v_{three}+0.25(v_{iv}-v_{iii})=3+0.25(4-3)=three.25.}$

Third variant, C = 0 [edit]

(The primary variant recommended by NIST.^[8] Adopted by Microsoft Excel since 2010 by ways of PERCENTIL.EXC function. Nevertheless, as the "EXC" suffix indicates, the Excel version excludes both endpoints of the range of p, i.eastward., ${\displaystyle p\in (0,1)}$ , whereas the "INC" version, the 2nd variant, does not; in fact, any number smaller than ${\displaystyle {\frac {1}{N+1}}}$ is besides excluded and would cause an error.)

${\displaystyle x=f(p,Due north)={\begin{cases}i{\text{, }}p\in \left[0,{\frac {i}{N+ane}}\correct]\\p(North+ane){\text{, }}p\in \left({\frac {1}{N+one}},{\frac {North}{North+1}}\correct)\\Northward{\text{, }}p\in \left[{\frac {N}{North+1}},1\right]\end{cases}}.}$

The inverse is restricted to a narrower region:

${\displaystyle p={\frac {x}{N+ane}}{\text{, }}x\in (0,Northward).}$

Worked example of the third variant [edit]

Consider the ordered list {15, 20, 35, 40, fifty}, which contains five data values. What is the 40th percentile of this list using the NIST method?

First we summate the rank of the 40th percentile equally follows:

${\displaystyle x={\frac {twoscore}{100}}(v+1)=2.4}$

So x=2.iv, which gives us ${\displaystyle \lfloor x\rfloor =2}$ and ${\displaystyle x{\bmod {1}}=0.4}$ . Then the value of the 40th percentile is calculated as:

${\displaystyle v(two.4)=v_{2}+0.4(v_{iii}-v_{2})=20+0.4(35-20)=26}$

And so the value of the 40th percentile of the ordered listing {15, twenty, 35, 40, fifty} using this variant method is 26.

The weighted percentile method [edit]

In addition to the percentile function, there is also a weighted percentile, where the per centum in the total weight is counted instead of the total number. There is no standard function for a weighted percentile. One method extends the in a higher place approach in a natural way.

Suppose nosotros have positive weights ${\displaystyle w_{1},w_{2},w_{3},\dots ,w_{N}}$ associated, respectively, with our N sorted sample values. Let

${\displaystyle S_{Due north}=\sum _{k=1}^{Due north}w_{k},}$

the sum of the weights. Then the formulas higher up are generalized by taking

${\displaystyle p_{n}={\frac {1}{S_{North}}}\left(S_{n}-{\frac {w_{n}}{2}}\right)}$ when ${\displaystyle C=1/2}$ ,

or

${\displaystyle p_{n}={\frac {S_{n}-Cw_{n}}{S_{N}+(i-2C)w_{northward}}}}$ for general ${\displaystyle C}$ ,

and

${\displaystyle v=v_{thousand}+{\frac {P-p_{k}}{p_{k+1}-p_{k}}}(v_{chiliad+i}-v_{m}).}$

The fifty% weighted percentile is known as the weighted median.

See besides [edit]

• Quantile
• Decile
• Summary statistics
• Percentile rank

References [edit]

1. ^ Johnson, Robert; Kuby, Patricia (2007), "Applied Instance ii.15, The 85th Percentile Speed Limit: Going With 85% of the Flow", Elementary Statistics (tenth ed.), Cengage Learning, p. 102, ISBN9781111802493 .
2. ^ "Rational Speed Limits and the 85th Percentile Speed" (PDF). lsp.org. Louisiana Country Police. Archived from the original (PDF) on 23 September 2018. Retrieved 28 October 2018.
3. ^ ^{a b} Hyndman, Rob J.; Fan, Yanan (November 1996). "Sample Quantiles in Statistical Packages". American Statistician. American Statistical Association. fifty (four): 361–365. doi:ten.2307/2684934. JSTOR 2684934.
4. ^ Lane, David. "Percentiles". Retrieved 2007-09-15 .
5. ^ ^{a b} Pottel, Hans. "Statistical flaws in Excel" (PDF). Archived from the original (PDF) on 2013-06-04. Retrieved 2013-03-25 .
6. ^ Schoonjans F, De Bacquer D, Schmid P (2011). "Interpretation of population percentiles". Epidemiology. 22 (5): 750–751. doi:10.1097/EDE.0b013e318225c1de. PMC3171208. PMID 21811118.
7. ^ Wessa, P (2021). "Percentiles in Free Statistics Software". Office for Inquiry Development and Education. Retrieved 13 November 2021.
8. ^ ^{a b c} "Engineering Statistics Handbook: Percentile". NIST. Retrieved 2009-02-18 .
9. ^ "Matlab Statistics Toolbox – Percentiles". Retrieved 2006-09-fifteen . , This is equivalent to Method 5 discussed hither
10. ^ Langford, E. (2006). "Quartiles in Uncomplicated Statistics". Journal of Statistics Education. 14 (3). doi:x.1080/10691898.2006.11910589.
11. ^ "NumPy 1.12 documentation". SciPy. Retrieved 2017-03-19 .

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Source: https://en.wikipedia.org/wiki/Percentile

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