percentile

百分位数 　 　

WordNet

1. (statistics) any of the 99 numbered points that divide an ordered set of scores into 100 parts each of which contains one-hundredth of the total (同)centile

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1. (統計で)百分位数 / 百分位数の

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出典(authority):フリー百科事典『ウィキペディア（Wikipedia）』「2014/01/30 11:42:59」(JST)

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A percentile (or a centile) is a measure used in statistics indicating the value below which a given percentage of observations in a group of observations fall. For example, the 20th percentile is the value (or score) below which 20 percent of the observations may be found. The term percentile and the related term percentile rank are often used in the reporting of scores from norm-referenced tests. For example, if a score is in the 86th percentile, it is higher than 86% of the other scores.

The 25th percentile is also known as the first quartile (Q₁), the 50th percentile as the median or second quartile (Q₂), and the 75th percentile as the third quartile (Q₃). In general, percentiles and quartiles are specific types of quantiles.

Applications

When ISPs bill "burstable" internet bandwidth, the 95th or 98th percentile usually cuts off the top 5% or 2% of bandwidth peaks in each month, and then 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 data throughput is that it gives a very accurate picture of the cost of the bandwidth. The 95th percentile says that 95% of the time, the usage is below this amount. Just the same, the remaining 5% of the time, the usage is above that amount.

Physicians will often use infant and children's 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 often used as a guideline in setting speed limits and assessing whether such a limits is too high or low.^{[citation needed]}

The normal distribution and percentiles

Representation of the 68–95–99.7 rule. The dark blue zone represents observations within one standard deviation (σ) to either side of the mean (μ), which accounts for about 68.2% of the population. Two standard deviations from the mean (dark and medium blue) account for about 95.4%, and three standard deviations (dark, medium, and light blue) for about 99.7%.

The methods given in the Definitions section are approximations for use in small-sample statistics. In general terms, for very large populations following a normal distribution percentiles may often be represented by reference to a normal curve plot. The normal distribution is plotted along an axis scaled to standard deviations, or 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 outside the −3 to +3 range. For example, with human heights very few people are above the +3 sigma height level.

Percentiles represent the area under the normal curve, increasing from left to right. Each standard deviation represents a fixed percentile. Thus, rounding to two decimal places, −3 is the 0.13th percentile, −2 the 2.28th percentile, −1 the 15.87th percentile, 0 the 50th percentile (both the mean and median of the distribution), +1 the 84.13th percentile, +2 the 97.72nd percentile, and +3 the 99.87th percentile. This is known as the 68–95–99.7 rule 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 as test results, natural lower and/or upper limits are enforced.

Definition

There is no standard definition of percentile, however all definitions yield similar results when the number of observations is very large.

Nearest rank

One definition of percentile, often given in texts, is that the P-th percentile () of N ordered values (arranged from least to greatest) is obtained by first calculating the (ordinal) rank

rounding the result to the nearest integer, and then taking the value that corresponds to that rank. (Note that the rounded value of n is just the least integer which exceeds .)

For example, by this definition, given the numbers

15, 20, 35, 40, 50

the rank of the 30th percentile would be

Thus the 30th percentile is the second number in the sorted list, 20.

The 35th percentile would have rank

so the 35th percentile would be the second number again (since 2.25 rounds down to 2) or 20

The 40th percentile would have rank

so the 40th percentile would be the third number (since 2.5 rounds up to 3), or 35.

The 100th percentile is defined to be the largest value. (In this case we do not use the above definition with P=100, because the rank n would be greater than the number N of values in the original list.)

In lists with fewer than 100 values the same number can occupy more than one percentile group.

Linear interpolation between closest ranks

An alternative to rounding used in many applications is to use linear interpolation between the two nearest ranks.

In particular, given the N sorted values , we define the percent rank corresponding to the n^th value as:

In this way, for example, if then the percent rank corresponding to the third value is:

The value v of the P-th percentile may now be calculated as follows:

If or , then we take or , respectively.

If there is some integer k for which , then we take .

Otherwise, we find the integer k for which , and take

Using the list of numbers above, the 40th percentile would be found by linearly interpolating between the 30th percentile, 20, and the 50th, 35. Specifically:

This is halfway between 20 and 35, which one would expect since the rank was calculated above as 2.5.

It is readily confirmed that the 50th percentile of any list of values according to this definition of the P-th percentile is just the sample median.

Moreover, when N is even the 25th percentile according to this definition of the P-th percentile is the median of the first values (i.e., the median of the lower half of the data).

Weighted percentile

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

Suppose we have positive weights associated, respectively, with our N sorted sample values. Let

the -th partial sum of the weights. Then the formulas above are generalized by taking

and

The 50% weighted percentile is known as the weighted median.

Alternative methods

Some software packages, including Microsoft Excel (up to and including version 2010, 2013 unverified) use the following method, noted as an alternative by NIST to estimate the value, , of the P-th percentile of an ascending ordered dataset containing N elements with values .

The rank is calculated:

and then split into its integer component k and decimal component d, such that . Then is calculated as:

$v_P = \begin{cases} v_1, & \text{for }k=0 \\ v_N, & \text{for }k=N \\ v_k+d(v_{k+1}-v_k), & \text{for }0 < k < N \end{cases}$

The primary method recommended by NIST is similar to that given above, but with the rank calculated as

These two approaches give the rank of the 40th percentile in the above example as, respectively:

and

The values are then interpolated as usual based on these ranks, yielding 29 and 26, respectively, for the 40th percentile.

Excel's Algorithm:

Let N be the number of values . P be the percentile where 0<P<1.

Then, (N-1)P=k+d , where k=integer part and d=decimal part (ex: 4.5 = 4+ 0.5)

Then, P th percentile is : v_{k+1} + d (v_{k+2} - v_{k+1})

Example : For the array 1,2,3,4 N=4 and P=.75(say) Then, (N-1)P = (4-1)*0.75 = 2.25 = 2+0.25

P th percentile = v_{2+1} + 0.25 (v_{2+2} - v_{2+1})

              = 3 + 0.25 (4-3)
              = 3.25

See also

• Quantile
• Decile
• Summary statistics
• Percentile rank

References

External links

• Free Online Software (Calculator) computes Percentiles for any dataset according to 8 different percentile definitions.

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English Journal

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• Soer R1, Hollak N2, Deijs M3, van der Woude LH4, Reneman MF3.Author information 1University of Groningen, University Medical Center Groningen, Center for Rehabilitation, The Netherlands; University of Groningen, University Medical Center Groningen, Groningen Spine Center, The Netherlands; Expertise Center of Health, Social Care and Technology, Saxion Universities of Applied Sciences, Enschede, The Netherlands. Electronic address: r.soer@saxion.nl.2Center for Human Movement Sciences, University of Groningen, Groningen, The Netherlands; Xsens Technologies B.V., Pantheon 6a, 7521 PR Enschede, The Netherlands.3University of Groningen, University Medical Center Groningen, Center for Rehabilitation, The Netherlands.4University of Groningen, University Medical Center Groningen, Center for Rehabilitation, The Netherlands; Center for Human Movement Sciences, University of Groningen, Groningen, The Netherlands.AbstractOBJECTIVES: To determine if functional capacity (FC) and physical work demands can be matched and to determine the validity of normative values for FC related to physical work demands as a screening instrument for work ability.
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• Treatment of familial hypercholesterolaemia in children and adolescents in the last three decades.

• Elis A1, Zhou R2, Stein EA3.Author information 11 Department of Medicine, Rabin Medical Center, Beilinson Campus, Petah-Tikva and Sackler School of Medicine, Tel Aviv University, Tel Aviv, Israel.22 Medpace Inc., Cincinnati, Ohio, United States of America.33 Metabolic & Atherosclerosis Research Center and Cholesterol Treatment Center, Cincinnati, Ohio, United States of America.AbstractBackground: This study evaluated the effectiveness of long-term intensive lipid-lowering therapy in children and adolescents with familial hypercholesterolaemia. Methods: The charts of 89 children and adolescents with heterozygous familial hypercholesterolaemia among ∼1000 patients treated from 1974 to 2008 were reviewed. Familial hypercholesterolaemia was defined as low-density lipoprotein cholesterol level >90th percentile in individuals with a history of familial hypercholesterolaemia. Results: Of the 89 patients, 51% were male; the mean age at diagnosis was 8 ± 4 years, and the mean follow-up was 13 ± 8 years. Baseline and most recent low-density lipoprotein cholesterol levels (mg/dl) under treatment were 250 ± 50 and 142 ± 49, respectively, reduced 43% from baseline (p < 0.0001). At the most recent visit, 39 patients received statin monotherapy, mainly atorvastatin or rosuvastatin, and 50 (56%) patients received combination therapy, mainly vytorin or rosuvastain/ezetimibe, 15 patients were >30 years of age, and none developed symptomatic cardiovascular disease or needed revascularisation. Conclusions: Long-term statin-based therapy can reduce low-density lipoprotein cholesterol levels in most children and adolescents with heterozygous familial hypercholesterolaemia and decrease cardiovascular risk significantly.
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• Correlation between the serum alkaline phosphatase level and the severity of coronary artery disease.

• Sahin I1, Karabulut A, Gungor B, Avci II, Okuyan E, Kizkapan F, Yildiz SS, Can MM, Dinckal M.Author information 1aDepartment of Cardiology, Bagcilar Public Education and Research Hospital bDepartment of Cardiology, Istanbul Medicine Hospital cDepartment of Cardiology, Siyami Ersek Cardiovascular and Thoracic Surgery Center, Istanbul, Turkey.AbstractOBJECTIVES: The serum alkaline phosphatase (ALP) level has shown to be a prognostic factor in myocardial infarction and peripheral vascular disease by its promoting effect on vascular calcification. A few recent studies also showed that elevated ALP levels were associated with mortality and unfavorable prognosis in coronary artery disease (CAD). Herein, we aimed to investigate the correlation between serum ALP levels and the severity of CAD by assessing the Gensini score.
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percentile

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パーセンタイル、パーセンタイル値

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• パーセンタイルとは、対象とする数値群を小さい順にソートし、指定された個数番目にある値を代表値とするもの

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percentile

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百分位数