What is a percentile and how is it different from a percentage?

A percentage is a ratio out of 100 applied to a single number (e.g. 75% of 200 = 150). A percentile is a positional measure in a dataset: the Nth percentile is the value below which N% of the data falls. Scoring 85% on a test means you answered 85% of questions correctly. Being at the 85th percentile means you scored higher than 85% of test-takers, regardless of the raw score.

How is the value at a percentile calculated?

The linear interpolation (inclusive) method sets index = (P/100) x (n-1) on the sorted array. If the index is an integer, that element is the answer. If not, the answer is interpolated between the floor and ceiling elements. For example, the 75th percentile of [1,2,3,4] gives index = 0.75 x 3 = 2.25, so value = 3 + 0.25 x (4-3) = 3.25. The nearest rank method instead takes the element at position ceil(P/100 x n).

What is the difference between inclusive and exclusive percentile methods?

The inclusive method (PERCENTILE.INC, used by Excel by default) can return the minimum at P=0 and the maximum at P=100. The exclusive method (PERCENTILE.EXC) excludes the endpoints and is undefined at P=0 and P=100. For most educational and practical purposes, the inclusive method is preferred. This calculator uses the inclusive interpolation method as the primary result.

How do I interpret a percentile rank?

A percentile rank of 70 means the value is higher than 70% of the data points. It does not mean the value scored 70% on anything. In standardised testing, a score at the 99th percentile is extremely high even if the raw score was, say, 75 out of 100, because 99% of other test-takers scored lower. Percentile ranks are always between 0 and 100 and describe relative position, not absolute performance.

What is the five-number summary shown by this calculator?

The five-number summary consists of: minimum (0th percentile), Q1 (25th percentile), median (50th percentile), Q3 (75th percentile), and maximum (100th percentile). Together with the IQR, these six values fully describe the spread and centre of a dataset without being sensitive to outliers. They are the basis for box-and-whisker plots used in exploratory data analysis.

Can a value be at the 0th or 100th percentile?

By convention, the minimum value in a dataset is at the 0th percentile and the maximum is at the 100th percentile when using the inclusive method. However, many textbook definitions exclude the endpoints (since no value is below the minimum, it can be considered at 0%, and no value is above the maximum). This calculator uses the midpoint formula for percentile rank, so the minimum gets PR = 0.5/n x 100 and the maximum gets PR = (n - 0.5)/n x 100.

How are percentiles used in child growth charts?

The CDC and WHO publish height, weight, and head circumference percentile charts for children aged 0-20. A child at the 50th percentile is average for their age and sex. A child at the 95th percentile is taller or heavier than 95% of children their age and sex. These charts use smoothed reference data from large population studies, not simple rank-based percentiles, but the interpretation is the same: relative position within a reference population.

What is the interquartile range (IQR) used for?

The IQR measures the spread of the middle 50% of data and is a robust alternative to range and standard deviation when outliers are present. It is used for outlier detection via Tukey fences: values below Q1 - 1.5*IQR or above Q3 + 1.5*IQR are flagged as outliers. The IQR is also used in box plots to set the whisker length. Unlike standard deviation, the IQR is not affected by extreme values.

Percentile Calculator

Q: What formula does this calculator use for percentile rank?

This calculator uses the midpoint formula: PR = ((B + 0.5 x E) / n) x 100, where B is the count of values strictly below the target, E is the count of values equal to the target, and n is the total sample size. This method places the target at the midpoint of its group and is recommended by most statistics textbooks because it handles ties gracefully.

Q: What are quartiles and how do they relate to percentiles?

Quartiles are special percentile positions that divide sorted data into four equal parts. Q1 is the 25th percentile (lower quartile), Q2 is the 50th percentile (median), and Q3 is the 75th percentile (upper quartile). The IQR (interquartile range) is Q3 minus Q1 and represents the spread of the middle 50% of values. Box plots use the five-number summary: min, Q1, Q2, Q3, max.

Find the percentile rank of any value, or find the value at any percentile, for any dataset you enter.

📊 What is a Percentile Calculator?

A percentile calculator is a tool that finds where a value stands relative to the rest of a dataset. It answers two distinct questions: "What percentile rank does this value have?" and "What value falls at the Nth percentile?" Both are fundamental operations in descriptive statistics, standardised testing, medical diagnostics, and data analysis.

Percentiles are used everywhere in everyday life. Standardised tests like the SAT, GRE, and USMLE report scores as percentile ranks so candidates can understand their standing relative to other test-takers, not just as raw numbers. Pediatric growth charts show height and weight as percentiles relative to a reference population of the same age and sex. Income distribution reports describe the 50th, 90th, and 99th income percentiles to characterise inequality. Quality control engineers use percentiles to set specification limits based on the distribution of measured dimensions.

A common source of confusion is the difference between a percentage and a percentile. A percentage is a ratio: 75% means 75 out of 100 units. A percentile is a rank: being at the 75th percentile means a value is higher than 75% of all values in a dataset. You could score 40% on a very difficult exam and still be at the 95th percentile if everyone else scored lower. The two concepts measure completely different things.

This calculator supports two modes. The Find Percentile Rank mode accepts a dataset and a specific value and returns the percentile rank using the midpoint formula, which handles ties fairly. The Find Value at Percentile mode accepts a dataset and a target percentile and returns the value at that position using both linear interpolation (the Excel PERCENTILE.INC method) and the nearest rank method. Both modes also display the full five-number summary (minimum, Q1, median, Q3, maximum, and IQR) and a sorted list with the relevant value highlighted, so you can visually verify the result.

📐 Formula

Percentile Rank = ((B + 0.5 × E) ÷ n) × 100

B = number of values strictly below the target value

E = number of values equal to the target value

n = total count of values in the dataset

Example: In [45, 52, 58, 63, 70, 75, 81, 88, 92, 97], for value = 70: B = 4, E = 1, n = 10. PR = ((4 + 0.5) / 10) × 100 = 45%

Value at Percentile = x⌊idx⌋ + frac × (x⌈idx⌉ − x⌊idx⌋)

idx = (P ÷ 100) × (n − 1) on the sorted array (0-indexed)

frac = fractional part of idx (idx − floor(idx))

x⌊idx⌋ = sorted array element at the floor index

x⌈idx⌉ = sorted array element at the ceiling index

Example: For P = 75 in [10, 20, 30, 40, 50, 60, 70, 80, 90, 100] (n=10): idx = 0.75 × 9 = 6.75, value = 70 + 0.75 × (80 − 70) = 77.5

📖 How to Use This Calculator

Steps

Choose your calculation mode - Select Find Percentile Rank to find where a specific value sits in your dataset, or Find Value at Percentile to find which value corresponds to a given percentile like the 75th.

Enter your dataset - Type or paste your numbers separated by commas or spaces into the dataset field. At least 2 values are required. The calculator accepts integers and decimals.

Enter the value or percentile - In Rank mode, type the value whose rank you want to find. In Value mode, type or drag the slider to set the percentile (0 to 100) you want to look up.

Click Calculate - Click Calculate to see the result, a sorted list with the relevant value highlighted, and the five-number summary table (min, Q1, median, Q3, max, IQR).

💡 Example Calculations

Example 1 - Test Score Percentile Rank

Class scores: 45, 52, 58, 63, 70, 75, 81, 88, 92, 97. Where does a score of 70 rank?

Sort the dataset: [45, 52, 58, 63, 70, 75, 81, 88, 92, 97]. Total n = 10.

Count values below 70: B = 4 (scores 45, 52, 58, 63). Count equal to 70: E = 1.

Apply formula: PR = ((4 + 0.5 × 1) / 10) × 100 = (4.5 / 10) × 100 = 45%

Percentile Rank = 45th percentile

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Example 2 - Height Percentile Rank

Heights in cm: 158, 162, 165, 167, 170, 172, 175, 178, 182, 190. What percentile is 172 cm?

Sorted: [158, 162, 165, 167, 170, 172, 175, 178, 182, 190]. n = 10.

Values below 172: B = 5 (158, 162, 165, 167, 170). Values equal: E = 1.

PR = ((5 + 0.5) / 10) × 100 = 55%. A height of 172 cm is at the 55th percentile in this group.

Percentile Rank = 55th percentile

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Example 3 - Find Value at 75th Percentile

Dataset: 10, 20, 30, 40, 50, 60, 70, 80, 90, 100. What is the 75th percentile value?

Sorted array (0-indexed): [10, 20, 30, 40, 50, 60, 70, 80, 90, 100]. n = 10.

Linear interpolation: idx = (75/100) × (10-1) = 6.75. Floor index = 6 (value = 70), ceiling index = 7 (value = 80).

Interpolate: 70 + 0.75 × (80 - 70) = 70 + 7.5 = 77.5. Nearest rank: ceil(0.75 × 10) = 8, sorted[7] = 80.

75th Percentile = 77.5 (interpolation) or 80 (nearest rank)

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Example 4 - Salary Distribution Q1 and Q3

Annual salaries (thousands): 32, 38, 42, 45, 48, 55, 62, 70, 85, 120. Find the 25th and 75th percentiles.

For Q1 (25th percentile): idx = 0.25 × 9 = 2.25. Values at index 2 and 3 are 42 and 45. Q1 = 42 + 0.25 × (45-42) = 42.75.

For Q3 (75th percentile): idx = 0.75 × 9 = 6.75. Values at index 6 and 7 are 62 and 70. Q3 = 62 + 0.75 × (70-62) = 68.

IQR = Q3 - Q1 = 68 - 42.75 = 25.25 thousand. Outlier fences: below 42.75 - 1.5×25.25 = 4.875 or above 68 + 1.5×25.25 = 105.875. The salary of 120 exceeds the upper fence and is flagged as an outlier.

Q1 = $42,750, Q3 = $68,000, IQR = $25,250

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❓ Frequently Asked Questions

What is the difference between a percentile and a percentage?+

A percentage is a fraction expressed out of 100 (e.g. 75% = 75/100). A percentile is a position in a sorted dataset: the 75th percentile is the value below which 75% of the data falls. You can score 40% on a test and still be at the 95th percentile if most other students scored below 40%. The two concepts are unrelated except in their use of the number 100 as a base.

What formula does this calculator use for percentile rank?+

This calculator uses the midpoint percentile rank formula: PR = ((B + 0.5 × E) / n) × 100, where B is the count of values strictly below the target, E is the count of values equal to the target, and n is the total sample size. This is the most widely taught method in statistics courses and handles tied values by placing them at the midpoint of their rank range.

What is the difference between linear interpolation and nearest rank methods?+

The linear interpolation (inclusive) method computes an index as (P/100) × (n-1) and interpolates between adjacent sorted values when the index is not an integer. This matches Excel's PERCENTILE.INC function and the NumPy default (method='linear'). The nearest rank method rounds up and returns the element at ceil(P/100 × n). For large datasets the two methods give very similar results; for small datasets they can differ noticeably. Both are shown so you can match whichever your textbook uses.

What are quartiles and how do they relate to percentiles?+

Quartiles are the three values that divide a sorted dataset into four equal parts: Q1 is the 25th percentile, Q2 (median) is the 50th percentile, and Q3 is the 75th percentile. They are special cases of percentiles that are used so frequently they have their own names. The IQR (interquartile range) = Q3 - Q1 measures the spread of the middle 50% of the data and is displayed in the five-number summary table this calculator generates.

How do I find the median using this calculator?+

The median is the 50th percentile. Switch to the Find Value at Percentile mode, enter your dataset, set the percentile to 50, and click Calculate. The result labeled "Value (Linear Interpolation)" is the median. Alternatively, every calculation in both modes displays the five-number summary table at the bottom of the results, which includes the median (labeled as "50th (Median / Q2)").

What is the IQR used for and how is it calculated?+

The interquartile range (IQR) = Q3 - Q1 measures the spread of the middle 50% of data. It is used for outlier detection using Tukey fences: values below Q1 - 1.5 × IQR or above Q3 + 1.5 × IQR are flagged as potential outliers. The IQR is also used to set whisker lengths in box plots. Unlike range and standard deviation, the IQR is resistant to extreme values because it ignores the top and bottom 25% of the data.

Can percentile rank be above 100 or below 0?+

No. Percentile rank is always between 0 and 100 by definition. The midpoint formula this calculator uses assigns the minimum value a percentile rank of (0.5/n) × 100 and the maximum a rank of ((n-0.5)/n) × 100. A value exactly at the minimum gets a small positive percentile rank rather than 0 because the formula accounts for the fact that half its "group" (itself) is at or below it. For practical purposes, the minimum is at the 0th percentile and the maximum is at the 100th.

How are percentiles used in standardised testing?+

In standardised tests like the SAT, GRE, and GMAT, raw scores are converted to percentile ranks based on the performance of a large reference group (the norming sample). A score at the 90th percentile means the candidate outperformed 90% of the reference group. Score reports include both the raw or scaled score and the percentile rank because the percentile communicates relative standing more clearly than a raw number alone. Percentile ranks for standardised tests use large norming samples and sophisticated equating, not simple rank-based calculations.

Why does my percentile rank differ from what Excel gives?+

Excel's PERCENTRANK function uses the formula (rank - 1) / (n - 1), which gives 0 to the minimum and 1 (100%) to the maximum and interpolates between. The midpoint formula this calculator uses gives 0.5/n to the minimum. For large datasets the two agree closely. For small datasets or when you are looking up the minimum or maximum values, they differ. If you need to match Excel's PERCENTRANK, use their formula; if you need the textbook midpoint formula, use this calculator.

What is a decile and how is it related to percentiles?+

Deciles divide sorted data into ten equal parts. D1 is the 10th percentile, D2 is the 20th percentile, and so on up to D9 at the 90th percentile. Deciles are used in income and wealth distribution analysis: the top decile (D9 to max) and bottom decile (min to D1) describe the extremes of a distribution. To find any decile using this calculator, switch to Find Value at Percentile mode and enter the corresponding percentile (10, 20, 30, etc.).

🔗 Related Calculators

📌 Quick Tips

💡Percentile rank tells you what percentage of values fall at or below a given data point. A score at the 80th percentile is higher than 80% of all scores in the dataset.

💡The two most common methods for finding a value at a given percentile are linear interpolation (used by Excel PERCENTILE.INC) and nearest rank. This calculator shows both so you can match whichever your textbook uses.

💡Q1 is the 25th percentile, Q2 (median) is the 50th percentile, and Q3 is the 75th percentile. These three values, together with min and max, form the five-number summary used in box plots.

💡The IQR (interquartile range) equals Q3 minus Q1 and represents the middle 50% of your data. Values more than 1.5 times the IQR below Q1 or above Q3 are often flagged as outliers (Tukey fences).

💡Percentiles are used in standardised testing (SAT, GRE, USMLE), growth charts for children, income distributions, and quality control processes. A baby at the 90th percentile for height is taller than 90% of babies of the same age and sex.