Section 6.10: Percentiles
- Page ID
- 216510
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)- Find a data value corresponding to a given percentile
- Compute the percentile rank for a data value
- Use percentile rank to compare values from different data sets
What is Measures of Position?
Measures of position are statistical values that tell you where a specific data point stands relative to the rest of the dataset. Unlike measures of center (mean, median, mode) that describe the "typical" value, measures of position describe the location or rank of values within a distribution.
Measures of position transform absolute values into relative comparisons, making it easier to:
- ✓ Understand where you stand
- ✓ Compare across different scales
- ✓ Identify outliers and unusual values
- ✓ Make informed decisions based on population distributions
In this section along with the next section, we will focus primarily on percentiles and quartiles, as they are the most intuitive and widely used measures of position in statistics.
Imagine you just took the SAT and received a score of 1360 out of 1600. The score of 1360 is called a raw score. The raw percentage is 85% (1360/1600). Your first questions might be: "Is this a good score?" "How do I compare to other students?" "Did I do better than most people, or worse?" It doesn't immediately answer these questions. We need context.
Now suppose you learn that your score is at the 90th percentile. What does this tell you? The 90th percentile means that you scored better than 90% of all test-takers. It also means that only 10% of students scored higher than you. Suddenly, your score has meaning. The percentile gives you a clear picture of where you stand relative to everyone else. Percentiles transform raw data into meaningful comparisons.
A percentile is a value that indicates the percentage of data that falls at or below that value.
If your score is in the P th percentile, you scored better than (or equal to) P% of the entire group.
The P th percentile is a value such that at least P% of the data values are less than or equal to that value, and at least (100 - P)% are greater than or equal to that value.
Key components:
- P is a number between 0 and 100
- The percentile is a position in the ordered dataset
- It divides the data into two groups: those below and those above
To find the position of a data set of sample size \(n\) given a percentile:
- Step 1 — Arrange the data in ascending order. (This isn't strictly necessary, but it makes the process easier and ensures your final plot is organized).
- Step 2 — Calculate the position by using the following formula:
\[\text{Position}=\frac{\text{Percentile }}{100}\cdot n \nonumber \]
- Step 3 — The result should be rounded to the nearest whole number.
- Step 4 — Locate the value.
To find the percentile of a data set of sample size \(n\) given a position:
- Step 1 — Arrange the data in ascending order. (This isn't strictly necessary, but it makes the process easier and ensures your final plot is organized).
- Step 2 — Calculate the percentile by using the following formula:
\[\text{Percent of a value }x=\frac{\text{Number of data values less than }x}{n}\cdot100 \nonumber \]
- Step 3 — The result should be rounded to the nearest whole number.
- Step 4 — Locate the value.
Note: There are multiple valid methods for calculating percentiles, and different textbooks, calculators, and software packages may use different formulas or rules (such as rounding, interpolation, or specific position formulas). As a result, it is possible to obtain slightly different percentile values for the same data set depending on the method used. For this course, we are following the methods demonstrated above.
Percentiles are essential for:
- Relative comparison: Understanding where a value stands compared to others
- Standardized interpretation: Comparing across different scales or distributions
- Outlier identification: Spotting unusual values
- Fair assessment: Evaluating performance considering the entire distribution
- Decision-making: Setting thresholds based on population characteristics
Percentiles are used in real-world applications, such as:
- Education: SAT/ACT scores, GPA rankings, class standings
- Medicine: Growth charts (height, weight for children), blood pressure ranges
- Finance: Income brackets, wealth distribution
- Business: Salary negotiations, performance reviews
- Quality control: Manufacturing tolerances, product specifications
Note: Interpolate, a few other methods exist.
The weights (in pounds) of 28 newborn babies are given below:
5.6, 5.7, 5.9, 6.0, 6.2, 6.3, 6.5, 6.7, 6.7, 6.9, 6.9, 6.9, 7.2, 7.2, 7.3, 7.4, 7.4, 7.5, 7.6, 7.7, 7.7, 7.7, 7.8, 8.0, 8.1, 8.3, 8.4, 8.7
What score is at the 80th percentile?
✅ Solution:
Step 1: Arrange the data in ascending order.
- Data was given in ascending order
Step 2: Calculate the position.
- Position = (Percentile/100) × \(n\)
- Position = (80/100) × 28 = 22.4
- Round up, so position = 23
Step 3: Locate the value.
- Find the 23rd value in the ordered list
- Counting from the left: 5.6, 5.7, 5.9, 6.0, 6.2, 6.3, 6.5, 6.7, 6.7, 6.9, 6.9, 6.9, 7.2, 7.2, 7.3, 7.4, 7.4, 7.5, 7.6, 7.7, 7.7, 7.7, 7.8, 8.0, 8.1, 8.3, 8.4, 8.7
- The 23rd value is in bold
- Thus, the 80th percentile is 7.8 pounds
This means that
- A weight of 7.8 pounds is at the 80th percentile.
- 80% of newborn weights are less than 7.8 pounds.
- 20% of newborn weights are greater than 7.8 pounds.
- If a newborn's weight was 7.8 pounds, then the baby's weight is higher than 80% of the newborns in that group.
The temperatures on a particular day across 25 states in the western U.S. was recorded:
94, 86, 75, 89, 92, 64, 78, 102, 77, 82, 69, 75, 86, 84, 81, 107, 97, 80, 83, 79, 72, 68, 74, 88, 71
Find the 28th percentile.
✅ Solution:
Step 1: Arrange the data in ascending order.
- Data was given in ascending order
Step 2: Calculate the position.
- Position = (Percentile/100) × \(n\)
- Position = (28/100) × 25 = 7 (No rounding up needed)
Step 3: Locate the value.
- Find the 7th value in the ordered list.
- Counting from the left: 64, 68, 69, 71, 72, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 86, 86, 88, 89, 92, 94, 97, 102, 107
- The 7th value is in bold.
- Thus, the 28th percentile is 75 degrees
This means that
- A temperature of 75 degrees is at the 28th percentile.
- 28% of temperatures are 75 degrees or below.
- 72% of temperatures are 75 degrees or above.
- If a temperature is 75, then its higher than 28% of the temperatures recorded.
The heights (in inches) of 20 students in math class are:
73, 60, 72, 61, 66, 65, 75, 68, 60, 66, 67, 70, 71, 72, 66, 68, 68, 64, 70, 66
Find the percentile rank for a height of 65.
✅ Solution:
Step 1: Arrange the data in ascending order.
- 60, 60, 61, 64, 65, 66, 66, 66, 66, 67, 68, 68, 68, 70, 70, 71, 72, 72, 73, 75 (Value is in bold)
Step 2: Calculate the total number of values to the left of the ordered data set.
- There are 4 values to the left of 65
Step 3: Use the formula.
\[\text{Percentile for the value }x=\frac{\text{Number of data values less than }x}{n}\cdot100 \nonumber \]
\[\text{Percentile for the value }65=\frac{\text{Number of data values less than }4}{20}\cdot100 \nonumber \]
\[\text{Percent for the value }65=20\% \nonumber \]
Thus, the value of 65 is the 20th percentile.
According to the 2025 Fuel Economy Guide, the driving range (in miles) for combined city/highway driving (55% city and 45% highway) of all 17 2025 BMW compact EV cars made is given here:
266, 244, 318, 295, 267, 227, 287, 268, 295, 278, 271, 253, 250, 239, 266, 262, 248
Find the percentile rank for a height of 278.
✅ Solution:
Step 1: Arrange the data in ascending order.
- 227, 239, 244, 248, 250, 253, 262, 266, 266, 267, 268, 271, 278, 287, 295, 295, 318 (Value is in bold)
Step 2: Calculate the total number of values to the left of the ordered data set.
- There are 12 values to the left of 278
Step 3: Use the formula.
\[\text{Percentile for the value }x=\frac{\text{Number of data values less than }x}{n}\cdot100 \nonumber \]
\[\text{Percentile for the value }278=\frac{\text{Number of data values less than }12}{17}\cdot100 \nonumber \]
\[\text{Percent for the value }278=70.58823529\% \nonumber \]
Thus, the value of 278 is the 71st percentile.
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The points scored of 11 players from the Los Angeles Clippers against the Philadelphia 76ers from the February 2nd, 2026 regular season NBA game was recorded.
15, 29, 8, 6, 17, 21, 13, 0, 2, 2, 0
What score is at the 85th percentile?
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The SAT Math scores from 22 seniors from a particular high school:
480, 500, 510, 530, 540, 550, 560, 570, 580, 590, 600, 610, 620, 630, 640, 650, 660, 670, 690, 700, 710, 720
What score is at the 22nd percentile?
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A random sample of 24 patient's systolic blood pressures (in mmHg) were obtained from a local hospital:
102, 108, 110, 112, 115, 118, 120, 122, 124, 126, 128, 130, 132, 134, 136, 138, 140, 142, 145, 148, 150, 152, 155, 160
Find the percentile rank for a blood pressure of 115.
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Daily customer counts for a café over four weeks are given:
48, 50, 52, 53, 55, 57, 58, 60, 62, 63, 65, 66, 68, 70, 72, 73, 75, 77, 78, 80, 82, 85, 88, 90, 92, 95, 100
Find the percentile rank for a count of 88.
- Answers
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1. 21; 2. 115; 3. 17th; 4. 82nd.