Section 6.11: Quartiles
- Page ID
- 216511
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)- Compute quartiles for a data set
While any percentile from 1 to 99 can be useful, certain percentiles are so important they have special names. Some of those percentiles are called quartiles.
A quartile is one of three values that divide a data set, ordered from lowest to highest, into four equal parts (or quarters).
The three quartiles are the following:
Q₁ - First Quartile (25th percentile):
- 25% of data falls below this value
- 75% of data falls above this value
- Also called the lower quartile
Q₂ - Second Quartile (50th percentile):
- 50% of data falls below this value
- 50% of data falls above this value
- This is the MEDIAN
- Also called the middle quartile
Q₃ - Third Quartile (75th percentile):
- 75% of data falls below this value
- 25% of data falls above this value
- Also called the upper quartile
Quartiles can quickly evaluate the center of your data (Q₂/median), the spread of your data, as well as the shape of your distribution (symmetric vs. skewed).
- Step 1 — Organize your data: Arrange your data values in order from smallest to largest. This isn't strictly necessary, but it makes the process easier and ensures your final plot is organized.
To find the quartiles of a data set:
- Step 1 — Arrange the data in ascending order.
- Step 2 — Find the median which is Q2.
- Step 3 — If the data is
- an even size, then split the data into two halves. A lower half and an upper half.
- an odd size, ignore the median, and split the data into two halves. A lower half and an upper half.
- Step 4 — Find the median of the lower half which is Q1.
- Step 5 — Find the median of the upper half which is Q3.
A statistics class of 20 students takes an exam. Here are the scores (out of 100), arranged in order:
52, 58, 63, 65, 68, 71, 73, 75, 78, 80, 82, 84, 85, 87, 89, 91, 93, 95, 97, 99
Find the quartiles of the data set.
✅ Solution:
Step 1: Arrange the data in ascending order.
- Data was given in ascending order.
Step 2: Find the median, which is Q2
- 52, 58, 63, 65, 68, 71, 73, 75, 78, 80, 82, 84, 85, 87, 89, 91, 93, 95, 97, 99 (The middle values are in bold)
- The median is \(\frac{80+82}{2}=\frac{162}{2}=81\). So, Q2 = 81
Step 3: The data is an even size, so split the data into two halves. A lower half and an upper half.
- Lower half: 52, 58, 63, 65, 68, 71, 73, 75, 78, 80
- Upper half: 82, 84, 85, 87, 89, 91, 93, 95, 97, 99
Step 4: Find the median of the lower half.
- Lower half: 52, 58, 63, 65, 68, 71, 73, 75, 78, 80 (The middle values are in bold)
- The median is \(\frac{68+71}{2}=\frac{139}{2}=69.5\). So, Q1 = 69.5
Step 5: Find the median of the upper half.
- Upper half: 82, 84, 85, 87, 89, 91, 93, 95, 97, 99 (The middle values are in bold)
- The median is \(\frac{89+91}{2}=\frac{180}{2}=90\). So, Q3 = 90
Thus, Q1 = 69.5, Q2 = 81, & Q3 = 90.
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 quartiles of the data set.
Step 1: Arrange the data in ascending order.
- 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
Step 2: Find the median, which is Q2
- 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 middle value is in bold)
- The median is 81. So, Q2 = 81
Step 3: The data is an odd size, so ignore the median, and split the data into two halves. A lower half and an upper half.
- Lower half: 64, 68, 69, 71, 72, 74, 75, 76, 77, 78, 79, 80
- Upper half: 82, 83, 84, 86, 86, 88, 89, 92, 94, 97, 102, 107
Step 4: Find the median of the lower half.
- Lower half: 64, 68, 69, 71, 72, 74, 75, 76, 77, 78, 79, 80 (The middle values are in bold)
- The median is \(\frac{74+75}{2}=\frac{149}{2}=74.5\). So, Q1 = 74.5
Step 5: Find the median of the upper half.
- Upper half: 82, 83, 84, 86, 86, 88, 89, 92, 94, 97, 102, 107 (The middle values are in bold)
- The median is \(\frac{88+89}{2}=\frac{177}{2}=88.5\). So, Q3 = 88.5
Thus, Q1 = 74.5, Q2 = 81, & Q3 = 88.5.
The heights (in inches) of 19 students in a 5th grade science class are:
48, 63, 51, 59, 61, 50, 52, 56, 60, 54, 52, 57, 59, 52, 58, 56, 52, 59, 52
Find the quartiles of the data set.
Step 1: Arrange the data in ascending order.
- 48, 50, 51, 52, 52, 52, 52, 52, 54, 56, 56, 57, 58, 59, 59, 59, 60, 61, 63
Step 2: Find the median, which is Q2
- 48, 50, 51, 52, 52, 52, 52, 52, 54, 56, 56, 57, 58, 59, 59, 59, 60, 61, 63 (The middle value is in bold)
- The median is 56. So, Q2 = 56
Step 3: The data is an even size, so split the data into two halves. A lower half and an upper half.
- Lower half: 48, 50, 51, 52, 52, 52, 52, 52, 54
- Upper half: 56, 57, 58, 59, 59, 59, 60, 61, 63
Step 4: Find the median of the lower half.
- Lower half: 48, 50, 51, 52, 52, 52, 52, 52, 54 (The middle value is in bold)
- The median is 52. So, Q1 = 52
Step 5: Find the median of the upper half.
- Upper half: 56, 57, 58, 59, 59, 59, 60, 61, 63 (The middle value is in bold)
- The median is 59. So, Q3 = 59
Thus, Q1 = 52, Q2 = 56, & Q3 = 59.
In late 2025, according to Realtor.com, the closest 17 housing prices (in millions) of single family homes near Orange Coast College was obtained.
1.42, 1.34, 1.69, 1.38, 3.29, 1.59, 1.45, 1.43, 3.38, 1.89, 2.39, 2.49, 1.49, 1.56, 1.73. 1.47, 1.39
Find the quartiles of the data set.
Step 1: Arrange the data in ascending order.
- 1.34, 1.38, 1.39, 1.42, 1.43, 1.45, 1.47, 1.49, 1.56, 1.59, 1.69, 1.73, 1.89, 2.39, 2.49, 3.29, 3.38
Step 2: Find the median, which is Q2
- 1.34, 1.38, 1.39, 1.42, 1.43, 1.45, 1.47, 1.49, 1.56, 1.59, 1.69, 1.73, 1.89, 2.39, 2.49, 3.29, 3.38 (The middle value is in bold)
- The median is 1.56. So, Q2 = 1.56
Step 3: The data is an even size, so split the data into two halves. A lower half and an upper half.
- Lower half: 1.34, 1.38, 1.39, 1.42, 1.43, 1.45, 1.47, 1.49
- Upper half: 1.59, 1.69, 1.73, 1.89, 2.39, 2.49, 3.29, 3.38
Step 4: Find the median of the lower half.
- Lower half: 1.34, 1.38, 1.39, 1.42, 1.43, 1.45, 1.47, 1.49 (The middle values are in bold)
- The median is \(\frac{1.42+1.43}{2}=\frac{2.85}{2}=1.425\). So, Q1 = 1.425
Step 5: Find the median of the upper half.
- Upper half: 1.59, 1.69, 1.73, 1.89, 2.39, 2.49, 3.29, 3.38 (The middle values are in bold)
- The median is \(\frac{1.89+2.39}{2}=\frac{4.28}{2}=2.14\). So, Q3 = 2.14
Thus, Q1 = 1.425, Q2 = 1.56, & Q3 = 2.14; (All values in millions).
A transportation researcher surveyed 22 employees at a mid-sized company to understand their daily commute times (in minutes, one-way) to help the company plan flexible work arrangements.
20, 70, 52, 15, 65, 25, 28, 40, 32, 30, 45, 38, 55, 50, 18, 85, 62, 58, 22, 42, 48, 35
Find the quartiles of the data set.
Step 1: Arrange the data in ascending order.
- 15, 18, 20, 22, 25, 28, 30, 32, 35, 38, 40, 42, 45, 48, 50, 52, 55, 58, 62, 65, 70, 85
Step 2: Find the median, which is Q2
- 15, 18, 20, 22, 25, 28, 30, 32, 35, 38, 40, 42, 45, 48, 50, 52, 55, 58, 62, 65, 70, 85 (The middle values are in bold)
- The median is \(\frac{40+42}{2}=\frac{82}{2}=41\). So, Q2 = 41
Step 3: The data is an even size, so split the data into two halves. A lower half and an upper half.
- Lower half: 15, 18, 20, 22, 25, 28, 30, 32, 35, 38, 40
- Upper half: 42, 45, 48, 50, 52, 55, 58, 62, 65, 70, 85
Step 4: Find the median of the lower half.
- 15, 18, 20, 22, 25, 28, 30, 32, 35, 38, 40
- The median is 28. So, Q1 = 28
Step 5: Find the median of the upper half.
- Upper half: 42, 45, 48, 50, 52, 55, 58, 62, 65, 70, 85 (The middle value is in bold)
The median is 55. So, Q3 = 55
Thus, Q1 = 28, Q2 = 41, & Q3 = 55.
For Exercises 1 through 6, find the quartiles of each data set.
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Data set: 1, 1, 2, 3, 4, 6
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Data set: 2, 2, 3, 4, 5, 7, 9
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Data set: 3, 4, 4, 5, 6, 7, 9, 11
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Data set: 4, 4, 5, 7, 8, 9, 10, 14, 15
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The owner of a supermarket recorded the number of customers who came into his store each hour in a day. The results were 7, 18, 10, 6, 11, 7, 5, 9, 12, 8, 15, 3, and 11.
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Here are the highest temperatures ever recorded \((^{\circ }F)\) in 32 different U.S. states:
93 , 105 , 105 , 105 , 106 , 106 , 107 , 107 , 108 , 110 , 110 , 112 , 112 , 112 , 114 , 114
114 , 115 , 116 , 117 , 118 , 118 , 118 , 118 , 118 , 119 , 121 , 124 , 127 , 129 , 132 , 134
- Answers
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1. Q1 = 1, Q2 = 2.5, & Q3 = 4; 2. Q1 = 2, Q2 = 4, & Q3 = 7; 3. Q1 = 4, Q2 = 5.5, & Q3 = 8; 4. Q1 = 4.5, Q2 = 8, & Q3 = 12; 5. Q1 = 6.5, Q2 = 9, & Q3 = 11.5; 6. Q1 = 107.5, Q2 = 114, & Q3 = 118.