2.4 Measures of Position
LEARNING OBJECTIVES
- Recognize, describe, calculate, and interpret the measures of the position of data: quartiles and percentiles.
The common measures of position are quartiles and percentiles. Previously, we learned that the median is a number that measures the “center” of the data. But the median can also be thought of as a measure of position because the median is the “middle value” of a set of data. The median is a number that separates ordered data into halves. Half of the values in the data are the same number or smaller than the median, and half of the values in the data are the same number or larger.
For example, consider the following data, already ordered from smallest to largest:
| 1 | 1 | 2 | 2 | 4 | 6 | 6.8 |
| 7.2 | 8 | 8.3 | 9 | 10 | 10 | 11.5 |
Because there are [latex]14[/latex] observations, the median is between the seventh value, [latex]6.8[/latex], and the eighth value, [latex]7.2[/latex]. To find the median, add the two values together and divide by two:
[latex]\displaystyle{\frac{6.8+7.2}{2}=7}[/latex]
The median of this data is [latex]7[/latex]. We can see that half (or [latex]50\%[/latex]) of the values are less than seven and half (or [latex]50\%[/latex]) of the values are larger than seven.
The median is an example of both a quartile and a percentile. The median is also the second quartile, [latex]Q_2[/latex], and the [latex]50[/latex]th percentile, [latex]P_{50}[/latex].
Quartiles
Quartiles are numbers that separate the data into quarters (four parts). Like the median, quartiles may or may not be an actual value in the set of data. To find the quartiles, order the data (from smallest to largest) and then find the median or second quartile. The first quartile, [latex]Q_1[/latex], is the middle value of the lower half of the data, and the third quartile, [latex]Q_3[/latex], is the middle value of the upper half of the data. To get the idea, consider the same (ordered) data set used above:
| 1 | 1 | 2 | 2 | 4 | 6 | 6.8 |
| 7.2 | 8 | 8.3 | 9 | 10 | 10 | 11.5 |
The median or second quartile is [latex]7[/latex]. The lower half of the data are:
| 1 | 1 | 2 | 2 | 4 | 6 | 6.8 |
The middle value of the lower half of the data is [latex]2[/latex]. The number [latex]2[/latex], which is part of the data, is the first quartile, [latex]Q_1[/latex]. One-fourth (or [latex]25\%[/latex]) of the values in the data are the same as or less than [latex]2[/latex], and three-fourths (or [latex]75\%[/latex]) of the values are more than [latex]2[/latex].
The upper half of the data are:
| 7.2 | 8 | 8.3 | 9 | 10 | 10 | 11.5 |
The middle value of the upper half of the data is [latex]9[/latex]. The third quartile, [latex]Q_3[/latex], is [latex]9[/latex]. Three-fourths (or [latex]75\%[/latex]) of the values in the data are the same or less than [latex]9[/latex]. One-fourth (or [latex]25\%[/latex]) of the values in the data set are greater than [latex]9[/latex].
The interquartile range is a number that indicates the spread of the middle half or the middle [latex]50\%[/latex] of the data. It is the difference between the third quartile ([latex]Q_3[/latex]) and the first quartile ([latex]Q_1[/latex]).
[latex]\displaystyle{IQR = Q_3 – Q_1}[/latex]
The [latex]IQR[/latex] can help to determine potential outliers. A value is suspected to be a potential outlier if it is less than [latex]1.5 \times IQR[/latex] below the first quartile or more than [latex]1.5 \times IQR[/latex] above the third quartile. Potential outliers always require further investigation.
NOTE
A potential outlier is a data point that is significantly different from the other data points. These special data points may be errors, some kind of abnormality, or they may be a key to understanding the data.
Video: “Median, Quartiles and Interquartile Range : ExamSolutions” by ExamSolutions [12:36] is licensed under the Standard YouTube License.Transcript and closed captions available on YouTube.
CALCULATING QUARTILES IN EXCEL
To find quartiles in Excel, use the quartile.exc(array, quartile number) function.
- For array, enter the array or cell range containing the data.
- For quartile number, enter the quartile (1, 2 or 3) being calculated.
The output from the quartile.exc function is the value of the corresponding quartile. For example, quartile.exc(array,1) returns the value of the first quartile where [latex]25\%[/latex] of the observations in the data are less than or equal to the value of the first quartile.
Visit the Microsoft page for more information about the quartile.exc function.
NOTE
We are using the quartile.exc function, and not the quartile.inc function. These two functions calculate the quartiles in slightly different ways.
- The quartile.exc function calculates the quartiles by first finding the median of the data set. The first quartile is the median of the lower half of the data, excluding the median value from the lower half of the data. The third quartile is the median value of the upper half of the data, excluding the median value from the upper half of the data.
- The quartile.inc function calculates the quartiles by first finding the median of the data set. The first quartile is the median of the lower half of the data, including the median value (if the median is a number in the data set) with the lower half of the data. The third quartile is the median of the upper half of the data, excluding the median value (if the median is a number in the data set) with the upper half of the data.
In some cases, the quartile.exc and quartile.inc will return the same values, depending on whether or not the median is a number in the data set.
Video: “How To Find Quartiles and Construct a Boxplot in Excel” by Joshua Emmanuel [4:13] is licensed under the Standard YouTube License.Transcript and closed captions available on YouTube.
EXAMPLE
For the following [latex]13[/latex] real estate prices, calculate the three quartiles and the [latex]IQR[/latex]. Determine if any prices are potential outliers. The prices are in dollars.
| 389,950 | 230,500 | 158,000 | 479,000 | 639,000 | 114,950 | 5,500,000 |
| 387,000 | 659,000 | 529,000 | 575,000 | 488,800 | 1,095,000 |
Solution
Enter the data into an Excel spreadsheet. For this example, suppose we entered the data in column A from cell A1 to A13.
For the first quartile [latex]Q_1[/latex]:
| Function | quartile.exc |
|---|---|
| Field 1 | A1:A13 |
| Field 2 | 1 |
| Answer | $308,750 |
For the second quartile [latex]Q_2[/latex]:
| Function | quartile.exc |
|---|---|
| Field 1 | A1:A13 |
| Field 2 | 2 |
| Answer | $488,800 |
For the third quartile [latex]Q_3[/latex]:
| Function | quartile.exc |
|---|---|
| Field 1 | A1:A13 |
| Field 2 | 3 |
| Answer | $649,000 |
For the [latex]IQR[/latex]: [latex]\displaystyle{IQR = 649,000 – 308,750 = \$340,250}[/latex]
To determine if there are any outliers:
[latex]\begin{eqnarray*}1.5\times IQR&=&1.5\times 340,250=510,375\\\\Q_1–1.5\times IQR&=&308,750–510,375=–201,625\\\\Q_3+1.5\times IQR&=&649,000+510,375=1,159,375\end{eqnarray*}[/latex]
No house price is less than [latex]–\$201,625[/latex]. However, [latex]\$5,500,000[/latex] is more than [latex]\$1,159,375[/latex]. Therefore, [latex]\$5,500,000[/latex] is a potential outlier.
NOTE
Quartiles have the same units as the data. In this case, the data is measured in dollars, so the quartiles are also in dollars.
TRY IT
For the following [latex]11[/latex] salaries, calculate the three quartiles and the [latex]IQR[/latex]. Are any of the salaries outliers? The salaries are in dollars.
| 33,000 | 72,000 | 54,000 |
| 64,500 | 68,500 | 120,000 |
| 28,000 | 69,000 | 40,500 |
| 54,000 | 42,000 |
Click to see Solution
Enter the data into an Excel spreadsheet. For this example, suppose we entered the data in column A from cell A1 to A11.
For the first quartile [latex]Q_1[/latex]:
| Function | quartile.exc |
|---|---|
| Field 1 | A1:A11 |
| Field 2 | 1 |
| Answer | $40,500 |
For the second quartile [latex]Q_2[/latex]:
| Function | quartile.exc |
|---|---|
| Field 1 | A1:A11 |
| Field 2 | 2 |
| Answer | $54,000 |
For the third quartile [latex]Q_3[/latex]:
| Function | quartile.exc |
|---|---|
| Field 1 | A1:A11 |
| Field 2 | 3 |
| Answer | $69,000 |
For the [latex]IQR[/latex]: [latex]\displaystyle{IQR = 69,000 – 40,500 = \$28,500}[/latex]
To determine if there are any outliers:
[latex]\begin{eqnarray*}1.5\times IQR&=&1.5\times 28,500=42,750\\\\Q_1–1.5\times IQR&=&40,500–42,750=-2,250\\\\Q_3+1.5\times IQR&=&69,000+42,750=111,750\end{eqnarray*}[/latex]
No salary is less than [latex]-\$2,250[/latex]. However, [latex]\$120,000[/latex] is more than [latex]\$111,750[/latex], so [latex]\$120,000[/latex] is a potential outlier.
TRY IT
Find the interquartile range for the following two data sets and compare them.
| Test Scores for Class A | |||||||||
|---|---|---|---|---|---|---|---|---|---|
| 69 | 96 | 81 | 79 | 65 | 76 | 83 | 99 | 89 | 67 |
| 90 | 77 | 85 | 98 | 66 | 91 | 77 | 69 | 80 | 94 |
| Test Scores for Class B | |||||||||
|---|---|---|---|---|---|---|---|---|---|
| 90 | 72 | 80 | 92 | 90 | 97 | 92 | 75 | 79 | 68 |
| 70 | 80 | 99 | 95 | 78 | 73 | 71 | 68 | 95 | 100 |
Click to see Solution
Enter the data into an Excel spreadsheet. For this example, suppose we entered the data for Class A into column A from cell A1 to A20 and the data for Class B into column B from cell B1 to B20.
Class A
For the first quartile [latex]Q_1[/latex]:
| Function | quartile.exc |
|---|---|
| Field 1 | A1:A20 |
| Field 2 | 1 |
| Answer | 70.75 |
For the third quartile [latex]Q_3[/latex]:
| Function | quartile.exc |
|---|---|
| Field 1 | A1:A20 |
| Field 2 | 3 |
| Answer | 90.75 |
For the [latex]IQR[/latex]: [latex]\begin{eqnarray*}IQR &=& 90.75-70.75=20 \end{eqnarray*}[/latex]
Class B
For the first quartile [latex]Q_1[/latex]:
| Function | quartile.exc |
|---|---|
| Field 1 | B1:B20 |
| Field 2 | 1 |
| Answer | 72.25 |
For the third quartile [latex]Q_3[/latex]:
| Function | quartile.exc |
|---|---|
| Field 1 | B1:B20 |
| Field 2 | 3 |
| Answer | 94.25 |
For the [latex]IQR[/latex]: [latex]\begin{eqnarray*}IQR &=& 94.25-72.25=22\end{eqnarray*}[/latex]
The data for Class B has a larger [latex]IQR[/latex], so the scores between [latex]Q_3[/latex] and [latex]Q_1[/latex] (the middle [latex]50\%[/latex] of the data) for the data for Class B are more spread out and not clustered about the median.
Percentiles
Percentiles are numbers that separate the (ordered) data into hundredths (100 parts). Like quartiles, percentiles may or may not be part of the data. The [latex]n[/latex]th percentile, [latex]P_n[/latex], is the value where [latex]n\%[/latex] of the observations in the data are less than or equal to the value of the [latex]n[/latex]th percentile. To score in the [latex]90[/latex]th percentile of an exam does not mean, necessarily, that the student received [latex]90\%[/latex] on a test. The [latex]90[/latex]th percentile means that [latex]90\%[/latex] of test scores are less than or equal to the student’s score and [latex]10\%[/latex] of the test scores are the same or greater than the student’s score. Percentiles are mostly used with very large data sets.
Quartiles are special percentiles. The first quartile, [latex]Q_1[/latex], is the same as the [latex]25[/latex]th percentile, and the third quartile, [latex]Q_3[/latex], is the same as the [latex]75[/latex]th percentile. The median is the [latex]50[/latex]th percentile.
Percentiles are useful for comparing values. For this reason, universities and colleges use percentiles extensively. One instance in which colleges and universities use percentiles is when SAT results are used to determine a minimum testing score that will be used as an acceptance factor. For example, suppose Duke accepts SAT scores at or above the [latex]75[/latex]th percentile. That translates into an SAT score of at least [latex]1220[/latex].
CALCULATING PERCENTILES IN EXCEL
To find the [latex]k[/latex]th percentiles in Excel, use the percentile.exc(array, percent) function.
- For array, enter the array or cell range containing the data.
- For percent, enter the percentile (as a decimal) being calculated. For example, if we are calculating the [latex]60[/latex]th percentile, we would enter [latex]0.6[/latex] for the percent in the percentile.exc function.
The output from the percentile.exc function is the value of the corresponding percentile. For example, percentile.exc(array,0.6) returns the value of the [latex]60[/latex]th percentile where [latex]60\%[/latex] of the observations in the data are less than or equal to the value of the [latex]60[/latex]th percentile.
Visit the Microsoft page for more information about the percentile.exc function.
NOTE
We are using the percentile.exc function, and not the percentile.inc function. Like the quartile functions, the percentile.exc and percentile.inc calculate the percentiles in different ways, and so give slightly different answers for the percentiles.
Video: “Percentiles – How to calculate Percentiles, Quartiles, …” by Joshua Emmanuel [3:44] is licensed under the Standard YouTube License.Transcript and closed captions available on YouTube.
EXAMPLE
Listed are twenty-nine ages (in years) for trees found in the Saint Louis Botanical Garden.
| 18 | 21 | 22 | 25 | 26 | 27 | 29 | 30 | 31 | 33 |
| 36 | 37 | 41 | 42 | 47 | 52 | 55 | 57 | 58 | 62 |
| 64 | 67 | 69 | 71 | 72 | 73 | 74 | 76 | 77 |
- Find the [latex]70[/latex]th percentile.
- Find the [latex]83[/latex]rd percentile.
Solution
Enter the data into an Excel spreadsheet. For this example, suppose we entered the data in column A from cell A1 to A29.
For the [latex]70[/latex]th percentile [latex]P_{70}[/latex]:
| Function | percentile.exc |
|---|---|
| Field 1 | A1:A29 |
| Field 2 | 0.7 |
| Answer | 64 years |
For the [latex]83[/latex]rd percentile [latex]P_{83}[/latex]:
| Function | percentile.exc |
|---|---|
| Field 1 | A1:A29 |
| Field 2 | 0.83 |
| Answer | 71.9 years |
NOTE
Percentiles have the same units as the data. In this case, the data is measured in years, so the percentiles are also in years.
TRY IT
Listed are [latex]29[/latex] ages (in years) for Academy Award-winning best actors.
| 18 | 21 | 22 | 25 | 26 | 27 | 29 | 30 | 31 | 33 |
| 36 | 37 | 41 | 42 | 47 | 52 | 55 | 57 | 58 | 62 |
| 64 | 67 | 69 | 71 | 72 | 73 | 74 | 76 | 77 |
Calculate the [latex]20[/latex]th percentile and the [latex]55[/latex]th percentile.
Click to see Solution
Enter the data into an Excel spreadsheet. For this example, suppose we entered the data in column A from cell A1 to A29.
For the [latex]20[/latex]th percentile [latex]P_{20}[/latex]:
| Function | percentile.exc |
|---|---|
| Field 1 | A1:A29 |
| Field 2 | 0.2 |
| Answer | 27 years |
For the [latex]55[/latex]th percentile [latex]P_{55}[/latex]:
| Function | percentile.exc |
|---|---|
| Field 1 | A1:A29 |
| Field 2 | 0.55 |
| Answer | 53.5 years |
Interpreting Percentiles and Quartiles
A percentile indicates the relative standing of a data value when data are sorted into numerical order from smallest to largest. Percentages of data values are less than or equal to the value of the [latex]n[/latex]th percentile. For example, [latex]15\%[/latex] of the data values are less than or equal to the value of the [latex]15[/latex]th percentile. Note that low percentiles always correspond to lower data values, and high percentiles always correspond to higher data values.
A percentile may or may not correspond to a value judgment about whether it is “good” or “bad.” The interpretation of whether a certain percentile is “good” or “bad” depends on the context of the situation to which the data applies. In some situations, a low percentile would be considered “good,” but in other contexts, a high percentile might be considered “good”. In many situations, there is no value judgment that applies.
Understanding how to interpret percentiles or quartiles properly is important not only when describing data but also when calculating probabilities in later chapters of this text. When writing the interpretation of a percentile or quartile in the context of the given data, the sentence should contain the following information:
- Information about the context of the situation being considered,
- The data value (value of the variable) that represents the percentile/quartile.
- The percent of individuals or items with data values less than or equal to the percentile/quartile.
EXAMPLE
On a timed math test, the first quartile for the time it took to finish the exam was [latex]35[/latex] minutes. Interpret the first quartile in the context of this situation.
Solution
- Interpretation: [latex]25\%[/latex] of students finished the exam in less than or equal to [latex]35[/latex] minutes.
- In this context, a low percentile could be considered good, as finishing more quickly on a timed exam is desirable. (If a student takes too long, they might not be able to finish.)
TRY IT
For the 100-meter dash, the third quartile for times for finishing the race was [latex]11.5[/latex] seconds. Interpret the third quartile in the context of the situation.
Click to see Solution
- Interpretation: [latex]75\%[/latex] of runners finished the race in less than or equal to [latex]11.5[/latex] seconds.
- In this context, a lower percentile is good because finishing a race more quickly is desirable.
EXAMPLE
On a [latex]20[/latex] question math test, the [latex]70[/latex]th percentile for the number of correct answers was [latex]16[/latex]. Interpret the [latex]70[/latex]th percentile in the context of this situation.
Solution
- Interpretation: [latex]70\%[/latex] of students answered less than or equal to [latex]16[/latex] questions correctly.
TRY IT
On a [latex]60[/latex] point written assignment, the [latex]80[/latex]th percentile for the number of points earned was [latex]49[/latex]. Interpret the [latex]80[/latex]th percentile in the context of this situation.
Click to see Solution
- Interpretation: [latex]80\%[/latex] of students earned less than or equal to [latex]49[/latex] points.
EXAMPLE
At a community college, it was found that the [latex]30[/latex]th percentile of credit units that students are enrolled for is [latex]7[/latex] units. Interpret the [latex]30[/latex]th percentile in the context of this situation.
Solution
- Interpretation: [latex]30\%[/latex] of students are enrolled in less than or equal to [latex]7[/latex] credit units.
- In this context, there is no “good” or “bad” value judgment associated with a higher or lower percentile. Students attend community college for varied reasons and needs, and their course load varies according to their needs.
TRY IT
During a season, the [latex]40[/latex]th percentile for points scored per player in a game is [latex]8[/latex]. Interpret the [latex]40[/latex]th percentile in the context of this situation.
Click to see Solution
- Interpretation: [latex]40\%[/latex] of players scored less than or equal to [latex]8[/latex] points.
Exercises
- How much time does it take to travel to work in a particular region? The table below shows the commute time for a sample of workers in the region who are at least [latex]16[/latex] years old and do not work at home.
24.0 24.3 25.9 18.9 27.5 17.9 21.8 20.9 16.7 27.3 18.2 24.7 20.0 22.6 23.9 18.0 31.4 22.3 24.0 25.5 24.7 24.6 28.1 24.9 22.6 23.6 23.4 25.7 24.8 25.5 21.2 25.7 23.1 23.0 23.9 26.0 16.3 23.1 21.4 21.5 27.0 27.0 18.6 31.7 23.3 30.1 22.9 23.3 21.7 18.6 - Find the first quartile.
- Interpret the first quartile.
- Find the third quartile.
- The middle [latex]50\%[/latex] of the travel times lie between what two numbers?
- Interpret the third quartile.
- Find the [latex]IQR[/latex].
- Find the [latex]45[/latex]th percentile.
- Interpret the [latex]45[/latex]th percentile.
Click to see Answer
- [latex]21.475[/latex] minutes
- [latex]25\%[/latex] of the workers take at most [latex]21.475[/latex] minutes to travel to work.
- [latex]25.55[/latex] minutes
- [latex]21.475[/latex] minutes to [latex]25.55[/latex] minutes
- [latex]75\%[/latex] of the workers take at most [latex]25.55[/latex] minutes to travel to work.
- [latex]4.075[/latex] minutes
- [latex]23.29[/latex] minutes
- [latex]45\%[/latex] of the workers take at most [latex]23.29[/latex] minutes to travel to work.
- The following data shows the lengths, in feet, of a sample of boats moored in a marina.
19 35 29 26 21 40 33 33 34 25 20 37 30 26 23 24 29 16 28 25 20 39 32 27 27 27 17 - Find the first quartile.
- Interpret the first quartile.
- Find the third quartile.
- Interpret the third quartile.
- The middle [latex]50\%[/latex] of boat lengths lie between what two numbers?
- Find the [latex]IQR[/latex].
- Find the [latex]63[/latex]rd percentile.
- Interpret the [latex]63[/latex]rd percentile.
Click to see Answer
- [latex]23[/latex] feet
- [latex]25\%[/latex] of the boats are less than or equal to [latex]23[/latex] feet in length.
- [latex]33[/latex] feet
- [latex]23[/latex] feet to [latex]33[/latex] feet
- [latex]75\%[/latex] of the boats are less than or equal to [latex]33[/latex] feet in length.
- [latex]10[/latex] feet
- [latex]28.64[/latex] feet
- [latex]63\%[/latex] of the boats are less than or equal to [latex]28.64[/latex] feet in length.
- The data below is the weight, in pounds, of all members of a particular NFL team.
177 210 270 275 212 185 200 241 250 220 259 185 210 272 285 212 250 302 205 232 280 285 184 265 215 223 265 260 278 185 228 273 242 185 241 290 210 276 290 206 174 286 247 190 215 245 205 178 290 280 188 230 260 - Find the first quartile.
- Interpret the first quartile.
- Find the third quartile.
- Interpret the third quartile.
- The middle [latex]50\%[/latex] of the player’s weights lie between what two numbers?
- Find the [latex]IQR[/latex].
- Find the [latex]87[/latex]th percentile.
- Interpret the [latex]87[/latex]th percentile.
Click to see Answer
- [latex]205.5[/latex] pounds
- [latex]25\%[/latex] of the football players weigh less than or equal to [latex]205.5[/latex] pounds.
- [latex]272.5[/latex] pounds
- [latex]75\%[/latex] of the football players weigh less than or equal to [latex]272.5[/latex] pounds.
- [latex]205.5[/latex] pounds to [latex]272.5[/latex] pounds
- [latex]67[/latex] pounds
- [latex]284.9[/latex] pounds
- [latex]87\%[/latex] of football players weigh less than or equal to [latex]284.9[/latex] pounds.
- A sample of 35 post-secondary institutions was taken from across the U.S. The data below shows the number of students enrolled at each institution.
6,414 1,550 2,109 9,350 21,828 4,300 5,944 5,722 2,825 2,044 5,481 5,200 5,853 10,012 6,357 27,000 9,414 7,681 3,200 17,500 9,200 7,380 18,314 6,557 13,713 17,768 7,493 2,771 2,861 1,263 7,285 28,165 5,080 11,622 2,750 - Find the first quartile.
- Interpret the first quartile.
- Find the third quartile.
- Interpret the third quartile.
- Find the [latex]IQR[/latex].
- Find the [latex]65[/latex]th percentile.
- Interpret the [latex]65[/latex]th percentile.
Click to see Answer
- [latex]3,200[/latex] students
- [latex]25\%[/latex] of the colleges enroll less than or equal to [latex]3,200[/latex] students.
- [latex]10,012[/latex] students
- [latex]75\%[/latex] of the colleges enroll less than or equal to [latex]10,012[/latex] students.
- [latex]6,812[/latex] students
- [latex]8,288.6[/latex] students
- [latex]65\%[/latex] of the colleges enroll less than or equal to [latex]8,288.6[/latex] students.
- Forty randomly selected students were asked the number of pairs of sneakers they owned. The data is recorded below
1 1 2 2 2 2 2 3 3 3 3 3 3 3 3 4 4 4 4 4 4 4 4 4 4 4 4 5 5 5 5 5 5 5 5 5 5 5 5 7 - Find the first quartile.
- Interpret the first quartile.
- Find the third quartile.
- Interpret the third quartile.
- Find the [latex]IQR[/latex].
- Find the [latex]90[/latex]th percentile.
- Interpret the [latex]90[/latex]th percentile.
Click to see Answer
- [latex]3[/latex] pairs of sneakers
- [latex]25\%[/latex] of the students own at most [latex]3[/latex] pairs of sneakers.
- [latex]5[/latex] pairs of sneakers
- [latex]75\%[/latex] of the students own at most [latex]5[/latex] pairs of sneakers.
- [latex]2[/latex] pairs of sneakers
- [latex]5[/latex] pairs of sneakers
- [latex]90\%[/latex] of the students own at most [latex]5[/latex] pairs of sneakers.
-
- For runners in a race, a low time means a faster run. The winners in a race have the shortest running times. Is it more desirable to have a finish time with a high or a low percentile when running a race?
- The [latex]20[/latex]th percentile of run times in a particular race is [latex]5.2[/latex] minutes. Write a sentence interpreting the [latex]20[/latex]th percentile in the context of the situation.
- A bicyclist in the [latex]90[/latex]th percentile of a bicycle race completed the race in [latex]1[/latex] hour and [latex]12[/latex] minutes. Is he among the fastest or slowest cyclists in the race? Write a sentence interpreting the [latex]90[/latex]th percentile in the context of the situation.
- For runners in a race, a higher speed means a faster run. Is it more desirable to have a speed with a high or a low percentile when running a race?
- The [latex]40[/latex]th percentile of speeds in a particular race is [latex]7.5[/latex] kilometres per hour. Write a sentence interpreting the [latex]40[/latex]th percentile in the context of the situation.
Click to see Answer
- A low percentile is more desirable because it means that the runner completed the race with a low time and ran a faster race.
- [latex]20\%[/latex] of the runners had run times of [latex]5.2[/latex] minutes or less.
- The cyclist is among the slowest. In this context, beginning in a high percentile means the cyclist had a high race completion time. [latex]90\%[/latex] of the racers completed the race in [latex]1[/latex] hour and [latex]12[/latex] minutes or less.
- A high percentile is more desirable because it means that the runner had a high speed and ran a faster race.
- [latex]40\%[/latex] of the speeds in the race are less than or equal to [latex]7.5[/latex] kilometres per hour.
- On an exam, would it be more desirable to earn a grade with a high or low percentile? Explain.
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A high percentile because it means a higher grade.
- Mina is waiting in line at the Department of Motor Vehicles (DMV). Her wait time of [latex]32[/latex] minutes is the [latex]85[/latex]th percentile of wait times. Is that good or bad? Write a sentence interpreting the [latex]85[/latex]th percentile in the context of this situation.
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Because Mina’s wait time is in a high percentile, it means that Mina’s wait time is among the longest. [latex]85\%[/latex] of the wait times are [latex]32[/latex] minutes or less.
- In a study collecting data about the repair costs of damage to automobiles in certain types of crash tests, a certain model of car had [latex]\$1,700[/latex] in damage and was in the [latex]90[/latex]th percentile. Should the manufacturer and the consumer be pleased or upset by this result? Explain and write a sentence that interprets the [latex]90[/latex]th percentile in the context of this problem.
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Because the cost is in a high percentile, it means that this model of car is among the most expensive to repair. [latex]90\%[/latex] of the cars have damage costs of [latex]\$1,700[/latex] or less.
- Suppose that you want to buy a house. You and your realtor have determined that the most expensive house you can afford is in the [latex]34[/latex]th percentile. The [latex]34[/latex]th percentile of housing prices in the town you want to move to is [latex]\$240,000[/latex]. In this town, can you afford [latex]34\%[/latex]of the houses or [latex]66\%[/latex] of the houses? Explain.
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[latex]34\%[/latex] because [latex]34\%[/latex] of the houses cost [latex]\$240,000[/latex] or less.
- Using the number of full-time equivalent students (FTES) each year at a local college for the past 40 years, the first quartile is [latex]528.5[/latex] FTES and the third quartile is [latex]1,447.5[/latex] FTES.
- [latex]75\%[/latex] of all years have an FTES at or below what value?
- [latex]75\%[/latex] of all years have an FTES above what value?
- What percent of the FTES were from [latex]528.5[/latex] to [latex]1447.5[/latex]? How do you know?
- What is the [latex]IQR[/latex]? What does the [latex]IQR[/latex] represent?
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- [latex]1,447.5[/latex]FTES
- [latex]528.5[/latex] FTES
- [latex]50\%[/latex]
- [latex]919[/latex] FTES. This is the spread of the middle [latex]50\%[/latex] of the data.
“2.5 Measures of Location” and “2.7 Exercices” from Introduction to Statistics by Valerie Watts is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License, except where otherwise noted.
