Measures of Position
At a checkup, the pediatrician tells a boy's parents that his height is at the $85$th percentile. It isn't a grade or a direct measurement: it is a relative position within the whole group of children his age. Measures of position extend the idea of the median to locate any data value within the distribution, not just the center. By the end of this page you will know what that $85$ means.
Median and quartiles with raw data
The interquartile range (IQR) is $Q_3-Q_1$: it measures how spread out the middle half of the data is. In the example, $IQR=28-23=5$ goals.
The number of goals scored by a handball team in their last 9 matches was: $24, 28, 21, 30, 25, 28, 22, 27, 26$. Calculate the lower quartile ($Q_1$), the median ($Me$)…
The number of goals scored by a handball team in their last 9 matches was: $24, 28, 21, 30, 25, 28, 22, 27, 26$. Calculate the lower quartile ($Q_1$), the median ($Me$) and the upper quartile ($Q_3$) of these data.
The battery life, in hours, of 10 cell phones after a full charge is: $14, 18, 12, 15, 19, 11, 16, 15, 13, 17$. Determine the median and the quartiles $Q_1$ and $Q_3$ of…
The battery life, in hours, of 10 cell phones after a full charge is: $14, 18, 12, 15, 19, 11, 16, 15, 13, 17$. Determine the median and the quartiles $Q_1$ and $Q_3$ of the battery life.
The maximum temperatures recorded in a city over 12 consecutive spring days were, in degrees Celsius: $18, 20, 22, 19, 21, 24, 25, 23, 20, 19, 26, 22$. Calculate the…
The maximum temperatures recorded in a city over 12 consecutive spring days were, in degrees Celsius: $18, 20, 22, 19, 21, 24, 25, 23, 20, 19, 26, 22$. Calculate the interquartile range (difference between the upper and lower quartiles) of the temperatures.
Percentiles and deciles
With raw data, order the list and calculate the position $\dfrac{k\times N}{100}$. If it is not a whole number, the percentile is the value at the next position; if it is a whole number, average that value with the one after it.
This closes the opening question: a boy's height being at $P_{85}$ means that $85\,\%$ of the children his age are the same height or shorter, and the remaining $15\,\%$ are taller.
The number of pages read in a day by a group of 15 students is: $12, 20, 8, 15, 30, 25, 10, 18, 22, 14, 35, 28, 16, 24, 19$. Find the percentile $60$ ($P_{60}$) and…
The number of pages read in a day by a group of 15 students is: $12, 20, 8, 15, 30, 25, 10, 18, 22, 14, 35, 28, 16, 24, 19$. Find the percentile $60$ ($P_{60}$) and interpret its meaning in the context of the problem.
At the annual medical check-up, the paediatrician tells a child's parents that his height is at the percentile $85$ ($P_{85}$) for his age. Explain in detail what this…
At the annual medical check-up, the paediatrician tells a child's parents that his height is at the percentile $85$ ($P_{85}$) for his age. Explain in detail what this statistical value means in relation to other children of his age.
Calculating with a frequency table
When the data comes in a table, the quartile (or percentile) is the first value $x_i$ whose cumulative frequency $F_i$ reaches the desired position.
With the same table, percentiles work the same way (extension): for $P_{20}$ the position is $0.20\times30=6$, which matches exactly $F=6$ (size $41$); average it with the next size: $P_{20}=\dfrac{41+42}{2}=41.5$. For $P_{80}$ the position is $24$ and the first $F_i\ge24$ is $26$ (size $44$): $P_{80}=44$.
In a survey carried out on 20 families about the number of pets they have at home, the following results were obtained: 0 pets: 6 families 1 pet: 8 families 2 pets: 4…
In a survey carried out on 20 families about the number of pets they have at home, the following results were obtained:
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0 pets: 6 families
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1 pet: 8 families
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2 pets: 4 families
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3 pets: 2 families
Calculate the median of the number of pets.
The 25 students in a class were asked how many days a week they play sport. The results are shown in the table: | Days | 0 | 1 | 2 | 3 | 4 | 5 |…
The 25 students in a class were asked how many days a week they play sport. The results are shown in the table:
| Days | 0 | 1 | 2 | 3 | 4 | 5 |
|---|---|---|---|---|---|---|
| Students | 3 | 5 | 7 | 6 | 2 | 2 |
Determine the lower and upper quartiles of this distribution.
The shoe sizes of the 30 players in a basketball club are distributed according to the following table: | Size | 40 | 41 | 42 | 43 | 44 | 45 | 46 |…
The shoe sizes of the 30 players in a basketball club are distributed according to the following table:
| Size | 40 | 41 | 42 | 43 | 44 | 45 | 46 |
|---|---|---|---|---|---|---|---|
| Frequency | 2 | 4 | 7 | 8 | 5 | 3 | 1 |
Calculate the percentiles $P_{20}$ and $P_{80}$ of the distribution and interpret the results obtained in the context of the problem.
The ages of a group of 6 friends are: $14, 15, 15, 16, 17, 18$. A new friend joins the group. Determine what age this new member must be so that the median age of the…
The ages of a group of 6 friends are: $14, 15, 15, 16, 17, 18$. A new friend joins the group. Determine what age this new member must be so that the median age of the whole group becomes exactly $16$ years.
Linear interpolation with grouped data
With data in intervals there isn't an exact data value at each position: we interpolate, assuming that the data is spread evenly within the interval.
The time, in minutes, that a group of 40 teenagers spend per day on social media has been grouped in the following table: | Time (min) | $[30, 60)$ | $[60, 90)$ |…
The time, in minutes, that a group of 40 teenagers spend per day on social media has been grouped in the following table:
| Time (min) | $[30, 60)$ | $[60, 90)$ | $[90, 120)$ | $[120, 150)$ | $[150, 180)$ |
|---|---|---|---|---|---|
| Frequency | 4 | 10 | 14 | 8 | 4 |
Identify the median class and calculate the exact value of the median using the linear interpolation formula.
The distance, in kilometers, traveled daily by 50 employees to get to their workplace is distributed as follows: | Distance (km) | $[0, 5)$ | $[5, 10)$ | $[10, 15)$ |…
The distance, in kilometers, traveled daily by 50 employees to get to their workplace is distributed as follows:
| Distance (km) | $[0, 5)$ | $[5, 10)$ | $[10, 15)$ | $[15, 20)$ | $[20, 25)$ |
|---|---|---|---|---|---|
| Employees | 12 | 18 | 10 | 6 | 4 |
Calculate the percentile $90$ ($P_{90}$) using the linear interpolation formula and explain what information this value provides to the company for planning parking spaces or transportation subsidies.
The weight, in grams, of 60 apples harvested in an orchard has been tabulated in intervals: | Weight (g) | $[120, 140)$ | $[140, 160)$ | $[160, 180)$ | $[180, 200)$ |…
The weight, in grams, of 60 apples harvested in an orchard has been tabulated in intervals:
| Weight (g) | $[120, 140)$ | $[140, 160)$ | $[160, 180)$ | $[180, 200)$ | $[200, 220)$ |
|---|---|---|---|---|---|
| Apples | 5 | 15 | 25 | 10 | 5 |
Calculate the interquartile range of the weights of the apples using linear interpolation to find $Q_1$ and $Q_3$ beforehand.
The monthly rainfall, in mm, recorded in a region over several years is grouped in the following table: | Rainfall (mm) | $[10, 30)$ | $[30, 50)$ | $[50, 70)$ |…
The monthly rainfall, in mm, recorded in a region over several years is grouped in the following table:
| Rainfall (mm) | $[10, 30)$ | $[30, 50)$ | $[50, 70)$ | $[70, 90)$ | $[90, 110)$ |
|---|---|---|---|---|---|
| Months | 8 | 15 | 22 | 10 | 5 |
Find the value of $k$ such that the percentile $P_k$ is exactly $65$ mm. What percentage of months had less rainfall than that amount?
The distribution of monthly income, in hundreds of pesos, of a residential neighborhood is grouped in the following table: | Income | $[200, 300)$ | $[300, 400)$ |…
The distribution of monthly income, in hundreds of pesos, of a residential neighborhood is grouped in the following table:
| Income | $[200, 300)$ | $[300, 400)$ | $[400, 500)$ | $[500, 600)$ | $[600, 800)$ |
|---|---|---|---|---|---|
| Families | 15 | 35 | 40 | 20 | 10 |
Calculate the first decile ($D_1$) and the ninth decile ($D_9$). Subsequently, find the ratio $\frac{D_9}{D_1}$ and explain what this quotient indicates about economic inequality in the neighborhood.
Reading graphs: the box-and-whisker plot
Extension: cumulative frequency polygon
In a cumulative frequency polygon, $P_k$ is read by drawing a horizontal line at height $kN/100$ to the curve and then dropping down to the value axis. For $Q_1$ the height is $N/4$; for the median, $N/2$; for $Q_3$, $3N/4$.
The following box plot represents the prices, in pesos, of tickets for various concerts in a city during the summer. Identify the minimum price, the maximum price, the…
The following box plot represents the prices, in pesos, of tickets for various concerts in a city during the summer. Identify the minimum price, the maximum price, the median and the quartiles $Q_1$ and $Q_3$.
The following cumulative frequency polygon shows the distribution of monthly salaries, in pesos, of the 50 employees of a tech company. From the graph, estimate…
The following cumulative frequency polygon shows the distribution of monthly salaries, in pesos, of the 50 employees of a tech company. From the graph, estimate approximately the value of the median ($Me$), the lower quartile ($Q_1$), and the upper quartile ($Q_3$).
The attached box plot represents the lifespan, in hours, of a sample of 200 LED light bulbs of a certain brand. Answer the following questions, justifying your answers:…
The attached box plot represents the lifespan, in hours, of a sample of 200 LED light bulbs of a certain brand. Answer the following questions, justifying your answers:
- a) What percentage of light bulbs last more than $1500$ hours?
- b) Between which values is the central $50\%$ of the distribution located?
- c) How many light bulbs exactly lasted less than $1200$ hours?
The following box plots show the marks obtained in a Science exam by two different classes, A and B. a) Which class has a greater dispersion in the central $50\%$ of its…
The following box plots show the marks obtained in a Science exam by two different classes, A and B.
- a) Which class has a greater dispersion in the central $50\%$ of its marks?
- b) In which class is the mark that marks the boundary of the lowest $75\%$ of marks higher?
- c) Which of the two classes do you consider to have had a more consistent performance overall? Justify your answer.
The following graph represents the cumulative frequency polygon of the charging time, in minutes, of the battery of 40 electric vehicles. From the graph, reconstruct the…
The following graph represents the cumulative frequency polygon of the charging time, in minutes, of the battery of 40 electric vehicles. From the graph, reconstruct the absolute frequency table by intervals and calculate analytically the upper quartile ($Q_3$). Compare the analytical result with the value observed directly on the graph.
The reaction times, in milliseconds, of 15 professional gamers in a video game are: $210, 245, 215, 230, 220, 250, 215, 225, 240, 235, 218, 222, 260, 228, 232$.…
The reaction times, in milliseconds, of 15 professional gamers in a video game are: $210, 245, 215, 230, 220, 250, 215, 225, 240, 235, 218, 222, 260, 228, 232$.
Calculate the five-number summary (minimum, $Q_1$, median, $Q_3$, maximum) for this group. If an amateur player has a box plot whose quartiles are $Q_1 = 240$ and $Q_3 = 280$, reason statistically whether it is possible to state that more than $25\%$ of professional gamers have worse (higher) reaction times than the fastest $25\%$ of amateur players.
Reasoning with positions
Measures of position don't change arbitrarily when the data is transformed. If all the data is multiplied by $a>0$ and $b$ is added, each quartile transforms the same way: $Q'=a\,Q+b$.
Why? That transformation preserves the order of the data, so the data value that was at each position stays at that position.
Mean, mode, and median constrain the data
To reconstruct a list of data, you can combine clues:
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The mean $\bar{x}$ fixes the sum: $\text{sum}=\bar{x}\times N$.
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The median fixes the middle value (or the average of the two middle ones if $N$ is even).
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The mode $Mo$ is the value that appears most often; if it is unique, no other value can appear as many times as it does.
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The quartiles fix which values fall in each half.
On a music app, the ratings (from 1 to 5 stars) given by a group of users have the following distribution: | Rating | 1 | 2 | 3 | 4 | 5 | |---|---|---|---|---|---| |…
On a music app, the ratings (from 1 to 5 stars) given by a group of users have the following distribution:
| Rating | 1 | 2 | 3 | 4 | 5 |
|---|---|---|---|---|---|
| Users | 4 | $x$ | 12 | 15 | 9 |
It is known that the median of the ratings is $3.5$ stars. Determine the value of the unknown frequency $x$, reasoning your answer, and then calculate the lower quartile ($Q_1$) of the distribution.
An electronics shop has a price list for 20 models of headphones. It is known that the lower quartile of the prices is $600$ pesos, the median is $900$ pesos and the…
An electronics shop has a price list for 20 models of headphones. It is known that the lower quartile of the prices is $600$ pesos, the median is $900$ pesos and the upper quartile is $1\,200$ pesos. For Black Friday week, the shop applies a $20\%$ discount to all headphones and, in addition, deducts an extra $100$ pesos at the till. Determine what the new values of $Q_1$, $Me$ and $Q_3$ will be after applying the promotion, justifying your answer mathematically.
The test scores of 6 students are integers from 1 to 10. It is known that the arithmetic mean is $6.5$, the median is $6.5$, the mode is $7$ (and is unique), and the…
The test scores of 6 students are integers from 1 to 10. It is known that the arithmetic mean is $6.5$, the median is $6.5$, the mode is $7$ (and is unique), and the first quartile ($Q_1$) is $5$. Reconstruct the exact scores of the 6 students, explaining step by step the reasoning followed.
The lifespan, in thousands of kilometers, of two brands of tires, X and Y, has been studied. For brand X: | Lifespan (thousand km) | $[30, 40)$ | $[40, 50)$ | $[50, 60)$…
The lifespan, in thousands of kilometers, of two brands of tires, X and Y, has been studied.
For brand X:
| Lifespan (thousand km) | $[30, 40)$ | $[40, 50)$ | $[50, 60)$ | $[60, 70)$ |
|---|---|---|---|---|
| Tires | 10 | 25 | 15 | 0 |
For brand Y:
| Lifespan (thousand km) | $[30, 40)$ | $[40, 50)$ | $[50, 60)$ | $[60, 70)$ |
|---|---|---|---|---|
| Tires | 5 | 15 | 20 | 10 |
Calculate the interquartile range of both brands and determine which of the two offers a more consistent lifespan (less spread out in the central $50\%$).