Measures of position
A paediatrician tells a child's parents that their son's height is at the $85$th percentile. It is not a mark or a direct measurement: it is a relative position within the whole reference group. Measures of position generalise the idea of the median so that we can place any data value within the distribution, not just the centre.
Median and quartiles with ungrouped data
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 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
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.
Calculation with a frequency table
With ungrouped data arranged in a frequency table, the value of a quartile or percentile is the first $x_i$ whose cumulative frequency $F_i$ reaches or exceeds the required position.
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.
Linear interpolation with grouped data
With data grouped into intervals there is no exact value at each position: we interpolate, assuming that the data are 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 kilometres, travelled daily by 50 employees to get to their workplace is distributed as follows: | Distance (km) | $[0, 5)$ | $[5, 10)$ | $[10, 15)$ |…
The distance, in kilometres, travelled 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 transport subsidies.
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?
Reading graphs: box plot and cumulative polygon
The following cumulative frequency polygon shows the distribution of monthly salaries, in euros, of the 50 employees of a tech company. From the graph, estimate…
The following cumulative frequency polygon shows the distribution of monthly salaries, in euros, 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 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.