Mean, median and mode
A human resources manager boasts that the average salary at his company is $2314$ €. But six employees earn between $1200$ and $1400$ € and the seventh, the owner, earns $8500$ €. Is $2314$ € a "typical" salary at that company? The mean, the median and the mode do not always tell the same story.
Review: mean, median and mode
In a botany experiment, the number of leaves on $8$ plants of the same species was measured after two weeks of growth. The results obtained are:…
In a botany experiment, the number of leaves on $8$ plants of the same species was measured after two weeks of growth. The results obtained are: $12, 15, 14, 12, 16, 12, 18, 13$.
Calculate the arithmetic mean, median and mode of these data.
Given the following two data sets concerning the number of books read in a year by two groups of young people: Set A: $2, 4, 5, 7, 9$ Set B: $1, 3, 4, 6, 8, 10$ Find the…
Given the following two data sets concerning the number of books read in a year by two groups of young people:
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Set A: $2, 4, 5, 7, 9$
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Set B: $1, 3, 4, 6, 8, 10$
Find the median for each of the data sets and explain the difference in the calculation procedure when having an even or odd number of data values.
Weighted mean
Not all data values carry the same weight. A final exam may be worth more than a project: in that case each value is multiplied by its weight before adding.
The final mark for Physics is calculated using a weighted mean. Laboratory practicals are worth $20\%$, research projects $30\%$ and the final exam $50\%$. If a student…
The final mark for Physics is calculated using a weighted mean. Laboratory practicals are worth $20\%$, research projects $30\%$ and the final exam $50\%$.
If a student scores $8$ in the practicals, $6$ in the projects and $7,5$ in the final exam, calculate her final mean mark.
A logistics company has two warehouses. At the Madrid warehouse, $45$ workers work with a mean salary of $1800 \text{ \euro}$. At the Barcelona warehouse, $30$ workers…
A logistics company has two warehouses. At the Madrid warehouse, $45$ workers work with a mean salary of $1800 \text{ \euro}$. At the Barcelona warehouse, $30$ workers work with a mean salary of $2100 \text{ \euro}$.
Calculate the mean salary of all the workers in the company as a whole.
Data grouped into intervals
With tables of intervals, each class mark (midpoint) represents the data in the interval. The mean calculated this way is an estimate: we have lost the individual values. The modal class identifies the most frequent interval in this table of intervals of equal width.
In a study on water consumption in a community, the monthly usage in cubic metres ($m^3$) of several households was recorded, grouped into the following intervals:…
In a study on water consumption in a community, the monthly usage in cubic metres ($m^3$) of several households was recorded, grouped into the following intervals:
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$[0, 10)$: $12$ households
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$[10, 20)$: $25$ households
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$[20, 30)$: $18$ households
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$[30, 40)$: $5$ households
Calculate the mean water consumption and state which is the modal class.
In a medical centre, the cholesterol level of $50$ patients was recorded. The arithmetic mean of the distribution is $194 \text{ mg/dL}$. Two frequencies, $f_1$ and…
In a medical centre, the cholesterol level of $50$ patients was recorded. The arithmetic mean of the distribution is $194 \text{ mg/dL}$. Two frequencies, $f_1$ and $f_2$, have accidentally been erased from the table.
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$[160, 180)$: $10$ patients
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$[180, 200)$: $f_1$ patients
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$[200, 220)$: $f_2$ patients
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$[220, 240)$: $8$ patients
Set up and solve a system of equations to find the absolute frequencies $f_1$ and $f_2$.
Grouping is not the same as rounding
The sets $1,2,11,12$ and $8,9,18,19$ both have two values in $[0,10)$ and two in $[10,20)$. Their marks $5$ and $15$ give the same estimate $10$, but their exact means are $6{,}5$ and $13{,}5$. The table does not allow us to recover a single exact mean.
When the mean misleads: outliers
The mean uses all the data, so an extreme value (an outlier) pulls it towards itself. The median, since it depends only on order, hardly moves.
In a quality control test, the time in seconds that a computer takes to boot up is measured. The following times are obtained: $12, 11, 13, 12, 14, 85$. Calculate the…
In a quality control test, the time in seconds that a computer takes to boot up is measured. The following times are obtained: $12, 11, 13, 12, 14, 85$.
Calculate the mean and the median. Which of the two measures best represents the central tendency of these data? Justify your answer.
In a small tech company, the monthly salaries of its $7$ employees are:…
In a small tech company, the monthly salaries of its $7$ employees are:
$1200 \text{ \euro}, 1250 \text{ \euro}, 1300 \text{ \euro}, 1200 \text{ \euro}, 1400 \text{ \euro}, 1350 \text{ \euro}, 8500 \text{ \euro}$.
Calculate the mean and the median. Write a short text explaining to future employees which of the two measures more realistically represents the typical salary they will earn and why.
Back to the beginning: $2314$ € is not a typical salary: six of the seven workers earn $1400$ € or less. The owner's salary ($8500$ €) acts as an outlier that inflates the mean, while the median ($1300$ €) describes the usual salary of the workforce far more faithfully.