Mean and variance
Two students have the same maths mean, a $6{,}8$. One always gets similar marks; the other alternates between top marks and fails. The mean on its own cannot tell these two situations apart: we need to complement it with a measure of how far the data lie from it, and to understand exactly what happens to the mean when the data change.
Arithmetic mean with frequencies
Given the following list of numerical data: $4, 6, 8, 10, 12$. Calculate the arithmetic mean and the variance of the distribution.
Given the following list of numerical data: $4, 6, 8, 10, 12$. Calculate the arithmetic mean and the variance of the distribution.
A group of people were asked about the number of times they go to the cinema per month. The results show that $3$ people go $1$ time, $5$ people go $2$ times and $2$…
A group of people were asked about the number of times they go to the cinema per month. The results show that $3$ people go $1$ time, $5$ people go $2$ times and $2$ people go $3$ times. Calculate the mean of this distribution.
A survey was carried out on $40$ families about the number of pets they have at home, obtaining the following results: $10$ families have no pets, $15$ families have $1$…
A survey was carried out on $40$ families about the number of pets they have at home, obtaining the following results: $10$ families have no pets, $15$ families have $1$ pet, $10$ families have $2$ pets, and $5$ families have $3$ pets. Organise the data in a table and calculate the mean, the variance and the standard deviation.
Algebraic properties of the mean
The mean age of a group of $15$ friends is currently $16$ years. What will the mean of their ages be in $4$ years' time? Justify your answer using the properties of the…
The mean age of a group of $15$ friends is currently $16$ years. What will the mean of their ages be in $4$ years' time? Justify your answer using the properties of the arithmetic mean.
Given the data $x_1, x_2, \dots, x_N$, prove algebraically that the sum of their deviations from the arithmetic mean is always equal to zero. That is, show step by step…
Given the data $x_1, x_2, \dots, x_N$, prove algebraically that the sum of their deviations from the arithmetic mean is always equal to zero. That is, show step by step that $\sum_{i=1}^N (x_i - \bar{x}) = 0$.
Variance and standard deviation
In a statistical study with $N = 10$ data values, it is known that the sum of all the data values is $\sum x_i = 50$ and the sum of their squares is $\sum x_i^2 = 300$.…
In a statistical study with $N = 10$ data values, it is known that the sum of all the data values is $\sum x_i = 50$ and the sum of their squares is $\sum x_i^2 = 300$. Calculate the variance of the distribution.
The time (in minutes) taken by $50$ students to get to school from home is grouped in the following frequency table: $[0, 10)$: $10$ students $[10, 20)$: $25$ students…
The time (in minutes) taken by $50$ students to get to school from home is grouped in the following frequency table:
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$[0, 10)$: $10$ students
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$[10, 20)$: $25$ students
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$[20, 30)$: $15$ students
Calculate the mean time taken to get to school and the standard deviation of the distribution.
A teacher calculated that the mean mark of their $20$ students in an exam was $6,5$. However, when reviewing the marking, they realised that they had accidentally…
A teacher calculated that the mean mark of their $20$ students in an exam was $6,5$. However, when reviewing the marking, they realised that they had accidentally recorded a $3$ for a student who had actually scored $8$. Calculate the correct mean mark without needing to add up all the individual marks again.
What happens to the variance under linear changes
The monthly salaries of the workers in a small company have a mean of $1200$ € and a standard deviation of $150$ €. For the coming year, the management decides to apply…
The monthly salaries of the workers in a small company have a mean of $1200$ € and a standard deviation of $150$ €. For the coming year, the management decides to apply an increase of $10\%$ to all salaries and, in addition, add a fixed bonus of $50$ €. Calculate the new mean and the new standard deviation of the salaries.
The maximum daily temperature in a city during a month has a mean of $25$ ºC and a standard deviation of $2$ ºC. If we convert these temperatures to degrees Fahrenheit…
The maximum daily temperature in a city during a month has a mean of $25$ ºC and a standard deviation of $2$ ºC. If we convert these temperatures to degrees Fahrenheit using the formula $F = 1,8 \cdot C + 32$, calculate the coefficient of variation in degrees Fahrenheit and compare it numerically with the original coefficient of variation in degrees Celsius. Does the relative dispersion change when changing scales?
Mean and variance of two combined groups
In a secondary school, class 4A has $20$ students with a mean mark in Mathematics of $6$. Class 4B has $30$ students with a mean mark of $7$. If we calculate the mark of…
In a secondary school, class 4A has $20$ students with a mean mark in Mathematics of $6$. Class 4B has $30$ students with a mean mark of $7$. If we calculate the mark of all the students from both classes combined, what will the overall mean mark be?
A group of $10$ boys has a mean height of $160$ cm and a variance of $20$ cm$^2$. A group of $10$ girls has a mean height of $150$ cm and a variance of $15$ cm$^2$.…
A group of $10$ boys has a mean height of $160$ cm and a variance of $20$ cm$^2$. A group of $10$ girls has a mean height of $150$ cm and a variance of $15$ cm$^2$. Calculate the combined variance of the $20$ students. (Hint: first calculate the overall mean and then the total sum of squares using the variance formulas for each group).
Back to the start: Although both students have a mean of $6{,}8$, the consistent student will have a variance close to $0$ (marks very close together), while the one who alternates extreme marks will have a high variance (large squared distances from the mean); the standard deviation $\sigma$ lets us measure and compare that consistency objectively.