Exercises: Variance and standard deviation

Calculate the variance and the standard deviation of the following set of data, corresponding to the ages of a group of five friends: $15, 16, 15, 17, 16$.

In a survey on the number of pets per household, the following results were obtained: $0$ pets (4 households), $1$ pet (5 households), $2$ pets (1 household). Calculate the variance of this distribution.

Two students have the following grades in math:

• Student A: $6, 7, 6, 8, 7$

• Student B: $4, 9, 5, 10, 6$

Without needing to do complex calculations, which of the two students do you think has a greater standard deviation? Confirm your answer by calculating the standard deviation for both.

In a statistical study with $N = 40$ data values, it is known that the arithmetic mean is $\bar{x} = 5,2$ and that the sum of the squares of all the data values is $\sum x_i^2 = 1250$. Using the short formula, calculate the variance and the standard deviation.

The standard deviation of the heights of a group of trees in a nursery is $12\text{ cm}$. If it is known that in a year's time all the trees will have grown exactly $15\text{ cm}$, what will be the new standard deviation of their heights? Justify your answer using the properties of measures of dispersion.

In a technology company, the variance of the monthly salaries of its employees is $40\,000\text{ \$}^2$. If the management decides to increase all salaries by $5\%$ (that is, multiply them by $1,05$), what will be the new variance of the salaries?

In a 10th grade class, the mean grade on a history exam was $6,5$ points and the coefficient of variation is $20\%$. From this data, calculate the variance of the grades of the class.

A set of data from a computing experiment consists exclusively of $20$ zeros and $30$ ones. Calculate the mean and the variance of this set of data.

The following table shows the time, in minutes, that the employees of a factory take to get to their workplace:

• $[0, 10)$: $12$ employees

• $[10, 20)$: $18$ employees

• $[20, 30)$: $14$ employees

• $[30, 40)$: $6$ employees

Calculate the variance and the standard deviation of the journey time using the class midpoints.

A coffee packaging machine must fill packets with a target weight of $250\text{ g}$. In a quality control check, a sample of $50$ packets was weighed, obtaining a mean of $252\text{ g}$ and a sum of the squares of the weights of $\sum x_i^2 = 3\,175\,450$. Calculate the coefficient of variation and determine if the process is acceptable, knowing that the quality standard requires that $CV < 1,5\%$.

The scores of four participants in a talent contest are $12, 15, x$ and $y$. It is known that the arithmetic mean of the four scores is $14$ and that $x = y$. Find the value of the unknown scores and calculate the standard deviation of the set.

In a basketball team, the mean height of the players is $1,95\text{ m}$ with a standard deviation of $0,08\text{ m}$. On the other hand, in a rhythmic gymnastics team, the mean height is $1,65\text{ m}$ with a standard deviation of $0,07\text{ m}$. Which team has a more dispersed height in relative terms? Justify your answer numerically.

A language school has two B2 level groups. The morning group has $20$ students with a mean grade of $7$ and a variance of $1,5$. The afternoon group has $30$ students with the same mean grade of $7$, but with a variance of $2,5$. Calculate the overall variance of the $50$ B2 level students combined.

The following bar chart represents the number of books read during the summer by a group of students.

From the graph, extract the frequencies corresponding to each value and calculate the standard deviation of the number of books read.

A biological experiment has been carried out measuring the length of some leaves, obtaining a variable $X$ with mean $\bar{x} = 12\text{ cm}$ and standard deviation $\sigma = 4\text{ cm}$. To facilitate calculations in a software program, a new "standardised" variable $Z = \frac{X - 12}{4}$ is defined. Prove numerically or by applying the properties of the mean and variance what the values of the mean and variance of the new variable $Z$ are.

The monthly returns of an investment portfolio during the last semester have been: $-2\%, 3\%, 1\%, -1\%, 4\%, 1\%$. Calculate the variance and the standard deviation of these returns expressed as a percentage.

In a class of $25$ students, the mean grade on a physics exam is $6,2$ and the standard deviation is $1,4$. When reviewing the exams, the teacher realizes that the grade of one student, which was a $2$, had been recorded by mistake as that student did not take the exam. If this erroneous grade is removed from the data set, what will be the new mean and the new standard deviation of the remaining $24$ students?

Two groups of students are combined to form a single class for a school trip. Group 1 has $15$ students with a mean height of $160\text{ cm}$ and a variance of $25\text{ cm}^2$. Group 2 has $25$ students with a mean height of $168\text{ cm}$ and a variance of $16\text{ cm}^2$. Calculate the variance of the height of the whole class formed by the $40$ students.

In the following grouped frequency table, two values are missing ($f_1$ and $f_2$):

• $[0, 20)$: $f_1$

• $[20, 40)$: $10$

• $[40, 60)$: $f_2$

• $[60, 80)$: $5$

It is known that the total number of data values is $N = 40$ and that the arithmetic mean is $\bar{x} = 37,5$. Calculate the values of the frequencies $f_1$ and $f_2$, and then find the variance of the complete distribution.

Show that the variance of the first $n$ natural numbers $\{1, 2, 3, \dots, n\}$ is given by the formula $\sigma^2 = \frac{n^2 - 1}{12}$. Apply this formula to quickly calculate the standard deviation of the integers from $1$ to $25$.

(Note: you can use the known sums $\sum_{i=1}^n i = \frac{n(n+1)}{2}$ and $\sum_{i=1}^n i^2 = \frac{n(n+1)(2n+1)}{6}$).

There are two sets of data $A$ and $B$. Set $A$ has $N_A$ elements, mean $10$ and variance $4$. Set $B$ has $N_B$ elements, mean $10$ and variance $9$. If, when joining both sets into a single group, the overall variance turns out to be $7$, determine the exact proportion between the number of elements in $A$ and in $B$ (that is, calculate the value of the fraction $\frac{N_A}{N_B}$).

Find a set consisting of exactly $4$ positive integers (which can be repeated) that simultaneously satisfy the following three conditions:

• Their arithmetic mean is $5$.

• Their median is $4,5$.

• Their variance is $4,5$.

A researcher measures the radius $r$ of a sample of $10$ metal spheres, obtaining a mean of $\bar{r} = 5\text{ cm}$ and a variance of $\sigma_r^2 = 2\text{ cm}^2$. They need to know the mean surface area of these spheres to calculate the amount of paint needed. Knowing that the area of a sphere is $A = 4\pi r^2$, and using the definition of variance $\sigma_r^2 = \frac{\sum r_i^2}{N} - \bar{r}^2$, calculate the arithmetic mean of the surface areas of the $10$ spheres.

Given a data distribution $x_1, x_2, \dots, x_N$, the function $E(k) = \frac{1}{N} \sum_{i=1}^N (x_i - k)^2$ is defined, which represents the mean squared error with respect to a constant value $k$. Expand the algebraic expression and show that the value of $k$ that minimizes this function is the arithmetic mean ($k = \bar{x}$), and that the minimum value reached by the function coincides exactly with the variance $\sigma^2$.