Exercises: Variance and standard deviation

Calculate the variance and the standard deviation of the following data set, 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 marks in mathematics:

• Student A: $6, 7, 6, 8, 7$

• Student B: $4, 9, 5, 10, 6$

Without doing complex calculations, which of the two students do you think has a larger standard deviation? Confirm your answer by calculating the standard deviation of 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 data values is $\sum x_i^2 = 1250$. Using the shortcut 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 one year all the trees will have grown by exactly $15\text{ cm}$, what will the new standard deviation of their heights be? Justify your answer using the properties of measures of dispersion.

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

In a 4th ESO class, the mean mark in a history exam was $6,5$ marks and the coefficient of variation is $20\%$. From these data, calculate the variance of the class's marks.

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

Lucas wants to calculate the variance of the maximum temperatures recorded over four days in his town: $18\text{ }^\circ\mathrm{C}$, $20\text{ }^\circ\mathrm{C}$, $22\text{ }^\circ\mathrm{C}$, and $24\text{ }^\circ\mathrm{C}$. First, he finds the mean, which is $\bar{x} = 21\text{ }^\circ\mathrm{C}$. Next, to measure dispersion, he calculates the distances from each data point to the mean and averages them directly:

Lucas concludes: "Since the result is $0$, the variance is $0\text{ }^\circ\mathrm{C}^2$ and it means there is no dispersion at all; every day had the same temperature."

1. a) Explain what conceptual mistake Lucas made and why the sum of deviations from the mean will always equal zero for any dataset.
2. b) Correct his procedure by calculating the correct variance ($\sigma^2$) and standard deviation ($\sigma$) for these four days, clearly indicating their units.

A municipal public transport company (EMT) analyzes the waiting times (in minutes) of passengers on two urban bus routes over five trips:

• Route A: $8, 9, 10, 11, 12$ minutes.

• Route B: $2, 6, 10, 14, 18$ minutes.

Both routes have a mean waiting time of $\bar{x} = 10$ minutes.

1. a) Before doing any numerical dispersion calculations, observe the two datasets and reasonably estimate which of the two routes will have a larger standard deviation. Is it possible for the standard deviation of Route A to be greater than $2$ minutes? Justify your answer considering what the standard deviation measures.
2. b) Check your estimate by calculating the exact variance and standard deviation for each route.
3. c) Based on the results, explain to users which route provides a more regular and predictable service.

The following table shows the time, in minutes, taken by employees of a factory to reach 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 travel time using the class marks.

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

The scores of four contestants in a talent show 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 values 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 more dispersed heights in relative terms? Justify your answer numerically.

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

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

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

You can do it on paper and send a photo with your answer.

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

The monthly returns of an investment portfolio over the last six months were: $-2\%, 3\%, 1\%, -1\%, 4\%, 1\%$. Calculate the variance and the standard deviation of these returns expressed as a percentage.

To calculate the variance of a dataset, two algebraically equivalent formulas can be used:

• Method 1 (direct formula): $\sigma^2 = \frac{\sum (x_i - \bar{x})^2}{N}$

• Method 2 (shortcut formula): $\sigma^2 = \frac{\sum x_i^2}{N} - \bar{x}^2$

You are presented with two calculation situations in the classroom:

• Situation A: You have $5$ integer grades ($5, 6, 7, 8, 9$) whose mean is exactly $\bar{x} = 7$.

• Situation B: A statistical software has processed the ages of the $40$ participants in a community run, whose mean is $\bar{x} = 34,6\text{ years}$, and directly provides you with the already calculated summations: $\sum x_i = 1384$ and $\sum x_i^2 = 49\,820$.

1. a) Analyze the advantages and disadvantages of each method and justify which one is faster and more convenient to apply in Situation A and which in Situation B (consider aspects such as rounding errors/carrying decimals and the number of operations).
2. b) Apply the chosen method in Situation B to calculate the variance and standard deviation of the ages in the race (round to two decimal places).

Based on historical records from the State Meteorological Agency (AEMET), the annual accumulated precipitation (in liters per square meter, $\text{mm}$) over a decade is compared in two Spanish provincial capitals:

• Bilbao: annual mean precipitation $\bar{x}_1 = 1130\text{ mm}$ with a standard deviation $\sigma_1 = 180\text{ mm}$.

• Almería: annual mean precipitation $\bar{x}_2 = 200\text{ mm}$ with a standard deviation $\sigma_2 = 70\text{ mm}$.

1. a) If someone only looks at the absolute value of the standard deviation, in which city would they state that the annual rainfall varies more from one year to another?
2. b) Explain why directly comparing standard deviations in this case is not appropriate if we want to assess the relative consistency of the climate.
3. c) Calculate the Coefficient of Variation ($CV$) for both cities and interpret the result: in which of the two cities is precipitation proportionally more erratic and unpredictable relative to its usual climate?

In a class of $25$ students, the mean mark in a physics exam is $6,2$ and the standard deviation is $1,4$. When checking the exams, the teacher realizes that a mark of $2$ for a student was entered by mistake, as that student was absent. If that erroneous mark is removed from the data set, what will the new mean and new standard deviation be for 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 full class made up of the $40$ students.

In the following frequency table grouped into intervals, 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 may use the known sums $\sum_{i=1}^n i = \frac{n(n+1)}{2}$ and $\sum_{i=1}^n i^2 = \frac{n(n+1)(2n+1)}{6}$).

Consider two data sets $A$ 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 combining both sets into a single group, the overall variance turns out to be $7$, determine the exact ratio between the number of elements of $A$ and $B$ (that is, calculate the value of the fraction $\frac{N_A}{N_B}$).

Find a set made up of exactly $4$ positive integers (which may repeat) 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 $\sigma_r^2 = 2\text{ cm}^2$. He needs to know the mean surface area of these spheres to calculate the amount of paint required. Knowing that the surface 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, representing the mean squared error with respect to a constant value $k$. Expand the algebraic expression and show that the value of $k$ that minimises this function is the arithmetic mean ($k = \bar{x}$), and that the minimum value attained by the function coincides exactly with the variance $\sigma^2$.

A school counselor records the weekly time (in whole hours) spent on home study by a group of $4$ students in 4th year of ESO. He has misplaced the individual records, but in his final report he noted three facts with certainty:

1. The mean of the group was exactly $\bar{x} = 6\text{ hours}$.

2. The range was $4\text{ hours}$ (the difference between the student who studied the most hours and the one who studied the least was $4$).

3. The standard deviation was exactly $\sigma = \sqrt{2}\text{ hours}$ (therefore, the variance is $\sigma^2 = 2\text{ hours}^2$).

1. a) Deduce step by step the hours spent by each of the four students, knowing that all of them are integers. (Hint: reason about the possible values of the deviations $x_i - \bar{x}$ and the sum of their squares).
2. b) Formally verify that the four values found satisfy the required mean, range, and standard deviation.

In a delivery company in Spain that has $30$ couriers, the average net monthly salary is $\bar{x} = 1400\text{ \euro}$ and the standard deviation is $\sigma = 160\text{ \euro}$. In view of the new collective bargaining agreement, two wage increase proposals are being discussed:

• Proposal A (flat linear increase): Increase each and every worker's salary by $100\text{ \euro}$ per month.

• Proposal B (percentage increase): Apply a $5\,\%$ raise to each worker's salary (which is equivalent to multiplying each salary by $1,05$).

Without needing to know the $30$ individual salaries:

1. a) Calculate the new mean, the new variance, and the new standard deviation of the staff under Proposal A and under Proposal B, justifying your results using the properties of translation and scaling of dispersion.
2. b) Calculate the Coefficient of Variation ($CV$) in the initial situation and in each of the two proposals.
3. c) A union representative states: "The flat increase is fairer because it does not widen the absolute wage gap, whereas the percentage increase makes salaries spread further apart." Justify this statement mathematically by analyzing how the absolute standard deviation and relative dispersion change in both options.