Exercises: Mean and variance

Given the following list of numerical data: $4, 6, 8, 10, 12$. Calculate the arithmetic mean and the variance of the distribution.

The marks obtained by five students in a mathematics exam are $5, 7, 6, 8$ and $9$. Calculate the mean mark and the standard deviation.

In one week, the maximum temperatures recorded from Monday to Thursday were $20$ ºC, $22$ ºC, $x$ ºC and $24$ ºC. Given that the mean temperature for those four days was $22$ ºC, find the unknown value of $x$.

The following bar chart shows the number of siblings of the students in a class. From the graph, calculate the mean number of siblings per student.

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

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.

A group of athletes has a mean weight of $70$ kg with a standard deviation of $7$ kg. Another group of gymnasts has a mean weight of $50$ kg with a standard deviation of $6$ kg. Calculate the coefficient of variation for both groups and state which of the two has a more spread-out weight in relative terms.

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.

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.

On a Year 11 (4.º de ESO) exam marked out of $10$ points, the group mean was $\bar{x} = 5,4$ and the standard deviation was $\sigma = 1,6$. The teacher decides to transform the marks to a scale out of $100$ and add $5$ extra points to everyone for cooperative work, applying the formula $y_i = 10 \cdot x_i + 5$ to each student's mark $x_i$. Upon finding out, Lucía calculates: "Then the new mean will be $\bar{y} = 10 \cdot 5,4 + 5 = 59$ points, and the new standard deviation will be $\sigma_y = 10 \cdot 1,6 + 5 = 21$ points". Identify the conceptual error Lucía made in calculating the dispersion, explain with justification why the $+5$ term does not alter the spread of the data about its mean, and calculate the correct values of the new mean $\bar{y}$, the new standard deviation $\sigma_y$, and the new variance $\sigma_y^2$.

The State Meteorological Agency (AEMET) has recorded the following minimum temperatures (in $^{\circ}\text{C}$) over five consecutive winter days in Soria: $-3$, $-1$, $0$, $2$, and $7$.

1. a) Before carrying out written calculations, make a reasoned estimate: state between which two integer values you intuitively expect the arithmetic mean to lie, and reason whether you consider the standard deviation will be less than $1^{\circ}\text{C}$, between $2^{\circ}\text{C}$ and $5^{\circ}\text{C}$, or greater than $8^{\circ}\text{C}$, looking at the range and apparent dispersion of the data.
2. b) Now calculate the exact value of the mean $\bar{x}$ and the variance $\sigma^2$, and obtain the standard deviation $\sigma$ rounded to the nearest tenth.
3. c) Check whether your initial estimate was correct and explain what practical information the value of $\sigma$ provides about the thermal variability of those days compared to knowing only the mean.

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.

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

• $[20, 30)$: $15$ students

Calculate the mean time taken to get to school and the standard deviation of the distribution.

To compare the dispersion of two different variables in the same group of people, weight and height were measured. The mean weight is $65$ kg with a standard deviation of $5$ kg. The mean height is $170$ cm with a standard deviation of $10$ cm. Which variable shows greater relative dispersion? Justify your answer.

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.

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?

The following histogram shows the time in minutes spent reading per day by several young people. From the graph, extract the frequency table (using the class midpoints) and calculate the mean reading time.

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

In a quick test of $3$ questions, the number of correct answers for a group of people was recorded. It is known that $4$ people had $0$ correct answers, $x$ people had $1$ correct answer, $5$ people had $2$ correct answers and $1$ person had $3$ correct answers. If the mean number of correct answers for the group is $1,2$, determine how many people had exactly $1$ correct answer.

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.

In a precision components factory in Navarra, an operator measures the length (in millimetres) of five metal parts taken from the assembly line and obtains: $501$, $498$, $500$, $503$, and $498$.

To calculate the sample variance, two strategies are available:

• Method A: Directly apply the shortcut formula with the raw data: $\sigma^2 = \frac{\sum x_i^2}{N} - \bar{x}^2$.

• Method B: Define a shifted variable $y_i = x_i - 500$, calculate the variance of $y_i$, and apply the properties of data shifting.

Compare both methods, arguing which is more advantageous if you want to calculate by hand without a calculator, avoiding operations with large numbers and reducing the risk of rounding errors. Then, apply the method you defended to calculate in a reasoned way the mean $\bar{x}$, the variance $\sigma^2$, and the standard deviation $\sigma$ of the length of the parts.

In a survey by the National Statistics Institute (INE) on sports habits, the number of weekly hours dedicated to training by four people in the same group is analysed. Two of them state that they trained for exactly $8$ and $12$ hours. The time for the other two people ($a$ and $b$) is unknown, but the report publishes that the group's mean is $\bar{x} = 10\text{ hours}$ and its variance is $\sigma^2 = 10\text{ hours}^2$.

1. a) From the property of the arithmetic mean, write an equation relating $a$ and $b$, and solve for one of the unknowns in terms of the other.
2. b) Using the variance formula from the squared deviations, $\sigma^2 = \frac{\sum (x_i-\bar{x})^2}{N}$, set up an equation and determine the exact values of $a$ and $b$ (assuming $a \ge b$). Justify with reasoning why it would not have been possible to determine these values having only the mean data.

In a study on daily water consumption (in litres) of a certain population, the data are grouped into three intervals: $[0, 10)$, $[10, 20)$ and $[20, 30)$. It is known that the absolute frequencies of the first two intervals are $5$ and $f$ respectively, and that of the third interval is $3$. If the approximate mean consumption is $13,75$ litres, set up an equation and find the unknown frequency $f$.

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 $\sum_{i=1}^N (x_i - \bar{x}) = 0$.

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).

The arithmetic mean of two unknown positive numbers is $5$ and their variance is $4$. Set up a system of equations using the formulas for the mean and variance, and determine what these two numbers are.

The marks obtained by three students in a project are proportional to their effort and are given by the expressions $a$, $2a$ and $3a$ (where $a > 0$). Knowing that the standard deviation of these three marks is exactly $\sqrt{\frac{2}{3}}$, calculate the value of parameter $a$ and the mean of the marks.

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?

In a small company of $5$ workers, $4$ of them earn exactly $1000$ € a month, while the manager earns $6000$ €. Calculate the mean and the standard deviation of the $5$ salaries. In light of the numerical results obtained, explain with reasons whether the arithmetic mean is a representative measure for this data set.

The following pie chart represents the results of a survey carried out on $40$ people about the number of times they play sport a week (variable $x_i$). Knowing that the full circle is $360^\circ$ and observing the angle of each sector, deduce the absolute frequencies of each value and calculate the variance of the distribution.

Two classes from a secondary school take part in the 4.º de ESO diagnostic assessments. Group 4.º A has $N_A = 20$ students, a mean mark of $\bar{x}_A = 7$, and a variance of $\sigma_A^2 = 2$. Group 4.º B has $N_B = 30$ students, a mean mark of $\bar{x}_B = 5$, and a variance of $\sigma_B^2 = 3$.

1. a) Calculate the overall weighted mean $\bar{x}$ of the $50$ students.
2. b) A teacher suggests that the total variance of the $50$ students is simply the weighted average of the two variances: $\frac{20 \cdot 2 + 30 \cdot 3}{50} = 2,6$. Reason why this approach is conceptually false, explaining what additional source of dispersion appears when the means of both groups do not coincide.
3. c) Reconstruct the sum of squared marks ($\sum x_i^2$) for each group using the relation $\sum x_i^2 = N(\sigma^2 + \bar{x}^2)$, calculate the actual combined variance $\sigma^2$ of the school and its standard deviation (to two decimal places), and verify analytically whether the value obtained is greater or less than the $2,6$ proposed by the teacher.

An INE report on the annual number of leisure trips taken by $10$ families from a region shows a mean of $\bar{x} = 4\text{ trips}$ and a standard deviation of $\sigma = 2\text{ trips}$. After publication, an eleventh family that had not been counted is added to the record.

1. a) First assume that this 11th family took exactly $4$ trips (matching the group mean). Without performing complete numerical calculations, argue whether the new standard deviation will increase, decrease, or remain equal to $2$. Then, prove it numerically by recovering the original sum of squares $\sum x_i^2$ and calculating the new variance.
2. b) Now consider an alternative case in which this 11th family had taken $10$ trips. Analytically reconstruct the new mean $\bar{x}_{nuevo}$ and the new variance $\sigma_{nuevo}^2$ for the set of $11$ families.
3. c) Compare the relative increase experienced by the mean with the one suffered by the variance in this second case, and reason mathematically why the presence of an outlier far from the centre destabilises quadratic dispersion measures much more than the mean.