Exercises: Measures of dispersion

The maximum temperatures recorded in a city during a winter week were: $12$, $14$, $9$, $11$, $15$, $10$ and $13$ degrees Celsius. Calculate the range of these temperatures.

A student's marks in five mathematics exams are $5$, $6$, $7$, $8$ and $9$. Knowing that the arithmetic mean is $7$, calculate the variance and standard deviation of their marks.

The mean weight of an industrially manufactured model of screw is $15$ grams, with a standard deviation of $0,45$ grams. Calculate the coefficient of variation and express it as a percentage.

In a pine forest, the mean height is $12$ metres with a standard deviation of $1,5$ metres. In an oak forest, the mean height is $8$ metres with a standard deviation of $1,2$ metres. Which tree species shows greater relative dispersion in its height?

In a statistical study on the monthly salaries of a company, it has been determined that the lower quartile is $1200$ euros and the upper quartile is $1800$ euros. What is the interquartile range of the salaries?

In a statistical study made up of $20$ data values, it is known that the sum of all values is $\sum x_i = 100$ and the sum of their squares is $\sum x_i^2 = 580$. Calculate the variance of the distribution.

The standard deviation of the ages of a group of friends is currently $2,5$ years. If in $4$ years' time we calculate the standard deviation of their ages again, what will the new value be? Justify your answer with reasons.

A frequency table groups the daily mobile phone usage time into the following intervals: $[0, 1)$, $[1, 2)$, $[2, 3)$ and $[3, 4)$ hours. Determine the maximum possible range of the original data that have been grouped in this table.

Lucía and Marcos analyze the results of a Mathematics exam in their 4th-year ESO class. The mean grade was $6,5$ and the standard deviation was $1,2$. To reward the submission of a voluntary project, the teacher decides to add $0,8$ points to the final grade of each student in the class.

Marcos claims: "Since $0,8$ is added to everyone, the new mean will be $7,3$ and the new standard deviation will be $1,2 + 0,8 = 2,0$, because all the grades are higher now."

1. a) Identify the mistake made by Marcos and explain with reasoning why dispersion does not change when adding a constant to all data points.
2. b) What would happen to the standard deviation if the teacher, instead of adding points, decided to multiply all original grades by $1,1$ (a $10\,\%$ increase)? Justify your answer by calculating the new value of $\sigma$.

Consider two groups of five students, $A$ and $B$, who have obtained the following marks in an exam from $0$ to $10$ points:

• Group $A$: $0$, $5$, $5$, $5$, $10$

• Group $B$: $0$, $1$, $5$, $9$, $10$

Notice that both groups have the same mean ($\bar{x} = 5$) and the same range ($10 - 0 = 10$).

1. a) Before calculating any variance, estimate with justification which of the two groups will have a greater standard deviation. Base your prediction on the meaning of standard deviation as the average of the distances from the data to the mean.
2. b) Calculate the variance and standard deviation of each group using the formula for deviations $\sigma = \sqrt{\frac{\sum (x_i - \bar{x})^2}{N}}$ and check whether your initial estimate was correct.

In a football team, the number of goals scored per match over a season is summarised in the following distribution:

• $0$ goals: $5$ matches

• $1$ goal: $8$ matches

• $2$ goals: $4$ matches

• $3$ goals: $3$ matches

Calculate the variance and the standard deviation of the number of goals per match.

The time, in minutes, taken by several customers to be served at a supermarket checkout is grouped into the following intervals:

• $[0, 5)$: $4$ customers

• $[5, 10)$: $10$ customers

• $[10, 15)$: $6$ customers

Find the variance and standard deviation of the waiting time.

Two cyclists, Ana and Carlos, have recorded their times (in minutes) in five identical training stages:

• Ana's times: $45$, $48$, $50$, $46$, $47$

• Carlos's times: $40$, $55$, $42$, $58$, $41$

Calculate the coefficient of variation for each of them and determine who is more consistent in their times.

Determine the range and the interquartile range of the distribution of scores represented in the following box plot.

The prices of a tech product in several shops in the United States have a standard deviation of $15$ dollars. If for an economic study we convert all prices to euros using a fixed exchange rate of $0,90$ euros per dollar, what will the variance and standard deviation of the prices expressed in euros be?

In a sample consisting of three data values, it is known that two of them are $4$ and $8$. Knowing that the arithmetic mean of the three data values is $6$, find the value of the third data value and calculate the sample variance.

Calculate the variance and standard deviation of the data represented in the following histogram, knowing that the horizontal axis represents age in years and the vertical axis the number of people. Use the class midpoints for your calculations.

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

A packaging machine A produces bags of rice with a mean weight of $500$ grams and a standard deviation of $5$ grams. Another machine B produces bags of lentils with a mean weight of $1000$ grams and a standard deviation of $8$ grams. Using the coefficient of variation, determine which machine is comparatively more precise in its packaging process.

Based on data published by the National Statistics Institute (INE) in its Annual Structure of Earnings Survey, the net monthly wages of two professional sectors in Spain are studied:

• Hospitality Sector: mean wage of $1250$ € with a standard deviation of $200$ €.

• IT Programming and Consulting Sector: mean wage of $2600$ € with a standard deviation of $390$ €.

1. a) State in which of the two sectors wages show a greater absolute dispersion and which statistical parameter allows you to state this directly.
2. b) A labour analyst states: "Wages in the IT sector are much more unequal than in hospitality because their standard deviation is almost double." Use the coefficient of variation ($CV$) to determine whether this statement is correct and explain in which of the two sectors there is greater relative dispersion in relation to its mean wage level.

You want to design a sample of four integer marks (values from $1$ to $10$) corresponding to four students in a working group, such that they simultaneously satisfy the following conditions:

1. The arithmetic mean must be exactly $6$.

2. The range must be exactly $4$.

3. The variance must be the lowest possible among all combinations satisfying the previous two conditions.

1. a) Find, with reasoning, the four marks that make up this sample, explaining why the central values must be placed as close as possible to the mean.
2. b) Calculate the variance and standard deviation of the sample you proposed.
3. c) Propose a different sample of four integer marks that still has a mean of $6$ and a range of $4$, but whose variance is strictly greater than the one obtained in the previous section, calculating its new value to justify it.

A statistical data set consisting of $10$ values has an arithmetic mean of $5$ and a variance of $2$. If a new value of $16$ is added to this set, calculate the new arithmetic mean and the new variance of the resulting $11$ values.

In a secondary school, class 4A has $20$ students with a mean mark in statistics of $6$ and a variance of $1,5$. Class 4B has $30$ students with a mean mark also of $6$ and a variance of $2$. If the students from both classes are combined into a single group, what will be the overall variance of the $50$ students?

The temperature in a city has been measured daily in degrees Celsius ($C$), obtaining a standard deviation of $5^\circ C$. Knowing that the mathematical formula to convert degrees Celsius to degrees Fahrenheit ($F$) is $F = \frac{9}{5}C + 32$, algebraically prove what the variance of the temperatures will be if all measurements are expressed in degrees Fahrenheit, and calculate its final value.

A discrete statistical variable takes the values $0$, $1$ and $2$ with absolute frequencies $2$, $k$ and $2$, respectively. Knowing that the arithmetic mean of the distribution is $1$, set up an equation to determine the value of the frequency $k$ such that the variance of the distribution is exactly $0,5$.

A statistical study has been carried out on a variable and it is known that the sample size is $N$. Furthermore, the following aggregated values have been obtained: $\sum x_i f_i = 40$ and $\sum x_i^2 f_i = 200$. If the variance of the distribution is $4$, determine the sample size $N$.

Observe the frequency polygons of distributions A and B shown in the graph below. Both distributions have the same arithmetic mean. Which of the two distributions has a larger standard deviation? Justify your answer by analysing the concentration of the data relative to the mean, without performing any numerical calculations.

When calculating the variance of a set of $10$ data values, a value of $4$ was obtained, with the arithmetic mean equal to $5$. Subsequently, upon checking the records, it was discovered that a value whose true value was $3$ had been incorrectly recorded as $8$. Calculate the correct variance of the distribution after correcting the error.

An investor is analysing the risk of two investment funds. The Alpha fund has obtained annual returns of $2\%$, $4\%$, $6\%$ and $8\%$, all with the same absolute frequency. The Beta fund has obtained returns of $4\%$, $5\%$, $5\%$ and $6\%$. Calculate the standard deviation of each fund and explain, using the coefficient of variation, which of the two carries a higher relative risk per unit of return obtained.

To calculate the variance of a statistical distribution, there are two algebraically equivalent expressions:

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

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

Imagine that you must solve the following two cases by hand and without a calculator:

• Case $A$: Five extremely high-precision measurements taken in a laboratory: $1002$, $1003$, $1005$, $1006$, and $1009$ mm.

• Case $B$: A frequency distribution where the mean is fractional, $\bar{x} = \frac{10}{3} = 3,333\dots$, with $N = 12$ observations and where the accumulated calculation $\sum x_i^2 f_i = 150$ is already available.

1. a) Reason which method is much more efficient and reliable for Case $A$. Explain what numerical difficulties would arise from using the other method by hand.
2. b) Justify which method is ideal for Case $B$ and what rounding drawback you would avoid.
3. c) Apply the method you selected in section a) to calculate the mean, variance, and standard deviation of Case $A$.

In a secondary school, two 4th year ESO groups take the same common Mathematics test:

• Group $4.º\text{ A}$ ($N_A = 20$ students): mean $\bar{x}_A = 6,0$ and standard deviation $\sigma_A = 1,0$.

• Group $4.º\text{ B}$ ($N_B = 30$ students): mean $\bar{x}_B = 8,0$ and standard deviation $\sigma_B = 1,0$.

The teaching team wants to determine the overall mean and standard deviation of all $50$ students combined.

1. a) A student claims: "Since both groups have identical standard deviation ($\sigma = 1,0$), the standard deviation of all $50$ students must remain $1,0$". Explain with reasoning why this conclusion is incorrect, identifying what dispersion factor arises when merging two groups with different means that the student has not taken into account.
2. b) Reconstruct, from the formulas for the mean and variance, the sum of the marks $\sum x_i$ and the sum of the squares of the marks $\sum x_i^2$ for each class.
3. c) Calculate the mean and standard deviation of all $50$ students combined. Compare the result obtained with the student's prediction from section a).