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 $12$ meters with a standard deviation of $1.5$ meters. In an oak forest, the mean height is $8$ meters with a standard deviation of $1.2$ meters. 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 $24\,000$ pesos and the upper quartile is $36\,000$ pesos. 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 cell 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.

In a soccer team, the number of goals scored per match over a season is summarized 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 stores 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.

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 high school, class 4A has $20$ students with a mean grade in statistics of $6$ and a variance of $1.5$. Class 4B has $30$ students with a mean grade 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.