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 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.

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.

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.

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.

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.

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.

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?

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?

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 $N$.