Mean and Standard Deviation
Ximena and Diego have the same average in math: $6.8$. Ximena always gets similar grades; Diego alternates between tens and fives. The average alone can't tell these two situations apart. We also need a measure of how far the data spread from the mean.
Mean, median, and mode
The median ($Me$) is the middle value of the ordered data. The mode ($Mo$) is the value that appears most often (in the pets table, $Mo=1$, the one with the highest frequency).
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 $2$…
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.
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…
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 $x$.
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 $1$…
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.
Grouped data and incomplete data
When data come in intervals (or in a histogram), we don't know the exact value of each one. We use the class mark as the representative of the interval.
If a frequency is missing, set up an equation with the mean formula and solve for it. For example, if the mean number of correct answers is $1.2$ in a group where $4$ people got $0$, $x$ people got $1$, $5$ people got $2$, and $1$ person got $3$: $\dfrac{x+10+3}{10+x}=1.2$.
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…
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 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…
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.
In a study on daily water consumption (in liters) of a certain population, the data are grouped into three intervals: $[0, 10)$, $[10, 20)$ and $[20, 30)$. It is known…
In a study on daily water consumption (in liters) 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 $13.75$ liters, set up an equation and find the unknown frequency $f$.
Properties of the mean
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…
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.
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…
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.
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…
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 $\sum_{i=1}^N (x_i - \bar{x}) = 0$.
Variance and standard deviation
Since the deviations cancel out, we square them before averaging.
If $N=10$, $\sum x_i=50$, and $\sum x_i^2=300$: $\bar{x}=5$ and $\sigma^2=30-25=5$.
In practice, a scientific calculator (statistics mode) or a spreadsheet gives you $\bar{x}$ and $\sigma$ directly. Be careful: choose $\sigma$ (population), not $s$.
Given the following list of numerical data: $4, 6, 8, 10, 12$. Calculate the arithmetic mean and the variance of the distribution.
Given the following list of numerical data: $4, 6, 8, 10, 12$. Calculate the arithmetic mean and the variance of the distribution.
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 $\sum x_i^2 = 300$.…
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.
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…
The time (in minutes) taken by $50$ students to get to school from home is grouped in the following frequency table:
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$[0, 10)$: $10$ students
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$[10, 20)$: $25$ students
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$[20, 30)$: $15$ students
Calculate the mean time taken to get to school and the standard deviation of the distribution.
Linear changes and relative spread
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…
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.
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…
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 $24\,000$ pesos and a standard deviation of $3\,000$ pesos. For the coming year, the management…
The monthly salaries of the workers in a small company have a mean of $24\,000$ pesos and a standard deviation of $3\,000$ pesos. For the coming year, the management decides to apply an increase of $10\%$ to all salaries and, in addition, add a fixed bonus of $1\,000$ pesos. Calculate the new mean and the new standard deviation of the salaries.
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…
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 $F = 1.8 \times C + 32$, 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?
Mean and variance of two groups
In a high school, class 4A has $20$ students with a mean grade in math of $6$. Class 4B has $30$ students with a mean grade of $7$. If we calculate the grade of all the…
In a high school, class 4A has $20$ students with a mean grade in math of $6$. Class 4B has $30$ students with a mean grade of $7$. If we calculate the grade of all the students from both classes combined, what will the overall mean grade be?
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$^2$.…
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…
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.
Interpreting: is the mean representative?
A mean or a standard deviation is only useful if it's interpreted. If the standard deviation is large compared with the mean, the mean describes the group poorly.
Other exercises ask you to set up systems using the mean and variance formulas, or to read data from a pie chart: the angle of each sector divided by $360^\circ$ gives the fraction of $N$ it represents.
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.
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.
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)…
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.
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…
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.
In a small company of $5$ workers, $4$ of them earn exactly $20\,000$ pesos a month, while the manager earns $120\,000$ pesos. Calculate the mean and the standard…
In a small company of $5$ workers, $4$ of them earn exactly $20\,000$ pesos a month, while the manager earns $120\,000$ pesos. 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 circle graph represents the results of a survey carried out on $40$ people about the number of times they play sports a week (variable $x_i$). Knowing that…
The following circle graph represents the results of a survey carried out on $40$ people about the number of times they play sports 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.
Back to the beginning: Ximena and Diego both have a mean of $6.8$, but Ximena's variance is close to $0$ (grades very close together) and Diego's is high (tens and fives far from the mean). The standard deviation lets us compare that consistency.