Measures of Spread
One machine packs $500$ g bags of rice with a standard deviation of $5$ g; another packs $1{,}000$ g bags of lentils with a standard deviation of $8$ g. Which of the two is more precise? The lentil machine deviates more in grams, but comparing without taking the size of the bag into account would be a mistake. At the end of the page we'll solve it.
Range
Two groups can have the same mean and still be very different: in one, everyone is alike, and in the other, there's a bit of everything. Measures of spread tell you how far apart the data are.
If the data are grouped into intervals, you no longer know the real minimum and maximum values. The most you can say is that the range is at most the upper end of the last interval minus the lower end of the first.
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…
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 cell phone usage time into the following intervals: $[0, 1)$, $[1, 2)$, $[2, 3)$ and $[3, 4)$ hours. Determine the maximum possible…
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.
Interquartile range
The range is fooled by the extremes. A more robust measure ignores the lowest $25\,\%$ and the highest $25\,\%$, and measures the spread of the middle $50\,\%$ of the data.
In a box-and-whisker plot, the IQR is the length of the box and the range is the distance between the ends of the whiskers.
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.…
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?
Determine the range and the interquartile range of the distribution of scores represented in the following box plot.
Determine the range and the interquartile range of the distribution of scores represented in the following box plot.
Variance and standard deviation
The idea: see how far each data value is from the mean, $x_i-\bar{x}$. If you add them up, they cancel out (some are positive and others negative). That is why they are squared before averaging.
The standard deviation $\sigma$ is in the same units as the data: it is "how far, on average, they stray from the mean." The larger $\sigma$ is, the more spread out the data are.
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…
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 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 $\sum x_i^2 = 580$.…
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.
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…
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.
Variance with tables and intervals
When the data are in a table, each $x_i$ counts $f_i$ times. If they are grouped into intervals, the class mark (midpoint of each interval) is used as the representative value; the result is only an approximation. You can also read the class mark in a histogram: it is the center of each bar.
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:…
In a soccer team, the number of goals scored per match over a season is summarized in the following distribution:
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$0$ goals: $5$ matches
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$1$ goal: $8$ matches
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$2$ goals: $4$ matches
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$3$ 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 $[5, 10)$: $10$…
The time, in minutes, taken by several customers to be served at a supermarket checkout is grouped into the following intervals:
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$[0, 5)$: $4$ customers
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$[5, 10)$: $10$ customers
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$[10, 15)$: $6$ customers
Find the variance and standard deviation of the waiting time.
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…
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.
Coefficient of variation
Is a deviation of 5 g in 500 g bags more "spread out" than 8 g in 1,000 g bags? To compare, you have to make it relative.
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…
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…
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?
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: $40$,…
Two cyclists, Ana and Carlos, have recorded their times (in minutes) in five identical training stages:
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Ana's times: $45$, $48$, $50$, $46$, $47$
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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.
Comparing precision, risk, and graphs
With the CV you can decide which process is more precise or which investment is riskier per unit of mean. You compute $\bar x$, $\sigma$, and $CV$ for each group, and choose the one with the smaller $CV$ as the more consistent.
You can also compare without calculating. If two distributions have the same mean, the one more concentrated around it (a tall, narrow polygon) has the smaller standard deviation; the more spread out one (a low, wide polygon) has the larger.
Now we close the opening problem: $CV_{\text{rice}}=1\,\%$ and $CV_{\text{lentils}}=0.8\,\%$, so the lentil packer is more precise despite its larger $\sigma$ in grams.
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…
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.
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…
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.
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…
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.
Changes of origin and scale
What happens to the spread if all the data increase by the same amount, or are multiplied by a number?
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…
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.
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…
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?
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…
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.
Adding data, correcting errors, and combining groups
If you know $N$, $\bar x$, and $\sigma^2$ for a group, you can recover the sums:
$$\sum x_i=N\bar x \qquad \sum x_i^2=N(\sigma^2+\bar x^2)$$With those sums you can add, remove, or correct data and recompute, without redoing everything.
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…
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…
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?
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,…
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.
Solving backward: the variance as an equation
If the mean or the variance is given and a frequency or a value is missing, the formulas turn into an equation with that unknown.
With $\sum x_i f_i$ and $\sum x_i^2 f_i$ known, the variance is a function of $N$: $\sigma^2=\dfrac{\sum x_i^2f_i}{N}-\left(\dfrac{\sum x_if_i}{N}\right)^2$. If $N$ is the unknown, a quadratic equation may result: check each solution in context.
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…
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:…
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 $N$.
Back to the beginning: $CV_{\text{rice}}=\frac{5}{500}=1\,\%$ and $CV_{\text{lentils}}=\frac{8}{1000}=0.8\,\%$: the lentil packer is more precise, even though its $\sigma$ in grams is larger.