Basic Statistics
Ximena studied 10 hours for the exam and got a 9. Diego studied 2 hours and got a 6. With only two students you can't conclude anything, but with the whole class you can: does studying more raise your grade, and how strong is that relationship? To answer, you need to measure two variables at once. First, you have to master the one-variable tools rigorously, because they are the building blocks of everything that comes after.
Mean, variance, and standard deviation
The mean minimizes the squared error
For any constant $c$, adding and subtracting $\bar x$ inside the square gives:
$$\sum (x_i-c)^2=\sum (x_i-\bar x)^2+n(\bar x-c)^2$$The cross term vanishes because $\sum(x_i-\bar x)=0$. The second summand is never negative and equals zero only when $c=\bar x$. That is why $f(c)=\sum (x_i-c)^2$ reaches its minimum at $c=\bar x$.
With derivatives you arrive at the same result: $f'(c)=-2\sum(x_i-c)=0$ gives $c=\bar x$, and $f''(c)=2n>0$ confirms it is a minimum. Later on, the regression line is also defined by "minimizing squared errors."
An agricultural engineer is calibrating a soil moisture sensor. The last 5 readings, as percentages, are: $22, 24, 23, 25, 21$. Calculate the arithmetic mean and the…
An agricultural engineer is calibrating a soil moisture sensor. The last 5 readings, as percentages, are: $22, 24, 23, 25, 21$. Calculate the arithmetic mean and the population standard deviation of these measurements.
A cycling team recorded the distances covered in their last 4 training sessions: $45, 50, x, 60$ kilometers. Given that the mean distance of the 4 training sessions is…
A cycling team recorded the distances covered in their last 4 training sessions: $45, 50, x, 60$ kilometers. Given that the mean distance of the 4 training sessions is exactly $52.5$ km, first determine the missing training session value ($x$) and then calculate the population standard deviation of the 4 training sessions.
Given a univariate dataset $x_1, x_2, \dots, x_n$, the squared error function $f(c) = \sum_{i=1}^n (x_i - c)^2$ is defined, measuring the sum of squared distances from…
Given a univariate dataset $x_1, x_2, \dots, x_n$, the squared error function $f(c) = \sum_{i=1}^n (x_i - c)^2$ is defined, measuring the sum of squared distances from each data point to a constant value $c$. Prove analytically (using derivatives or completing the square) that the value of $c$ that minimises this function is exactly the arithmetic mean $\bar{x}$.
Combining samples and correcting data
Sometimes you don't have the individual data, only means and variances. From the variance formula, the mean of the squares is $\sigma^2+\bar x^2$. With that quantity you can reconstruct almost everything.
A quality control analyst observes manufacturing errors in 10 production batches. The mean number of errors is $6.5$ and the variance is $2.25$. Later, during an audit,…
A quality control analyst observes manufacturing errors in 10 production batches. The mean number of errors is $6.5$ and the variance is $2.25$. Later, during an audit, they notice that a typographical error recorded $2$ errors instead of $8$ for one of the batches. Calculate the corrected new mean and variance without needing to know the individual data for the other 9 batches.
Two independent epidemiological studies on body mass index (BMI) were carried out in a region. The first study, with a sample of $40$ people, obtained a mean of $22$ and…
Two independent epidemiological studies on body mass index (BMI) were carried out in a region. The first study, with a sample of $40$ people, obtained a mean of $22$ and a variance of $4$. The second study, with a sample of $60$ people, obtained a mean of $25$ and a variance of $5$. If both studies are combined into a single pooled sample of $100$ people, calculate the mean and variance of this combined sample.
Coefficient of variation
Is a $150$ cm fish with $\sigma=12$ cm more "irregular" than a $40$ cm fish with $\sigma=4$ cm? The standard deviation alone is no good for comparing, because it depends on the units and on the size of the data.
A marine biologist compares the length of two fish species. Species Alpha has a mean length of $40$ cm with a standard deviation of $4$ cm. Species Beta has a mean…
A marine biologist compares the length of two fish species. Species Alpha has a mean length of $40$ cm with a standard deviation of $4$ cm. Species Beta has a mean length of $150$ cm with a standard deviation of $12$ cm. Calculate the coefficient of variation for both species and state which one shows greater relative dispersion.
Two aeronautical component factories produce parts whose lifespan is a statistical variable. Factory Alpha produces components with a mean lifespan of $1200$ hours and a…
Two aeronautical component factories produce parts whose lifespan is a statistical variable. Factory Alpha produces components with a mean lifespan of $1200$ hours and a variance of $14400$ hours$^2$. Factory Beta produces components with a mean lifespan of $800$ hours and a variance of $10000$ hours$^2$. By calculating a suitable measure of relative dispersion, justify which factory has a more homogeneous production.
Position and grouped data
Quartiles divide the ordered data into four equal parts. $Q_1$ leaves $25\%$ below it, the median $50\%$, and $Q_3$ $75\%$. The mode is the value with the highest frequency.
If $n$ is odd, the median is not included in either half when calculating $Q_1$ and $Q_3$.
Data grouped in intervals
When the data come in intervals $[a,b)$, each interval is represented by its class mark $\frac{a+b}{2}$. The midpoint is used because, since the exact data are unknown, the errors from overestimating and underestimating tend to cancel out.
The scores obtained by the 9 members of a rhythmic gymnastics team in a flexibility test are: $16, 17, 16, 18, 16, 17, 19, 16, 17$. Determine the mode, the median and…
The scores obtained by the 9 members of a rhythmic gymnastics team in a flexibility test are: $16, 17, 16, 18, 16, 17, 19, 16, 17$. Determine the mode, the median and the lower quartile ($Q_1$) of this dataset.
At a weather station, the maximum daily wind speed ($X$, in km/h) has been grouped into the following intervals: $[0, 10)$, $[10, 20)$, $[20, 30)$, $[30, 40)$, with…
At a weather station, the maximum daily wind speed ($X$, in km/h) has been grouped into the following intervals: $[0, 10)$, $[10, 20)$, $[20, 30)$, $[30, 40)$, with absolute frequencies $5, 12, 8, 5$ respectively. Using the class midpoints of each interval, calculate the arithmetic mean and variance of the wind speed.
Two-way tables
When you measure $X$ and $Y$ on the same individuals, each individual gives a pair $(x_i,y_j)$. The two-way table puts the values of $X$ in the rows, the values of $Y$ in the columns, and in each cell the joint frequency $n_{ij}$: how many times that pair appears.
In a maternity ward, the weight in kg ($X$) and length in cm ($Y$) of 6 newborn babies were measured, obtaining the following pairs of values $(x_i, y_i)$: $(3.2; 50)$,…
In a maternity ward, the weight in kg ($X$) and length in cm ($Y$) of 6 newborn babies were measured, obtaining the following pairs of values $(x_i, y_i)$: $(3.2; 50)$, $(3.5; 52)$, $(3.2; 51)$, $(3.8; 52)$, $(3.5; 50)$ and $(3.2; 50)$. Construct the two-way table (joint absolute frequencies) for these two variables.
You can do it on paper and send a photo with your answer.
In a sociological study on reading habits, 50 people were surveyed about their sex ($X$: $1$ for Male, $2$ for Female) and whether they read more than one book a month…
In a sociological study on reading habits, 50 people were surveyed about their sex ($X$: $1$ for Male, $2$ for Female) and whether they read more than one book a month ($Y$: $1$ for Yes, $2$ for No). The joint absolute frequency table $n_{ij}$ is as follows: $n_{11} = 10$, $n_{12} = 15$, $n_{21} = 20$, $n_{22} = 5$. Construct the marginal distribution table for the variable $Y$ and calculate its relative frequencies.
You can do it on paper and send a photo with your answer.
Marginal and conditional distributions
From a joint frequency table of two variables $X$ (number of siblings) and $Y$ (number of pets), it is known that the marginal absolute frequency of $X=1$ is…
From a joint frequency table of two variables $X$ (number of siblings) and $Y$ (number of pets), it is known that the marginal absolute frequency of $X=1$ is $n_{1\bullet} = 40$. Of those 40 people, 15 have $Y=0$ pets, 20 have $Y=1$ pet and 5 have $Y=2$ pets. Calculate the relative frequency distribution of the variable $Y$ conditional on $X=1$.
A veterinary clinic classifies the dogs seen in a month according to their size ($X$: $1$ = Small, $2$ = Medium, $3$ = Large) and the number of visits to the vet in the…
A veterinary clinic classifies the dogs seen in a month according to their size ($X$: $1$ = Small, $2$ = Medium, $3$ = Large) and the number of visits to the vet in the last year ($Y$: $1, 2, 3$). The joint frequency table $n_{ij}$ is:
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For $X=1$: $n_{11}=12$, $n_{12}=8$, $n_{13}=5$
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For $X=2$: $n_{21}=10$, $n_{22}=15$, $n_{23}=10$
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For $X=3$: $n_{31}=4$, $n_{32}=6$, $n_{33}=10$
Calculate the arithmetic mean and the variance of the marginal distribution of the number of visits ($Y$).
In an agronomy study, the amount of fertiliser applied in kg/ha ($X$) and the crop yield in metric tons/ha ($Y$) are analysed. The following two-way table for the joint…
In an agronomy study, the amount of fertiliser applied in kg/ha ($X$) and the crop yield in metric tons/ha ($Y$) are analysed. The following two-way table for the joint frequencies $n_{ij}$ is available:
| $X$ \ $Y$ | $2$ | $4$ | $6$ |
|---|---|---|---|
| 10 | $1$ | $2$ | $0$ |
| 20 | $0$ | $3$ | $1$ |
| 30 | $0$ | $1$ | $2$ |
Calculate the mean and standard deviation of the crop yield conditional on $20$ kg/ha of fertiliser having been applied (that is, from the distribution $Y | X=20$).
Statistical independence
Two qualitative variables are analysed in a sample of $200$ tourists: continent of origin ($X$: European, American, Asian) and type of accommodation ($Y$: Hotel,…
Two qualitative variables are analysed in a sample of $200$ tourists: continent of origin ($X$: European, American, Asian) and type of accommodation ($Y$: Hotel, Apartment). The marginal relative frequencies are: European $50\%$, American $30\%$, Asian $20\%$. For accommodation: Hotel $60\%$, Apartment $40\%$. Assuming that both variables are statistically independent, how many tourists in the sample are expected to be American and stay in an apartment at the same time?
In a study on the relationship between the number of weekly hours spent playing sports ($X$) and the level of work stress ($Y$), the following joint absolute frequency…
In a study on the relationship between the number of weekly hours spent playing sports ($X$) and the level of work stress ($Y$), the following joint absolute frequency table is obtained, where $k$ is an unknown value:
| $X$ \ $Y$ | $2$ | $5$ | $8$ |
|---|---|---|---|
| 0 | $1$ | $2$ | $3$ |
| 3 | $2$ | $4$ | $k$ |
| 6 | $1$ | $2$ | $3$ |
Mathematically determine the value of $k$ so that the variables $X$ and $Y$ are strictly statistically independent.
Scatter plot
For two numerical variables, each pair $(x_i,y_i)$ is drawn as a point in the Cartesian plane. The cloud of points shows whether there is a relationship and what kind.
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Direct (positive): as $X$ increases, $Y$ tends to increase.
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Inverse (negative): as $X$ increases, $Y$ tends to decrease.
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Strong: the points hug a line. Weak: the cloud is wide and scattered.
To calculate means from the graph, read the exact coordinates of each point (at the grid intersections) and build the table.
Simpson's paradox
Sometimes a trend that appears in every subgroup reverses when the data are combined. Look at two departments: within each one, more experience is associated with lower income; but department B has, on average, more experience and higher incomes than A.
With all the points together, the overall correlation would be positive (dotted line); separately, each department's is negative. The cause is a hidden third variable, the department. That's why you should always look at the graph before drawing conclusions.
Look at the following scatter plot relating weekly training hours ($X$) and time in seconds in a 100-meter sprint ($Y$) for a group of athletes. State whether the…
Look at the following scatter plot relating weekly training hours ($X$) and time in seconds in a 100-meter sprint ($Y$) for a group of athletes. State whether the correlation between both variables is positive or negative, and reason visually whether the linear dependence appears to be strong or weak.
The following scatter plot represents the price in thousands of pesos ($Y$) against age in years ($X$) of a specific model of used van. From the points shown exactly on…
The following scatter plot represents the price in thousands of pesos ($Y$) against age in years ($X$) of a specific model of used van. From the points shown exactly on the grid intersections of the graph, construct the bivariate data table and calculate the marginal mean age $\bar{x}$ and the marginal mean price $\bar{y}$.
The following scatter plot shows the relationship between years of experience ($X$) and annual income in thousands of pesos ($Y$) for a group of professionals, divided…
The following scatter plot shows the relationship between years of experience ($X$) and annual income in thousands of pesos ($Y$) for a group of professionals, divided into two different departments (Department A in blue and Department B in green). Explain the concept of Simpson's paradox based on what you would observe if you calculated the overall correlation of all points combined, versus the local correlations for each department separately.
Covariance
Covariance measures whether $X$ and $Y$ move together. It is the mean of the products of the deviations from the means:
Reconstructing a table using the covariance
The human resources department of an engineering firm has collected data on the years of experience ($X$) and monthly salary in thousands of pesos ($Y$) of its…
The human resources department of an engineering firm has collected data on the years of experience ($X$) and monthly salary in thousands of pesos ($Y$) of its employees. Given that the mean experience is $\bar{x} = 5$ years, the mean salary is $\bar{y} = 2.5$ thousand pesos, the sample size is $N = 20$ and the sum of cross-products is $\sum_{i=1}^{20} x_i y_i = 280$, calculate the covariance $\sigma_{xy}$.
The following scatter plot shows the relationship between the force applied to a spring in Newtons ($X$) and its length in centimeters ($Y$). Given that the exact pairs…
The following scatter plot shows the relationship between the force applied to a spring in Newtons ($X$) and its length in centimeters ($Y$). Given that the exact pairs of values are $(1, 3)$, $(2, 5)$, $(3, 7)$ and $(4, 9)$, calculate the covariance $\sigma_{xy}$ and check analytically whether there is an exact functional relationship between both variables.
A bivariate table of two variables $X$ and $Y$ taking the values $X \in \{1, 2\}$ and $Y \in \{0, 1\}$ is given. The following overall statistical data are known: the…
A bivariate table of two variables $X$ and $Y$ taking the values $X \in \{1, 2\}$ and $Y \in \{0, 1\}$ is given. The following overall statistical data are known: the sample size is $N=100$, the marginal mean of $X$ is $\bar{x} = 1.4$, the marginal mean of $Y$ is $\bar{y} = 0.6$, and the covariance is $\sigma_{xy} = 0.06$. Using the definitions of these parameters, completely reconstruct the joint absolute frequency table $n_{ij}$.
Properties of covariance and of a sum
Changes of scale and origin
If $U=aX+b$ and $V=cY+d$, then $u_i-\bar u=a(x_i-\bar x)$ and $v_i-\bar v=c(y_i-\bar y)$. Multiplying and averaging:
$$\sigma_{uv}=ac\,\sigma_{xy}$$Adding constants ($b$ and $d$) does not change the covariance: it only shifts the cloud. Multiplying by $a$ and $c$ rescales it. If $a$ and $c$ have opposite signs, the covariance changes sign.
Mean and variance of a sum
$$\bar z=\bar x+\bar y\qquad \text{Var}(X+Y)=\sigma_x^2+\sigma_y^2+2\sigma_{xy}$$This comes from expanding $(x_i-\bar x+y_i-\bar y)^2$. If $\sigma_{xy}=0$, the variances simply add.
Back to the beginning: to find out whether studying more raises your grade, measure hours ($X$) and grade ($Y$), draw the cloud, and calculate $\sigma_{xy}$. If it is positive and the cloud hugs a line, studying more goes hand in hand with better grades. With the correlation coefficient in the next section you'll be able to say how strong that relationship is.
Let $X$ and $Y$ be two bivariate statistical variables with means $\bar{x}, \bar{y}$, standard deviations $\sigma_x, \sigma_y$ and covariance $\sigma_{xy}$. If a linear…
Let $X$ and $Y$ be two bivariate statistical variables with means $\bar{x}, \bar{y}$, standard deviations $\sigma_x, \sigma_y$ and covariance $\sigma_{xy}$. If a linear transformation of the form $U = 3X - 2$ and $V = -2Y + 5$ is applied to both variables, prove analytically what the value of the new covariance $\sigma_{uv}$ will be, expressed in terms of $\sigma_{xy}$.
In a tech company, the monthly base salary ($X$, in pesos) has a mean of $1500$ and a standard deviation of $200$. Commissions for completed projects ($Y$, in pesos)…
In a tech company, the monthly base salary ($X$, in pesos) has a mean of $1500$ and a standard deviation of $200$. Commissions for completed projects ($Y$, in pesos) have a mean of $300$ and a standard deviation of $100$. Given that the covariance between base salary and commissions is $\sigma_{xy} = 15000$, calculate the mean and variance of the total monthly salary $Z = X + Y$.
(Note: Recall the property of the variance of the sum of two correlated variables: $\text{Var}(X+Y) = \text{Var}(X) + \text{Var}(Y) + 2\text{Cov}(X,Y)$).