Let’s recapitulate. We read data with

birth <- read.table("birth.txt", header=T)

which results in a data frame like this:

head(birth)

    id birth apgar5 sex weight head  age parity weeks
1 4347     1      8   F   1610 41.0 28.5      1    31
2 4346     1      9   F   3580 51.0 35.0      1    39
3 4300     1      9   F   3350 52.0 37.0      1    40
4 4345     1      9   F   3230 50.5 35.0      1    38
5 4349     1      8   F   3650 52.0 36.5      1    40
6 4315     2      8   F   3900 51.0 35.0      1    38

The easiest to understand are minimum and maximum

min(birth$weight)

[1] 1100

max(birth$weight)

[1] 5000

Which sometimes can be useful together

range(birth$weight)

[1] 1100 5000

If m is the median of a vector v, then

• half of the values in v are smaller than m
• half of the values in v are bigger than m

median(birth$weight)

[1] 3250

The median is the value \(m\) minimizes the absolute error \(\sum_{i=1}^n \vert v_i-m\vert\).

What if the number of elements is even?

Quart means one fourth in latin.

If we split the set of values in four subsets of the same size

Which are the limits of these sets?

\(Q_0\): Zero elements are smaller than this one
\(Q_1\): One quarter of the elements are smaller
\(Q_2\): Two quarters (half) of the elements are smaller
\(Q_3\): Three quarters of the elements are smaller
\(Q_4\): Four quarters (all) of the elements are smaller

It is easy to see that \(Q_0\) is the minimum, \(Q_2\) is the median, and \(Q_4\) is the maximum

Generalizing, we can ask, for each percentage p, which is the value on the vector v which is greather than p% of the rest of the values.

The function in R for that is called quantile()

By default it gives us the quartiles

quantile(birth$weight)

    0%    25%    50%    75%   100% 
1100.0 2972.5 3250.0 3580.0 5000.0 

quantile(birth$weight, seq(0, 1, by=0.1))

  0%  10%  20%  30%  40%  50%  60%  70%  80%  90% 100% 
1100 2630 2906 3020 3140 3250 3388 3530 3660 3850 5000 

The mean value of the vector v is \[\mathrm{mean}(v) = \frac{1}{n}\sum_{i=1}^n v_i\] where \(n\) is the length of v. Sometimes it is written as \(\bar{v}\)

mean(birth$weight)

[1] 3243.905

This value is usually called average, but the correct name is mean

This value minimizes the quadratic error \(\sum_{i=1}^n (v_i-\bar{v})^2\)

if \(n\) is the length of \(v\) and \(\bar{v}\) is its mean, then the mean of the quadratic error is \[\frac{1}{n}\sum_{i=1}^n (v_i-\bar{v})^2\] This number is called variance of the sample

This is a number in squared units, so it is hard to compare with the mean value

mean(birth$weight)

[1] 3243.905

var(birth$weight)

[1] 273438

To understand easily the variability of the data, we take the square root of the variance

The standar deviation of the sample is the square root of the variance \[\sqrt{\frac{1}{n}\sum_{i=1}^n (v_i-\bar{v})^2}\]

In many cases, including in R, people uses a slightly different formula \[\mathrm{sd}(v) = \sqrt{\frac{1}{n-1}\sum_{i=1}^n (v_i-\bar{v})^2}\] Explaining the reason is for a next course.
(It is because of the bias of the expected value of the expected value)

This value is called standard deviation of the population

The difference is small, especially when \(n\) is big

v <- birth$weight
sd(v)

[1] 522.913

sqrt(sum((v-mean(v))^2)/length(v))

[1] 522.5361

sqrt(sum((v-mean(v))^2)/(length(v)-1))

[1] 522.913

Sometimes the best way to tell the story of the data is with a graphic

plot(birth$weight)

each individual has a position in the x axis

plot(birth$head)

par(mfrow = c(1,2))
plot(birth$apgar5)
plot(as.factor(birth$apgar5))

par(mfrow = c(1,2))
plot(birth$head)
hist(birth$head)

par(mfrow = c(1,2))
plot(birth$head)
plot(birth$head, col="red")

par(mfrow = c(1,2))
plot(birth$head, cex=2)
plot(birth$head, cex=0.5)

par(mfrow = c(1,2))
plot(birth$head, pch=16)
plot(birth$head, pch=".")

par(mfrow = c(1,2))
plot(birth$head, type = "l")
plot(birth$head, type = "o")

par(mfrow = c(1,2))
plot(birth$head, type = "l", xlim=c(1,100))
plot(birth$head, type = "o", xlim=c(1,100))

par(mfrow = c(1,2))
plot(birth$head, type = "b", xlim=c(1,100))
plot(birth$head, type = "n", xlim=c(1,100))

plot(birth$weight, main = "Weight at Birth", sub = "694 samples", ylab="weight [gr]")

plot(birth$head, type="l", ylim = c(22,55))
lines(birth$age, col="red")

plot(birth$weight[birth$sex=="F"], ylim=range(birth$weight), ylab = "weight [gr]")
points(birth$weight[birth$sex=="M"], col="blue")