setwd("C:/Users/emontenegro1/Documents/MEGA/stanStateDocuments/PSYC3000/lecture5")
rum <- read.csv("ruminationComplete.csv")
mean(rum$age)[1] 15.34906Central tendency and variability
APA Style
Central tendency measures III
Let’s do it by hand:
\[\begin{align}
\bar{X} &= \frac{\sum X}{n}\\
&= \frac{18+21+24+23+22+24+25}{7}\\
&= \frac{157}{7}\\
&= 22.43\\
\end{align}\]
Median to the rescue
As you might remember, the mean is strongly affected by the extreme cases, whereas the median is more “robust” to extreme cases. This means the median is less affected by extreme values.
Let’s use simulation to find out if it is true, imagine you have data related to a depression score:
set.seed(1256)
M <- 25
SD <- 1
n <- 50
## Simulated depression score
depressionScore <- rnorm(n = n, mean = M, sd = SD)
hist(depressionScore)
- Let’s check the mean of these generated values:
- Now, let’s add a very extreme case, imagine one person has a diagnosis of bipolar disorder, and that person is experiencing a depressive episode when she or he filled out your test:
- Let’s compare the mean on both cases:
Mean before the extreme case: 24.97403Mean after the extreme case: 26.48716- We can check now if the median was heavely affected by th extreme case:
Median before the extreme case: 24.82834Median after the extreme case: 24.89818- Great the median is robust enough! It remain practically intact!
The fashionable Mode II
- However, things can get messy when you have two modes!
- When you have a bimodal distibution you are dealing with a mixture of two or more distributions.
Summary() function as a good option
- In
Rwe can count on a handy function to describe a distribution, this function issummary().
- This function shows the minimum value, the 1st quantile, the median, the 3rd quantile, mean and maximum value.
library(ggplot2) ### package to create pretty plots
dens <- density(rum$age)
df <- data.frame(x=dens$x, y=dens$y)
probs <- c(0, 0.25, 0.5, 0.75, 1)
quantiles <- quantile(rum$age, prob=probs)
df$quant <- factor(findInterval(df$x,quantiles))
figure <- ggplot(df, aes(x,y)) +
geom_line() +
geom_ribbon(aes(ymin=0, ymax=y, fill=quant)) +
scale_x_continuous(breaks=quantiles) +
scale_fill_brewer(guide="none") +
geom_vline(xintercept=mean(rum$age), linetype = "longdash", color = "red") +
annotate("text", x = 14, y = 0.2, label = "Q1 = 14 years") +
annotate("text", x = 17, y = 0.3, label = "Median = 16 years") +
annotate("text", x = 15.35, y = 0.33, label = "Mean = 15.35 years") +
ylab("Likelihood") +
xlab("Age in years")+
ggtitle("Quantiles and mean of Age")+
theme_classic()
The Standard Deviation will be a better horse II
As always we can study an example:
- Remember when we were simulating data days ago? Well, we’ll do it again!
Hospital Example
library(ggplot2) ### <- this is a package in R to create pretty plots.
set.seed(359)
### Non-hospital observations
### Mean or average in Kg
Mean <- 65
## Standard Deviation
SD <- 1
## Number of observations
N <- 300
### Generated values from the normal distribution
data_1 <- rnorm(n = N, mean = Mean, sd = SD )
data_1
### Hospital group
### Mean or average in Kg
Mean <- 90
## Standard Deviation
SD <- 10
## Number of observations
N <- 300
### Generated values from the normal distribution
data_2 <- rnorm(n = N, mean = Mean, sd = SD )
data_2
dataMerged <- data.frame(
group =c(rep("College", 300),
rep("Hospital", 300)),
weight = c(data_1, data_2))
ggplot(dataMerged , aes(x=weight, fill=group)) +
geom_density(alpha=.25) +
theme_bw()+
labs(title = "College and Hospital Weight Density Function") +
xlab("Weight (kg)") +
ylab("p(y) or likelihood")
The Standard Deviation will be a better horse IV
Another example related to quantiles II
- We can estimate the percentiles using the formula showed before, this time we will find the estimate for the quantiles in
mpgvariable inside the datamtcars:
Cumulative Density Function (CDF)
As the name says, we are like “stacking” the whole density, therefore it changes the shape of the curve, but at the end is the same information in a different metric.
In fact, you get the derivative of a CDF, th calculation will give you the PDF back.
But no worries, I won’t ask you to do it… you are safe!
Cumulative Density Function (CDF)
All continuous distributions will have a CDF, and we are going to use very often the normal CDF.
The normal distribution is also called “Gaussian Distribution” , I prefer this name instead of “normal distribution”.
Anyhow, let’s check some properties here.
We can also understand the importance of the Gaussian CDF using R:
- When we assume that the Gaussian distribution has a mean = 0 and standard deviation = 1, the CDF looks like this:
## sequence of x-values
justSequence <- seq(-4, 4, .01)
#calculate normal CDF probabilities
prob <- pnorm(justSequence)
#plot normal CDF
plot(justSequence ,
prob,
type="l",
xlab = "Generated Values",
ylab = "Probability",
main = "CDF of the Standard Gaussian Distribution")
abline(v=1.96, h = 0.975, col = "red")
- We can see that the probability of observing a value less or equal than 1.96 is 0.975.
Cumulative Density Function (CDF) II
Let’s do something more intersting, remember the example of weight where we simulated the weight of two groups: hospital patients vs. college students?
We could now get the probability of observing a particular value.
Let’s imagine again that the distribution of weight among college students has a mean of 65 kg, and standard deviation of 20 kg.
- A weight of 76 kg has a probability of 0.71, it is likely to see this weight in the college students under my assumptions.
Skewness
I left some concepts behind because I got excited talking about the CDF.
One important concept to describe a distribution is skewness.
Skewness II
- We say that a distribution is right skewed when the tail is longer to the right:
set.seed(5696)
N <- 1000
### Number of times people check Instagram
weight <- rnbinom(N, 10, .5)
plot(density(weight, kernel = "gaussian" ),
ylab = "p(y) or likelihood",
xlab = "How many times people check Instagram?",
main = "Density plot of How many times people check Instagram?")
- We say that a distribution is left skewed when the left tail is longer:
