7. Expectation, Averages, Variability
7.1
Summarizing Data on Random Variables
When we return midterm tests, someone almost always asks what the average was. While we could
list out all marks to give a picture of how students performed, this would be tedious. It would also
give more detail than could be immediately digested. If we summarize the results by telling a class
the average mark, students immediately get a sense of how well the class performed. For this reason,
“summary statistics” are often more helpful than giving full details of every outcome.
To illustrate some of the ideas involved, suppose we were to observe cars crossing a toll bridge, and
record the number,
X
, of people in each car. Suppose in a small study
10
data on 25 cars were collected.
We could list out all 25 numbers observed, but a more helpful way of presenting the data would be in
terms of the
frequency distribution
below, which gives the number of times (the “frequency”) each
value of
X
occurred.
X
Frequency Count
Frequency
1
   

6
2
   
  
8
3
   
5
4
  
3
5
 
2
6

1
We could also draw a
frequency
histogram of these frequencies:
Frequency distributions or histograms are good summaries of data because they show the variability
in the observed outcomes very clearly. Sometimes, however, we might prefer a singlenumber sum
10
"Study without desire spoils the memory, and it retains nothing that it takes in." Leonardo da Vinci
93
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document