Summarise a small data set and distinguish location from spread.
Notes and key methods
Put observations in ascending order before locating the median. For n observations, \(\bar x=\sum x/n\), \(\operatorname{Var}(X)=\sum x^2/n-\bar x^2\), and the range is the largest minus the smallest observation.
A stem-and-leaf diagram retains every observation: sort leaves within each stem and provide a key that includes the units. For a box plot, locate the median and quartiles before plotting the five-number summary.
Check the conditions
Use population variance when describing the complete listed data set.
For odd n the median is the middle observation; quartile conventions must be stated.
Three worked examples
Example 1 · Data summaries
The complete data set, measured in minutes, is \(6,9,9,12,15,18,18,21,27\). Treat these nine observations as a population.
Stem
Leaf
0
6 9 9
1
2 5 8 8
2
1 7
Key: 2 | 7 represents 27 minutes.
Find the mean in minutes.
Find the population variance in square minutes.
Find the median in minutes.
Find the range in minutes.
Show the reasoning
(a)
\(\bar x=15\)
Answer: \(15\).
(b)
\(\operatorname{Var}(X)=\sum x^2/9-\bar x^2=40\)
Answer: \(40\).
(c)
\(x_{(5)}=15\)
Answer: \(15\).
(d)
\(27-6=21\)
Answer: \(21\).
Example 2 · Data summaries
The complete data set, measured in minutes, is \(7,11,11,15,19,23,23,27,35\). Treat these nine observations as a population.