The 5 number summary is a very useful measure of the variability within a data
set. It has the advantage of being quick and easy to calculate, while giving us an
accurate picture of the spread of our data. As you might gather from the name,
the five number summary consists of five numbers that summarize the data set. The
five numbers are: minimum, quartile 1, median, quartile 3, and maximum. The
minimum is the smallest number in the data set. The maximum is the largest
number in the data set. The median is calculated using a median split as we
have done before. Quartile 1 is the first quartile (aka the 25th percentile) the
middle of the scores between the minimum and the median. Quartile 3 is the third
quartile, also called the 75th percentile, it's the middle of the scores between
the median and the maximum. Using data set D and what we have just learned
about the interquartile range, we can see that the minimum is one. Quartile 1 is 3.
The median is 3.5. Quartile 3 is 9, and the maximum is 20. We can see that the
outlier affects the range, but not the interquartile range. If you are
particularly clever, you may recognize that the 5 number summary gives you the
same information as a box plot. The box plot is the graphic representation of
the 5 number summary. Using the data set world95.sav, we will now create the 5
number summary and a box plot using SPSS. The data set world95.sav is demographic
data from around the world from the year 1995. To begin our analysis, we will go
to analyze > descriptive statistics > frequencies. We currently see our
variables as variable names. To change that, right click or on a Mac, ctrl click
on any of the variable names and choose display variable labels. We are
interested in the variables average female life expectancy and average male
life expectancy. Drag those into the variables box. Now click on statistics. In
order to calculate a five number summary, we will need the minimums, the maximum
quartiles, and the median. Actually we don't need to choose the median, it is
the same as quartile too, but I want to prove that to you. So click continue. Now
unclick display frequency tables and click OK. Before we spend too much time
with the output however, let's also create a box plot. Go to graphs > chart
builder. In the gallery, choose "box plot". Drag simple box plot up into the canvas.
Move average female life expectancy into the y-axis drop zone, and now click OK.
There's our box plot. Alternately, we could create a box plot using the
Explorer command, so go to analyze > descriptive statistics > explore. Move
average female life expectancy and average male life expectancy into the
dependent list, and click on plots. Check "normality plots with tests" and unclick
"stem and leaf" click continue and ok. Scroll down a little more. . .
and there's another box plot. Using the output from SPSS, let's create the 5
number summary. First of all, notice that the quartiles are actually listed as
percentiles. The 25th percentile is the same as the first quartile. The 75th
percentile is the third quartile. So continuing with our five number summary,
the minimum is 43. The first quartile is 66.5, and the median is 74. Notice that
the median is the same as the second quartile. The third quartile is 78, and
the maximum is 82, so the 5 number summary for female life expectancy is 43,
66.5, 74, 78, 82. To reinforce the idea that we can display our data as numbers or
pictures, let's look at the box plot I've edited this a little bit to fit on the
slide, but you can see that the minimum, maximum and the lines of the box match
the numbers of our five number summary. From this output, we can tell that in
half of countries for 1995 female life expectancy was 74. The upper fifty
percent of countries were between 74 and 82 years, an eight year gap, indicating
the life expectancy is reaching its maximum. In the lower half of the
distribution, there is much more variability. In the lowest quartile, the
bottom whisker of the box-plot some countries have life expectancies between
ages 43 and 66.5. This suggests that female life expectancy is unnaturally
brief in a handful of countries. We could look at our data set to find out what
countries those are, but now it's your turn. Create a box plot and a 5 number
summary for average male life expectancy. Do some comparisons with what you find
from average female life expectancy. Are the life expectancy similar or different?
Who lives longer on average? And if you do it right, your answer will look like this.