Video Transcript
The table represents the distance
needed to take off and the distance needed to land for several aircraft. Find the Spearman′s rank
correlation coefficient and round your answer to three decimal places.
We′re asked to find Spearman′s rank
correlation coefficient. And to do this, we′ll use the
formula shown. In this formula, 𝑛 corresponds to
the number of data pairs. And we use the term paired data,
since each pair of data corresponds to a single aircraft. So for example, the aircraft
needing a takeoff distance of 893 meters has a landing distance of 724. And this accounts for one data
pair. 𝑑 𝑖 in our formula corresponds to
the difference in ranks for a single pair, where 𝑖 takes values from one to 𝑛. And we take the sum of the
differences squared.
And since we want to find the
difference in ranks for each pair, the first thing we need to do is to rank the data
in each of the takeoff and landing data sets. We should rank these both in the
same direction, that is, either low to high or high to low, so ranking low to high
and adding lines to our table. If we begin with our takeoff
distance, the lowest takeoff distance is 893. And so we rank this one. The next lowest takeoff distance is
956 meters, which we rank two. 975 is our next lowest, which we
rank three. 980 meters is next, which is ranked
four. And 1036 is our longest takeoff
distance, which we rank fifth.
Our shortest landing distance is
677 meters. So this comes first. Our second is 724, which is ranked
second, and so on. So that′s 741 meters is third, 770
meters is fourth, and 853 meters is fifth.
Now we′re going to need the
difference in the ranks for each pair. So we subtract the landing ranks
from the takeoff ranks for each pair. In our first column of data then,
we have one minus two is negative one. In our second column, we have two
minus one, which is equal to one, in our third column, three minus three, which is
zero, followed by four minus four, which is zero, and five minus five, which is also
zero.
Now if we′ve done this correctly,
we should find that the sum of the differences is equal to zero. And in fact, this is the case,
since negative one plus one plus zero plus zero plus zero is indeed equal to
zero.
Our next step is to find the
differences squared. We can then sum these and put this
into our formula. In our first column of data, we
have negative one squared, which is equal to one. In our second column, one squared
is equal to one. And since zero squared is equal to
zero, in our final three columns we have zero. The sum of our differences squared
is then one plus one plus zero plus zero plus zero, which is equal to two.
Since we have five pairs of data,
𝑛 is equal to five. So now we have everything we need
to use the formula to find Spearman′s rank correlation coefficient for this
data. And with our sum of differences
squared equal to two ,𝑛 is equal to five, we have one minus six times two over five
times five squared minus one. That is one minus 12 over 120,
which is one minus 0.100. And since one minus 0.100 is 0.900,
we have the Spearman′s rank correlation coefficient for this data is equal 0.900 to
three decimal places.
This is close to positive one. So this means there′s a very strong
association between takeoff distance needed and landing distance needed. That is the longer the distance
needed for takeoff, the longer the distance needed for landing.
