David’s history test scores are 74,
96, 85, 90, 71, and 98. Determine the upper and lower
quartiles of his scores.
In order to calculate the upper and
lower quartiles for a data set, we firstly need to sort the data into ascending
order. In this case, the lowest score was
71. The next lowest score was 74. The remainder of David’s scores in
ascending order were 85, 90, 96, and 98. We have six test scores in total,
and we know that the median is the middle value.
One way to calculate the median
with a small data set is to cross off numbers from either end. We cross off the smallest number
and the largest number. We then cross off 74 and 96. This means we’re left with two
middle numbers, 85 and 90. The median will be the midpoint of
these two numbers. We could work this out on a number
line. Alternatively, we can find the
average or midpoint of two numbers by finding their sum and dividing by two. This is equal to 87.5. The median of David’s test scores
An alternative way of finding the
median, which is useful if we have a large data set, is by using the formula 𝑛 plus
one divided by two. This gives us the median position
on the list. As there were six values in this
question, 𝑛 is equal to six. Six plus one is equal to seven, and
dividing by two gives us 3.5. This means that the median will be
halfway between the third and fourth value. This confirms that our answer of
87.5 was correct.
As we had six values in total,
there are three values less than the median and three values greater than the
median. We know that the lower quartile is
the center of the bottom half of our data set. As there are three values here, the
lower quartile, or Q one, will be the middle one. This is equal to 74. The upper quartile will be the
center of the top half of our data set. Once again, we have three numbers
above the median. The center number will be the
middle one. This is equal to 96.
We can therefore conclude that the
upper quartile of David’s history scores was 96 and the lower quartile was 74. Before moving on from this
question, let’s consider how we could find the lower quartile and upper quartile
position. The position of the lower quartile
can be calculated using the formula 𝑛 plus one divided by four or a quarter of 𝑛
plus one. Seven divided by four is equal to
1.75. As this is more than halfway
between one and two, we round up to two. The lower quartile will be the
second value in our list.
We can calculate the position of
the upper quartile using a similar method. Three-quarters of 𝑛 plus one, or
three multiplied by 𝑛 plus one divided by four. This is equal to 5.25, which we
notice is three times 1.75. As this is less than halfway
between five and six, we round down to five. The fifth number in our list will
be the upper quartile. This method is particularly useful
if we have a large data set.