A group of 14 students were asked
to log the number of friends they each had on the new social media site Nosebook,
one month after joining the site, with the following results. Find the median number of
friends. Find the quartiles 𝑄 one and 𝑄
three of the data.
Before being able to calculate the
median or quartiles from any data set, we need to list the values in ascending
order. In this case, the smallest value is
32. The next smallest is 44. The rest of the values are listed
in order as shown up to 95. The median is the middle value. Whilst we could cross off a number
from either end to find this middle value, we can also use the formula 𝑛 plus one
divided by two to find the middle position.
As there are 14 values in this
question, we need to add one to 14 and then divide by two. This is equal to 7.5. Therefore, the median value is
between the seventh and eighth value. The seventh value is 68, and the
eighth value is 70. The median is the midpoint or
average of these two values. The midpoint of 68 and 70 is
69. Therefore, the median number of
friends is 69.
𝑄 one or the lower quartile is the
center of the bottom half of the data set. We have seven values that are below
the median. The middle or center of these will
be the fourth value as it has three values on either side. As the fourth value is 53, 𝑄 one
or the lower quartile is 53. In a similar way, we can find 𝑄
three or the upper quartile by finding the center of the top half of data. Once again, there are seven values
that are greater than the median. These range from the eighth value
to the 14th, the middle of which is the 11th, which has three values on either
side. As this is equal to 82, the upper
quartile or 𝑄 three is 82.
We will now look at the definition
of the interquartile range and how we can use these values to calculate it. The interquartile range is a
measure of the middle 50 percent of the data and gives us an indication of how
spread out the data is. We can calculate the interquartile
range or IQR of any data set by subtracting our 𝑄 one value from 𝑄 three; we
subtract the lower quartile from the upper quartile.
In our previous question, 𝑄 one or
the lower quartile was equal to 53, and 𝑄 three, the upper quartile, was equal to
82. The interquartile range would,
therefore, be equal to 82 minus 53. This is equal to 29.