Measures of spread in a numerical data set seek to describe whether the scores in a data set are very similar and clustered together, or whether there is a lot of variation in the scores and they are very spread out.
There are several methods to describe the spread of data, which vary in complexity. We can simply look at the numerical range of the entire data set - the difference between the largest and smallest value, or we can break the data into chunks to examine the range of smaller sections within the data.
Remember, the range only changes if the highest or lowest score in a data set is changed. Otherwise it will remain the same. This will mean it will be significantly affected by an outlier being present in the data.
In this lesson, we will look at an alternative to the range called the interquartile range.
Whilst the range is very simple to calculate, it is based on only two numbers in the data set, it does not tell us about the spread of data within these two values. To get a better picture of the internal spread in a data set, it is often more useful to find the set's quartiles, which can be used for a measure of spread called interquartile range (IQR).
Quartiles are scores at particular locations in the data set-similar to the median, but instead of dividing a data set into halves, they divide a data set into quarters. Let's look at how we would divide up some data sets into quarters now.
Make sure the data set is ordered before finding the quartiles or the median.
Now there are four scores in each half of the data set, so split each of the four scores in half to find the quartiles.
Now let's look at a situation with 9 scores:
Finally, let's look at a set with 10 scores:
Each quartile represents 25\% of the data set. The least score to the first quartile is approximately 25\% of the data, the first quartile to the median is another 25\%, the median to the third quartile is another 25\%, and the third quartile to the greatest score represents the last 25\% of the data. We can combine these sections together, for example, 50\% of the scores in a data set lie between the first and third quartiles.
These quartiles are sometimes referred to as percentiles. A percentile is a percentage that indicates the value below which a given percentage of observations in a group of observations fall. For example, if a score is in the 75\text{th} percentile in a statistical test, it is higher than 75\% of all other scores. The median represents the 50\text{th} percentile, or the halfway point in a data set.
Here are Ray's scores from his last 13 rounds of golf played: 66,\,66,\,68,\,68,\,70,\,78,\,80,\,84,\,106,\,116,\,126,\,130,\,132
What is his median?
What is the lower quartile?
What is the upper quartile?
Q_{1} is the first quartile (sometimes called the lower quartile). It is the middle score in the bottom half of data and it represents the 25\text{th} percentile.
Q_{2} is the second quartile, and is usually called the median. It represents the 50\text{th} percentile of the data set.
Q_{3} is the third quartile (sometimes called the upper quartile). It is the middle score in the top half of the data set, and represents the 75\text{th} percentile.
The interquartile range (IQR) is the difference between the third quartile and the first quartile. 50\% of scores lie within the IQR because it contains the data set between the first quartile and the median, as well as the median and the third quartile.
Since it focuses on the middle 50\% of the data set, the interquartile range often gives a better indication of the internal spread than the range does, and it is less affected by individual scores that are unusually high or low, which are the outliers.
Subtract the first quartile from the third quartile. That is, \text{IQR} = Q_{3}-Q_{1}
Answer the following given the frequency table:
Score | Frequency |
---|---|
5 | 1 |
14 | 1 |
18 | 3 |
24 | 2 |
32 | 1 |
38 | 2 |
50 | 5 |
Find the number of scores.
Find the median.
Find the lower quartile of the set of scores.
Find the upper quartile of the set of scores.
Find the interquartile range.
Consider the dot plot below:
Find the total number of scores.
Find the median.
Find the lower quartile of the set of scores.
Find the upper quartile of the set of scores.
Find the interquartile range.
To calculate the interquartile range: