Video: Interquartile Range

In this video, we will learn how to find the interquartile range given different representations of data.

17:55

Video Transcript

In this video, we will learn how to find the interquartile range given different representations of data. The interquartile range of a data set is a measure of how the data values are spread out around the center. We will begin by recalling how to calculate the median as well as the lower and upper quartiles. Once we have done this, we will show how we can use these values to calculate the interquartile range.

The median, otherwise known as 𝑄 two or the second quartile, marks the middle of a data set. 50 percent of the data is below the median, and 50 percent is above it. The lower or first quartile, known as 𝑄 one, marks the center of the bottom half of a data set. 25 percent of the data is below 𝑄 one. The upper or third quartile, 𝑄 three, marks the center of the top half of a data set. 25 percent of the data is above the upper quartile, and 75 percent is below it.

When dealing with a small data set, it is easy to calculate the median and quartiles by inspection. However, when dealing with larger data sets, we can use formulas to find the median position and 𝑄 one and 𝑄 three position in our data set. The median position is equal to 𝑛 plus one divided by two, where 𝑛 is the number of data values in our set. The lower quartile position is equal to a quarter of 𝑛 plus one or 𝑛 plus one divided by four. The upper quartile position is equal to three-quarters of 𝑛 plus one or three multiplied by 𝑛 plus one divided by four.

In some cases, particularly when dealing with an even number of data values, we need to round these answers up or down. We’ll now look at a quick example where we need to calculate the median as well as the lower and upper quartile.

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. We will now look at some questions where we need to calculate the interquartile range.

A set of data’s minimum is 3.0, its lower quartile is 4.5, its median is 6.4, its upper quartile is 7.9, and its maximum is 10.1. Determine its interquartile range.

We know that the interquartile range of any data set is equal to the upper quarter minus the lower quartile. We are told that the lower quartile is equal to 4.5. The upper quartile is equal to 7.9. This means that the IQR or interquartile range is equal to 7.9 minus 4.5. This is equal to 3.4. The interquartile range of the data is 3.4.

We will now look at a question where we need to compare the interquartile range for two data sets.

In this question, we’re given two data sets. Calculate the interquartile range for each data set. What do the interquartile ranges reveal about the two data sets? Is it (A) the spread of the middle 50 percent of the values is similar for both data sets? (B) The difference between the minimum and maximum values is similar for both data sets? (C) The median of the two data sets should be the same? (D) The mean of the two data sets should be the same? Or (E) the mode of the two data sets should be the same?

We will begin by clearing some space to calculate the interquartile range for each data set. Let’s begin by considering data set one. We begin by writing our seven values in ascending order, starting with 22 and ending with 51. The median of any data set is the middle value. In this case, this will be the fourth value as there are three values on either side of this. The median of data set one is 28.

The lower quartile or 𝑄 one is the center of the bottom half of our data set. The bottom half of the data set contains three values, 22, 25, and 26. The middle one of these is 25. This means that the lower quartile of data set one is 25. The upper quartile or 𝑄 three is the center of the top half of our data set. This contains the numbers 28, 29, and 51. The middle one of these is equal to 29. Therefore, the upper quartile is 29.

The interquartile range or IQR is equal to 𝑄 three minus 𝑄 one. We subtract the lower quartile value from the upper quartile value. 29 minus 25 is equal to four. The interquartile range of data set one is equal to four. We will now repeat this method for data set two.

As there are also seven values in data set two, the position of the quartiles and median will remain the same. The lowest value of data set two is 19, and the highest value is 28. We can see from our list that the median is equal to 24; the lower quartile, 21; and the upper quartile, 27. This means that the interquartile range is equal to 27 minus 21, which is equal to six. The interquartile range of data set two is six.

We will now move on to the second part of the question. In the second part of the question, we are asked to consider what the interquartile ranges reveal about the two date sets. The interquartile range does not rely on the median, mean, or mode. Therefore, we know that options (C), (D), and (E) are all incorrect. The maximum and minimum values also have no impact on the interquartile range as these are used to calculate the range of the entire data.

The interquartile range does contain the middle 50 percent of the values from the lower quartile to the upper quartile. As our values of four and six are quite close, we can conclude that the spread of the middle 50 percent of the values is similar for both data sets. The interquartile range only gives us information about those middle values.

Our next question involves calculating the range and interquartile range from a frequency table.

The table shows some non-English languages spoken by some of the U.S. population. Determine the range and interquartile range of the data.

The range of any data set can be calculated by subtracting the minimum value from the maximum. Whereas the interquartile range or IQR is equal to the upper quartile minus the lower quartile, also known as 𝑄 three minus 𝑄 one. Our first step is to sort our eight values into ascending order. The smallest value is equal to 216,300. This is the number of people that speak Hebrew. Next, we have 246,900 people that speak Armenian. We can continue to list these in order all the way up to the number of Spanish speakers, which is 37,580,000.

This is the maximum value. We can now calculate the range by subtracting the minimum value from the maximum one. This is equal to 37,363,700. This is the range of the data in the frequency table. As we have eight values in total, and the median is the middle number, this will lie halfway between the fourth and fifth value. Whilst we don’t need the median to calculate the interquartile range, it makes it easier to find the lower and upper quartiles.

The lower quartile is the center of the bottom half of our data. As there are four values that are less than the median, the lower quartile will lie halfway between 246,900 and 304,900. We can find the midpoint of these two values by adding them and then dividing by two. This gives us 275,900. We can repeat this process for the upper quartile or 𝑄 three. As there are four values above the median, the center of this will lie halfway between 800,000 and 1,410,000. This is equal to 1,105,000. We can then calculate the interquartile range by subtracting 275,900 from this. This is equal to 829,100, which is the interquartile range of the data.

The final question in this video involves calculating the interquartile range from a line plot.

The given line plot shows the magnitudes of the earthquakes that recently took place around the world. Determine the range and interquartile range of the data.

One way of approaching this question would be to write out all of the values in order, two, 2.1, 2.6, 2.6, 2.8, and so on. This should be very time consuming, so it is easier to work out how many earthquakes of each magnitude we have first. There was one earthquake of magnitude two. There was also one earthquake of magnitude 2.1. There were two earthquakes of magnitude 2.6, four of 2.8, all the way up to four of 3.5.

We can also calculate a running total or cumulative frequency of these to calculate the total number of earthquakes. This gives us values of one, two, four, eight, 11, 15, 17, 21, 23, and 27. There were 27 earthquakes that took place altogether. When dealing with a large data set, we can calculate the position of the median and the quartiles as follows.

The median position can be calculated by dividing 𝑛 plus one by two, where 𝑛 is the total number of data values. In this question, we have 27 plus one divided by two. This is equal to 14. So, the median is the 14th number. Whilst we don’t need to calculate the median in this case, it helps us work out the position of the quartiles. The 12th to 15th values all had a magnitude of three. This means that the median equals three.

The lower quartile or 𝑄 one position will be half of this. As the seventh number is 2.8, the lower quartile or 𝑄 one is 2.8. The upper quartile or 𝑄 three position will be the 21st value. This means that 𝑄 three is equal to 3.3. The range of values is calculated by subtracting the minimum from the maximum. 3.5 minus two is equal to 1.5. So, this is the range. The interquartile range or IQR is equal to 𝑄 three minus 𝑄 one. 3.3 minus 2.8 is 0.5. So, the interquartile range is 0.5.

We will now summarize the key points from this video. The median, lower quartile, and upper quartile of a data set can all be calculated from a list of data, a frequency table, or a line plot. The interquartile range or IQR is a measure of the middle 50 percent of the data. It is equal to the upper quartile or 𝑄 three minus the lower quartile, 𝑄 one. We also use the fact that the range is equal to the minimum value subtracted from the maximum value of the data set.

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