Possible answer: 36
If you also want to know the steps here scroll down.
Step 1: Order the data
In order to calculate the IQR, we need to begin by ordering the values of the data set from the least to the greatest. Likewise, in order to calculate the median, we need to arrange the numbers in ascending order (i.e. from the least to the greatest).
Let's sort an example data set with an odd number of values into ascending order.
{Odd data set}: 9, 3, 2, 5, 6, 11, 4, 3, 2
{Odd data set (ascending)}: 2, 2, 3, 3, 4, 5, 6, 9, 11
Now, let's perform this task with another example data set that is comprised of an even number of values.
Even data set}: 11, 2, 4, 3, 8, 1, 2, 7, 4, 9
Rearrange into ascending order.
{Even data set (ascending)}: 1, 2, 2, 3, 4, 4, 7, 8, 9, 11
Step 2: Calculate the median
Next, we need to calculate the median. The median is the "center" of the data. If the data set has an odd number of data points, then the mean is the centermost number. On the other hand, if the data set has an even number of values, then we will need to take the arithmetic average of the two centermost values. We will calculate this average by adding the two numbers together and then dividing that number by two.
First, we will find the median of a set with an odd number of values. Cross out values until you find the centermost point
{Odd data set}: {2}, {2}, {3}, {3}, { 4}, {5}, {6}, {9}, {11}
The median of the odd valued data set is four.
Now, let's find the mean of the data set with an even number of values. Cross out values until you find the two centermost points and then calculate the average the two values.
{Even data set}: {1}, {2}, {2}, {3}, {4}, {4}, {7}, {8}, {9}, {11}
Find the average of the two centermost values.
{Average}={4+4}{2}
{Average}={8}{2}
{Average}=4
The median of the even valued set is four.
Step 3: Upper and lower medians
Once we have found the median of the entire set, we can find the medians of the upper and lower portions of the data. If the data set has an odd number of values, we will omit the median or centermost value of the set. Afterwards, we will find the individual medians for the upper and lower portions of the data.
{Odd data set}: 2, 2, 3, 3, 4, 5, 6, 9, 11
Omit the centermost value.
{Odd data set}: 2, 2, 3, 3, 5, 6, 9, 11
Find the median of the lower portion.
{Odd data set}: 2, 2, 3, 3, 5, 6, 9, 11
Calculate the average of the two values.
{Average}={2+3}{2}
{Average}={5}{2}
{Average}=2.5
The median of the lower portion is 2.5
Find the median of the upper portion.
{Odd data set}: 2, 2, 3, 3,{5}, {6}, {9}, {11}
Calculate the average of the two values.
{Average}={6+9}{2}
{Average}={15}{2}
{Average}=7.5
The median of the upper potion is 7.5
If the data set has an even number of values, we will use the two values used to calculate the original median to divide the data set. These values are not omitted and become the largest value of the lower data set and the lowest values of the upper data set, respectively. Afterwards, we will calculate the medians of both the upper and lower portions.
{Even data set}: 1, 2, 2, 3, 4, 4, 7, 8, 9, 11
Find the median of the lower portion.
{Even data set}: {1}, {2}, {2}, {3}, {4} 4, 7, 8, 9, 11
The median of the lower portion is two.
Find the median of the upper portion.
{Even data set}: 1, 2, 2, 3, 4, 4, 7, 8}, {9}, {11}
The median of the upper portion is eight.
Step 4: Calculate the difference
Last, we need to calculate the difference of the upper and lower medians by subtracting the lower median from the upper median. This value equals the IQR.
Let's find the IQR of the odd data set.
{IQR of the odd data set}=7.5-2.5
{IQR}=5
Finally, we will find the IQR of the even data set.
{IQR of the even data set}=8-2
{IQR}=6
In order to better illustrate these values, their positions in a box plot have been labeled in the provided image.
How to find iqr boxplot image
Now that we have solved a few examples, let's use this knowledge to solve the given problem.
Solution:
First, we need to put the data in order from smallest to largest.
21,25,27,31,32,37,43,45,49
Q2 = median of the overall data set
Q1 = median of the lower half of the data
Q3 = median of the upper half of the data
Q2=32
32 is the overall median, leaving 21,25,27,31 as the lower half of the data and 37,43,45,49 as the upper half of the data.
The median of the lower half falls between two values.
Q1=25+272=26
The median of the upper half falls between two values.
Q3=43+452=44
The interquartile range is the difference between the third and first quartiles.
Q3−Q1=44−26=18
Multiply by 2 to find the answer:
2∗18=36