The five-number summary of a data set consists of the minimum, the first quartile, the median, the third quartile, and the maximum. With the five-number summary, you can then construct a box plot to represent the data set. To find the five-number summary of this data set, we should begin by putting it in order from least to greatest, and that gets us here.
We see the minimum is 0, the maximum is 10. The median is the number in the middle, or in this case, because we have an even data set, it's the average of the two numbers in the middle. So that's just 3 plus 4 divided by 2. You can also just think of this as the number that's halfway between the two numbers in the middle. If we had one more number in this data set, the median would just be the number in the middle. In this case, that would have been 4. So we found the minimum, the maximum, and we just found the median is 3.5.
To find the first and third quartiles, we need to split the data in half, just like that. We have our lower half of data, the median of that is Q1, and the upper half of data, the median of that is Q3. We see the median of the lower half of data is 2, so that's Q1.
The median of the upper half of data... is 5, so that's Q3. To use the five number summary to construct a box plot, we need a number line that goes from our minimum, 0, to our maximum, 10. There's our number line, then we just draw vertical lines for all the numbers in the five number summary, the minimum, the maximum, and the quartiles and median. We draw a box between Q1 and Q3 with the median in the middle, the distance between these values, or the width of the box is the interquartile range, and then we draw what are called whiskers going out to the maximum and the minimum.
And that's how you make a box plot. We can also label the values specifically if we need to be extra clear. You may also want to consider a modified box plot where we can use our interquartile range, the distance between Q1 and Q3, to find outliers which we represent differently in a modified box plot.
We see that our maximum of 10 is pretty far from the rest of the data, and we can determine if it's an outlier using this rule of thumb. A value is an outlier if it's less than Q1 minus 1.5 times the IQR, or greater than Q3 plus 1.5 times the IQR. The IQR in our case, Q3 minus Q1, is 3. Once we run the calculations, we get a lower fence of negative 2.5. and none of our data is below that, so there are no lower outliers.
But our upper fence is 9.5, so we see 10 is indeed an upper outlier. To represent that on our box plot, we can make it so the maximum whisker only goes to the maximum excluding 10, and then we can just draw a dot to represent 10 as an outlier. And that's what this would look like as a modified box plot.
But again, you can also just use a normal box plot. Thanks for watching, and let me know in the comments if you have any questions. Check the description for links to related videos for extra practice.