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We can construct a box-and-whisker plot to represent a five statistical summary. To do this plot, we need first to find:
1. Minimum of the data.
2. First quartile.
3. Median.
4. Third quartile.
5. Maximum of the data.
To do this:
1. We first need to order the data from least to greatest. It has already been done.
Then, we have that the minimum is 10. We also know that the maximum is 42. We have these two points to graph in the plot.
Median
2. Now we need to find the first quartile, the median, and the third quartile.
We can find the median first. The data in ascending order is:
10, 11, 14, 15, 16, 18, 20, 27, 27, 30, 31, 33, 33, 37, 38, 42
The median is the value for which 50 percent of the data is below it, and 50 percent of the data is above this value. We have here, an even number of data (16 values). Then, we have that the median is:
a. We have that the median is 27 because we have an even number of data, we have two central values:
10, 11, 14, 15, 16, 18, 20, 27, 27, 30, 31, 33, 33, 37, 38, 42
We need to sum these numbers and then divide them by 2. Then, we have:
(27 + 27)/2 = 27.
First Quartile
3. The first quartile is a value for which 25% of the values are below it, and 75% of the values are above it. We can proceed in a similar way (as we do for the median) to find the first quartile having into account the first eight numbers:
10, 11, 14, 15, 16, 18, 20, 27
Because we have an even number of values, we take the two central numbers, and then we need to find an average between them:
(15 + 16)/2 = 15.5
Third Quartile
In this case, we can do the same to find the first quartile, but because we need to find the third quartile, we need to take the last eight values:
27, 30, 31, 33, 33, 37, 38, 42
Then, the third quartile is:
(33 + 33)/2 = 33
Now, we have these five statistics:
Minimum = 10
First Quartile = 15.5
Median = 27
Third Quartile = 33
Maximum = 42
Now, we can plot these values as follows:
1. We can plot in a line the given numbers.
2. We need to graph these five numbers in this line.
3. The box starts in the First Quartile, and finish in the Third Quartile.
4. We draw the median inside the box.
5. Minimum and maximum will be outside this box.
Then, this graph is:
And this is the box and whisker plot for these data.
