Using Technology to Find Quartiles and Draw Graphs In Exercise...

Question

Using Technology to Find Quartiles and Draw Graphs In Exercises 23–26, use technology to draw a box-and-whisker plot that represents the data set.

Vacation Days The number of vacation days used by a sample of 20 employees in a recent year

3 9 2 1 7 5 3 2 2 6

4 0 10 0 3 5 7 8 6 5

Accepted Answer

Hello there. Today we're gonna solve the following practice problem together. So first off, let us read the problem and highlight all the key pieces of information that we need to use in order to solve this problem. The ages in years of 14 participants in a workshop are 22, 25, 28, 21, 24, 27, 23, 26, 29, 22, 25, 28, 24, 27. Draw a box and whisker plot for this data set. Awesome. So it appears for this particular problem, we're ultimately trying to create a box and whisker plot for this particular data set. So now that we know that we're ultimately trying to create a box and box and whisker plot for this particular problem. Our first step in order to solve for this problem is we need to sort our data. So we need to sort our data in ascending order, meaning from smallest to largest value. And when we do that, we will achieve the following. So as you can see in ascending order, we have it be 21, 22, 22, 23, 24, 24, 25, 25, 26, 27, 27, 28, 28, 29. Awesome. Our second step now is we need to find the median, which is our Q2 value. So once again we need to note that Q2 is our median value. So since there are 14 data points, which makes it an even number, the median is the average of the 7th and 8th values. So that means that our 7th value is 25 and our 8th value is 25. So that means that our median, our Q2 value is going to be equal to 25 plus 25 divided by 2, which will be equal to 25. Awesome. Our 3rd step now is we need to find Q1, our first port tile. So focusing our attention on Q1, as we should recall, Q1 is the median of the lower half, which is the first seven numbers. So noting once again that our Q1, which is the median of the lower half, our first seven numbers, meaning our first seven numbers are, as we should recall, 21, 22, 22, 23, 24, 24, 25. Fantastic. So that means that the median of these 7 numbers, which will be the 4th value, so let's make a note of this, the 4th value. We need to be 23. So once again we need to note that our fourth value is 23, so that means that Q1 will be also equal to 23. Fantastic. So now, our 4th step is we need to find Q3, our 3rd port tile, and as we should recall, Q3 is the median of the upper half, which is the last 7 numbers. So that means that our last seven numbers, as we should recall, are 25, 26, 27, 27, 28, 28, and 29. Awesome. And the median of these seven numbers, which we need to know in this particular case, the 4th number in this case is 27, so that will mean that Q3, or 3rd quartile, is also equal to 27. Fantastic. So now our 5th step is we need to identify whether there are any outliers, and in order to do that we need to recall and use our equation to solve for the interquartile range or the IQR for short, and the IQR is equal to Q3 minus Q1, which is equal to 27. -23 in this case, which give us which will give us an answer of 4. And we need to recall and note that if a value is less than Q1, So let's make a note, less than Q1 minus 1.5 multiplied by our IQR value. Which will mean that, once again, so if our value is less than Q1 minus 1.5 multiplied by IQR or greater than Q3 + 1.5 multiplied by Our IQR value, then it will be an outlier value. So with that in mind, let's plug in our known variables. So in this case, for Q1, we have 23, so 23 minus 1.5 multiply. By our IQR, which is 4, which will give us a value of 17. Our Q3 value is 27, so 27 plus 1.5 multiplied by 4 will give us an answer of 33. And since there is no value less than 17 or greater than 33, our particular data set for this problem has no outliers. So I'll put an X next door data, so that means there's no outliers. Fantastic. So moving right along here, so our 6th step. we need to find the minimum and the maximum. So our minimum is 21, and our maximum is 29, means our minimum is the smallest number and our maximum is the biggest number. So now, we need to perform a 5 number summary. So we need to note that our minimum is 21. And our Q1 value is going to be 23, so Q1 value is 23. Our Q2 value is 25. And our Q3 value. 27 and our maximum is 29. So now, We can start on our 7th and final step, which is to draw or to create our box and whisker plot. So we need to create the box and whisker plot using our 5 number summary, which once again is our minimum maximum in our Q1 through Q3 values. So we need to construct a horizontal axis and label the axis with the values that will cover the range of the data, and then we need to plot our 5 numbers above the axis. So we need to draw a box above the horizontal axis from Q1 to Q3 and draw a vertical line segment in the box to indicate our Q2 value, and then we need to draw the whiskers from the box to the minimum and maximum values. So what does that all mean? So, looking up at our problem itself, I've gone ahead and and used a computer to create a computer generated box and whisker plot. And as we could see, For our box and whisker plot. We have our boxes, so as you could see, since we know that our Q So once again, our Q1 value is 23, so that's our edge of our main box. So we have our main box represented in purple. And we have Q1 indicated at 23. Then we have our Q3 value at 27, and then for our median is as a line cutting our purple box. Almost in half, but it has a line indicating our median, so it cuts our box into two, and it indicates at 25. And then our whiskers, which resemble like sideways Ts, like T shapes, indicating our minimum at 21, and our maximum at 29. And that's it. That is how you draw a box and whisker plot. And it's also very important whenever you do a box and whisker plot that you label your axi, your horizontal axis, which in this case it's age and units of years, so you need to make sure you put the units, and it's always helpful to add a title to, which in this case it's ages of workshop participants. And to make sure we have all the data encompassed in our box and whisker plot, I had, even though we didn't have any values of 19, I started with 19 and then I ended with 31, so we have two values extra on either side to really show clearly our data points for our box and whisker plot. And that's it. That is how you solve for this problem. Thank you so much for watching. Hopefully that helped, and I can't wait to see you in the next video. Bye.

Using Technology to Find Quartiles and Draw Graphs In Exercises 23–26, use technology to draw a box-and-whisker plot that represents the data set.

Vacation Days The number of vacation days used by a sample of 20 employees in a recent year
3 9 2 1 7 5 3 2 2 6
4 0 10 0 3 5 7 8 6 5

Key Concepts

Quartiles

Box-and-Whisker Plot

Technology in Statistics

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