IQ Inter-Quartile Range And Box Plots How do you describe the spread of this data?

How many times have you gotten 100% on a spelling test this year? 0 1 2 3 4 5 6 7 8 9 10 How many times have you gotten 100% on a spelling test this year? How do you describe the spread of this data? 2

0 2 4 6 8 10 12 14 16 18 20 22 Number of Books How many books did 6th graders at Taylor Middle School read this summer? Let’s review how to read a line plot. The title tells us that this graph is about how many books the 6th graders at Taylor Middle School read this summer. Each circle represents one student’s response and it is placed on the line according to how many books they said they read. So 2 people said they read 1 book, 4 people said they read 2 books, and all the way up to 1 students who read 21 books.

Mean Absolute Deviation Center Mean Median Mode Spread Range Interquartile Range Mean Absolute Deviation A common mistake that students make is confusing the measures of center and spread. The measures of center include the mean, median, and mode. The measures of spread include the range, interquartile range, and mean absolute deviation. This lesson will focus on the spread.

You need 5 piece of information for a Box Plot the median Q1 – first quartile Q3 – third quartile lower extreme (the smallest value) upper extreme (the largest value)

Make a box plot 5, 6, 6, 7, 8, 8, 12, 13, 13, 15, 15

Make a box plot 20, 22, 23, 25, 25, 30

Make a box plot 100, 110, 110, 120,140, 140, 160, 160

1 0 - 0 0 1 2 3 4 5 6 7 8 9 10 How many times have you gotten 100% on a spelling test this year? Let’s begin our description of the spread by beginning with the range. To find the range of a data set, take the least data point and subtract it from the greatest data point. 10 minus 0 equals 10, so 10 is the range of this data. This tells us that no matter what the data points are, the smallest point is 10 whole spaces away from the highest point. For example, in another graph the least data point could be 580 and the greatest data point could be 590, but they still have a range of 10, because the greatest data point is 10 whole spaces away from the least.

9.5 - 5 4.5 0 1 2 3 4 5 6 7 8 9 10 How many times have you gotten 100% on a spelling test this year? Next we will find the interquartile range. To do this we are going to draw a box plot for the data. We start by finding the median, by crossing out the least and greatest data points until we reach the middle. The median of this data set is 8, so we will draw a line at 8. Let’s also draw a circle around the median so we don’t forget which point it is. Next we need to find the 25th percentile, which will the median in the first half of the data. So we will cross out the least data point, followed by the greatest in this set, and keep going until we find the median of this group. The median is 5, so we’ll draw a line for the 25th percentile at 5. Now we need the 75th percentile, which is the median in the upper half of the data points. So we begin by crossing out the least data point following the median, followed by the greatest data point, and continue until we reach the middle. Since it is between the 9 and 10 we will draw a line for the 75th percentile at 9.5. It’s not necessary for this measurement, but we can complete the box plot and connect the least data point and the greatest data point to the box. The box surrounds the interquartile range, which is the data from the 25th to the 75th percentile. To find the interquartile range, we need to take the 25th percentile value of 5 and subtract it from the 75th percentile value of 9.5. 9.5 minus.5 = 4.5, so the interquartile range is 4.5. How does this help us describe the spread? Half of all of the data points are inside the interquartile range, and they have a spread of 5. That means that the middle half of the data points have a distance of 5 whole steps between the 25th percentile, or the number that is ¼ of the way from least to greatest, and the 75th percentile, or the number that is ¾ of the way from least to greatest. Does this sound a lot like the range? It is! But imagine you have a set of data with one point that is much much higher than the others, also known as an outlier. The range would be huge, but the interquartile range would not include this outlier, so it can be a better way of measuring the spread.

Books Read this Summer by 6th Graders 0 2 4 6 8 10 12 14 16 18 20 22 Number of Books Books Read this Summer by 6th Graders Now try a problem on your own. Find the range, interquartile range, and mean absolute deviation of this data set. Hit pause while you work on the problem, and then push play again to hear my answers. To find the range we take the greatest data point of 21 and subtract the least data point of 0, to get a range of 21. To find the interquartile range we need to start by finding the median. The median here is 3 so we draw a line there and circle the median. Next we find the 25th percentile at 2 so we mark a line at 2. The 75th percentile is at 6 so we mark a line there. We can continue to draw the box plot. Finally we take the 25th percentile value of 2 and subtract it from the 75th percentile value of 6 to get an interquartile range of 4. Finally we will find the mean absolute deviation. We start by finding the mean, so we add all of the data points together to get a sum of 60. Next we divide by the number of data points, 15, to get a mean of 4. Now we need to find the distance from each point to the mean of 4, so we have a 4, 2 3s, 4 2s, 3 1s, 2 0s, 2 2s, and 1 17. When we sum these numbers together we get 38. Divide this by the number of data points, 15, and we get a mean absolute deviation of approximately 2.5. What do these numbers mean? The range is 21, the interquartile range is 4, and the mean absolute deviation is approximately 2.5. Since the range is so different from the interquartile range it tells us there is an outlier. Because the interquartile range and mean absolute deviation are both very small they tell us that most of the data points are clustered together, although we don’t know which end has the cluster without looking at the data.

How many letters are in your first and last name? 7 8 9 10 11 12 13 14 15 16 17 How many letters are in your first and last name? 7 7 8 9 9 9 9 10 10 11 11 12 14 14 15 17 17 17 Median 10.5 (10 and 11 in middle) Q1 = 9 Q3 = 14 IQ 5