How do you find the range of a set of numbers? The good news is you don’t have to rearrange the whole set of numbers to find the range. You do however have to find the biggest and smallest number in the set. Let’s take a look at how to find the range.
How to Find the Range
Very simply, the range is the difference between the largest value in the set of numbers and the smallest. The only work you have to do is find the lowest and highest values, then you simply subtract the lowest from the highest. This shows us how much variation there is in the set of numbers.
It doesn’t matter how many numbers the set contains, or how many times the lowest OR highest numbers show up. Just find the highest, even if it shows up multiple times. Same for the lowest. Subtract the two and you’ve done it. One simple subtraction calculation gives you the range.
For a shortcut on how to quickly find the highest and lowest numbers without putting them all in order, check out the video above at 0:43
The highest number in the set is 29 and the smallest number in the set is 10. 29 minus 10 equals 19, which is our range. This was short and sweet.
Even if this data set had 10,000 numbers in it, the process is still the same. Highest minus lowest equals range.
Another Range Example Problem
Let’s find the range of the following set:
57, 43, 65, 55, 39, 50, 65, 64, 52, 61, 49
Remember we are going to subtract the highest and lowest numbers in this set. To quickly find the highest and lowest, check out the video at 2:40. The highest value is 65, the lowest is 39. We subtract 39 from 65 to get 26, our range.
This process is the same even if we had a list of decimal numbers or anything else you can think of. I hope this helps you quickly find the range of a set.
To learn how to find other values of a set of numbers, check out the other articles in this series:
How do you find the midrange of a set of numbers? Midrange and Range are similar, but different. They’re easy to get confused so we’ll discuss the differences, and give you two examples of how to quickly find the midrange.
Differences Between Range and Midrange
Range is the difference between the largest value in the set of numbers and the smallest. To find the range you do not have to put the numbers in order, but you do have to identify the largest and the smallest. Then you simply subtract the lowest from the highest. This shows us how much variation there is in the set of numbers.
The key difference is that for midrange we want to find the MIDDLE of the RANGE (hence midrange). We do this by finding the average of the lowest and highest values.
Realize that I used the word average instead of “mean” because I don’t want you to get confused between the mean of the set and the mean of the lowest and highest value.
So to find midrange we still find the lowest and highest value but we ADD them and DIVIDE by two. This gives us the midrange.
For a shortcut on how to quickly find the highest and lowest numbers without putting them all in order, check out the video above at 1:08
The highest number in the set is 29 and the smallest number in the set is 10. We add them, which gives us 39. We divide 39 by 2, which gives us our midrange value which is 19.5. It is ok to have a decimal answer, so don’t worry.
Another Midrange Example Problem
Let’s find the midrange of the following:
57, 43, 65, 55, 39, 50, 65, 64, 52, 61, 49
Remember we are going to take an average of the highest and lowest numbers in this set. To quickly find the highest and lowest, check out the video at 2:47. The highest value is 65, the lowest is 39. They add to give us 104. Then we divide 104 by 2 which gives us our midrange. The midrange is 52.
To learn more check out the other articles in this series:
A way to graphically represent a data set in ascending order by categorizing the data in which part of the number is shown on the left and called a “Stem” and the last digit is shown on the right and called a “Leaf.”
Blah, blah, blah, blah, blah
This is probably what your textbook says.
So, What is a Stem and Leaf Plot?
Here’s a better definition of a stem and leaf plot:
It takes a long list of numbers and puts them in order from smallest to largest. Draw a vertical line. Take all but the last digit of each number and list them in order, top to bottom, on the left side of the line. Only write each number once, this creates the stem. Now, take the last digit of each number and put them in order from left to right on the other side of the line. The catch is, you must put the number on the same line as it’s “stem.”
This is more like what your teacher or professor might say. This is more descriptive, but still not good enough.
Here’s OUR definition of stem and leaf plot:
Put the numbers in order, from least to greatest
Break off the last digit from each number
The left digits are the stems, if there is no left digit, use zero. We list these ONCE.
Right digits are the leaves, we will list them ALL
“Stemplot” is just another way to say Stem and Leaf Plot. The Stem is main part of the diagram, hence it has become shorthand to just mention a stemplot. Regardless, you treat them the same. Stemplots are a very robust and effective way to illustrate data, even though they look quite simple. Stemplots can take a long, complicated list of many data points and categorize them in an easy to read format.
Stemplot with Decimals
Stemplots are actually more useful than some give them credit for. They are usually only shown indicating two digit numbers. Not only can they be used to show three digit numbers, they can also be used with decimals. To use stemplots with decimals, the leaves become the decimal points.
For the data set: 23.2, 23.4, 23.8, 24.1, 24.9, 25.0, 25.1, 25.1, 25.1, 25.3, and 25.8, the stem and leaf plot would look as follows.
Note that the stems are the two digit numbers and the leaves are the decimal points. It is crucial when you use stemplots to include a key. In this instance, the key indicates that the sample data point is in fact a decimal number, not a three digit number.
Can You Split the Stem with Stemplots with Decimals?
Absolutely! If you have a particularly large data set, you can still split the stem. The same rules apply.
Is it OK to Use a Back to Back Stemplot With Decimals?
Yes. Similar to splitting the stem, you can use a back to back stemplot if you have a second data set.
This video will give us the definition of median. In any given set of numbers, the median is the number that is in the middle of the set, when the numbers are ordered from least to greatest.
Median is the Middle Number
You will deal with the median any time you have a set of numbers. This can be when you are in the topic of statistics, or you could even pull numbers from a histogram or line plot, order them, and find the median of the graph. The median is a good way to describe the middle of a list of numbers.
Finding the median should be easy right? Well, there’s a catch.
For an Odd Amount of Numbers, Median is Easy
For an Even Amount of Numbers, You must calculate Median
If you have a group of numbers like this: 23, 26, 26, 26, 29, 36, 37, 38, 42, 42, 48, all you have to do is find the number in the middle. The numbers are already listed from least to greatest, so the first step is done for you. Since there are 11 numbers, we find the middle number, which is the sixth number. Our median is 36.
If you have a group of numbers like this: 23, 26, 26, 26, 29, 36, 39, 40, 42, 42, 48, 49, finding the median takes an extra step. Since there are 12 numbers, there is no single number in the middle. We take the sixth and seventh numbers, which are 36 and 39, and we must calculate the median.
To calculate the median, we add the two middle numbers and divide by two. So basically we split the difference between the two middle numbers to find the median of the set. 36 plus 39 is 75, and 75 divided by 2 is 37.5.
In Review:
The Median is the middle number in a set. First you order them from least to greatest, and either choose the middle number or calculate the median between the two middle numbers.
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In any given set of numbers the mean is the sum of those numbers all divided by how many numbers there are in a given set. Mean is a statistical term but it is also commonly referred to as the average.
Mean Is The Same As Average
When will you deal with mean? Mean will be used as a way to try and describe any set or group of numbers. This can be in statistics, or you can even pull data from charts, like histograms and line plots and calculate the mean. This will teach you more about the data set.
An Example of Finding the Mean
For the data set: 15, 10, 19, 19, 7, 11, 15, 19, 20, 12, 17, and 18, how would we find the mean? Unlike the mode, you can’t just look at this data set and “see” the mean. But, on the other hand, unlike the median, you don’t have to order the numbers from least to greatest. You can leave them how they are. But we do have to do some calculation.
The sum of all the numbers is 182, and we count that there are 12 individual numbers represented. 182 divided by 12 gives us a mean of 15.17. Don’t freak out when you see a decimal number for the mean. It’s actually more common to get a decimal number than a whole number when you calculate the mean of a set of numbers.
In Review:
The mean is the average of all the numbers in a data set. We add the numbers and divide by how many numbers were in the group.
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Today we will discuss the definition of Mode. Mode is a term that is used any time we have a group of numbers.
Mode is the number in a given set of numbers that occurs with the greatest frequency (or most often). A good way to remember this is to always think:
MOde happens MOst often
When will you deal with mode? Mode will be used as a way to try and describe any set or group of numbers. This can be in statistics, histograms, or line plots. Any time you have a group of numbers you can find the mode.
Example of Mode
Let’s look at an example. For the numbers: 15, 10, 19, 19, 7, 11, 15, 19, 20, 12, 17, and 18, find the mode. We see that the number 19 occurs 3 times, so it is the mode of the data set. The number 15 occurs twice, and all other numbers only occur once. Note that the numbers do not have to be in any order to find the mode. It might help, but isn’t necessary.
In Review:
Always remember the mode is the number that shows up most often (greatest frequency) in a set of numbers.
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You probably know the basics of Stem and Leaf Plots. You take some boring list of data points, order them from smallest to largest, and slap them on a plot. If you’re really on top of your game you’ll include a title and a key. It’s not too bad. But then you have to learn about split stems and back to back plots… I can tell you’re already getting sleepy. Wake up and make learning fun with these Stem and Leaf Plot Games.
That’s right, GAMES! Actual group activities that involve you interacting with others all while applying this to learning stem and leaf plots.
Maybe you’re a teacher and you’re looking for a way to teach stem and leaf plots.
Or maybe you’re a tutor trying to come up with ideas to help a group of students.
Or maybe you’re just looking for a fun way to learn while interacting with others.
There are very fews ways to learn something as effectively as hands on interaction with others. And pretty much anyone will say that group activities help break the monotony of learning something from a lecture, textbook, or computer screen. Get the worksheet at the link below:
In this worksheet you will find teacher/instructor directions and student worksheets. It is an all in one worksheet that you can use to teach or learn basic and advanced stem and leaf plot applications.
P.S. – If you are a teacher, professor, or school/university official I would LOVE to know if this was helpful to you. Please email me at contact[at]learnalgebrafaster[dot]com. I would love to chat with you and possibly send you other instructional material that I’ve developed.
Having trouble learning stem and leaf plots? Try some activities!
If you are trying to learn stem and leaf plots, here are a few activities you can do to help understand the topic better.Whether you are a student trying to grasp a topic, a teacher trying to engage with your students, or a parent or tutor trying to give good examples, activities are always a way you can make a topic stick with a student.
Stem and Leaf Plot Individual Activity Ideas
Plot the scores of your favorite sport events
Grab a television remote control and tune in to your favorite sports news channel. Alternatively you can pull your favorite sports website. Pick a sport, and begin writing down each team’s score in their latest games/matches. After you have recorded the scores from your favorite division, league, etc. put them on a stem and leaf plot.
Note: You must use ONE sport only. Sports differ greatly in their typical scores, so be consistent.
Roll or toss a small object at a target
Find something sturdy in the room with you. If you cannot find anything specific, just use the wall itself. This is your target. Take a small object (preferably something that is NOT round or bouncy) and use as your marker. Now move to the other side of the room and mark a rolling or tossing point. The object of this activity is to toss or roll the marker object as close to the target as possible (similar objective as bocce ball, lawn darts, etc.). Each time you your marker lands, use a ruler to mark how far it was from the target. Do this 10, 20, or 30 times. Plot each measurement on a stem and leaf plot.
Note: It makes no difference how specific you are with your measurements. You can use inches or centimeters, and you can even use decimals. Be creative but be consistent!
Stem and Leaf Plot Group Activity Ideas
Measure the height of everyone in the group
This activity works best with a classroom of people, but any group will work. The objective is simple, measure everyone’s height with a ruler, meter stick, etc. Plot the results on a stem and leaf plot. If there are more than 20 people in the group, mix it up by having half the people plot their results in inches, the other half in centimeters. Or, if you have a smaller group, have everyone measured twice and create two plots.
Measure the distance that everyone in the group can jump (from a standstill)
Elite athletes often measure their athletic performance in standard ways so that they can judge their overall athletic improvement. One such method is called a “broad jump.” Have each group member stand with their toes at one line and have them jump as far forward as they can. In order for the jump to count, they must jump from both feet and land without falling or taking extra steps. Measure from the launch point to the back of their closest foot. If you have a small group, have them jump again. Record all measurements on a stem and leaf plot. If you have a large marker board, have each member write their own measurements on the plot (allowing room for other people to place their scores). This encourages teamwork and allows them to understand the importance of placing the data in order.
Stem and Leaf Plot Activity Tips
When you are doing these activities, notice how important it is to be consistent. Be consistent with what you measure and record. A plot that combines last night’s soccer scores (typically 0, 1, 2, or 3 points per team) with last night’s basketball scores (typically between 70 – 95 points) will look silly and not tell you much. There will be many empty stems and the plot is not consistent.
Also, with measurements it is important to include the units of measure. It makes no difference if your measurements are in inches, meters, centimeters, etc. as long as you are consistent. A plot that combines inch measurements with centimeter measurements is incorrect and confusing.
Title is important! The way you title your stem and leaf plot is crucial. You must describe what the plot represents. You cannot label a graph “sports scores” or “distance.” You must say “basketball scores” or “jumping distance measurements.” The goal is for someone to be able to know exactly what the plot represents. Remember, a person who reads the plot later on probably did not witness the measurement or data collection.
Do not forget the key! A proper key tells the reader what a typical stem and leaf entry represents. Just like the title, it must be descriptive enough that anyone will know what the leaves mean (and units).
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So you know that Stem and Leaf Plots are great at taking a big list of numbers and organizing them in an easy to read fashion. You might even know about how if there are lots of data points you can use a Split Stem and Leaf Plot to separate the data even further, giving you a more readable plot.
But what if you could double all this?
Back to Back Stem and Leaf Plots give you the ability to take two separate data sets and put them on the same plot. This allows you to compare the different data sets on one plot.
How Does a Back to Back Stem and Leaf Plot Work?
On a normal plot, the stem is on the left and all the leaves are on the right. There is a vertical line separating the two. On a back to back plot, the stem remains the same. But to add another set of data points, we begin adding leaves to the LEFT side.
Just like on a typical plot, the smallest leaves are placed closest to the stem, and larger leaves are further away. The stem now serves a double purpose. It anchors both sets of data points, keeping them separate but it still organizes both.
In a back to back stem and leaf plot, you can compare two sets of data, and still be able to find the statistical measurements of each set. It also retains the same pros and cons of a normal plot. In the picture above, you can see that the stems work for each side of the plot, yet the data is separate. We added the points for Team B, but we started building at the center of the plot. It is pointed out on the last line of Team B’s points that the smaller leaves are closest to the stem and the larger leaves are farther away. In this picture you can see that Team B scored 92, 92, 92, 93, 96, and 96, respectively. Note the locations of the “2” leaf and the “6” leaf.
What is different about a Back to Back Stem and Leaf Plot?
You do not have to have the same number of data points on each side of the plot.
Add stems as necessary to accommodate the new data points.
You can still split the stems if you need to!
As you can see in the picture to the right, Team A has more data points than Team B. This is ok! It is not critical for them the have the same number of data points on the graph. Do not go crazy with this though. If one side of the plot has many more data points than the other, it will look odd. The purpose of the plot is to organize the data.
The second point is that the key must work for both sides of the plot. DO NOT mix units or try to save space by combining plots when they do not measure different things. Note that this plot compares Team A and Team B. They are playing the same game. If one team was playing basketball and the other was playing football, a back to back stem and leaf plot CANNOT be used.
Some times a back to back plot is avoided because the stems are not identical. Don’t abandon a back to back plot because of this. You can add stems to accommodate more data. But, as in the first point, do not go overboard with this. In this second example you will see how we added a “3” stem for Team B, but did not violate rules #1 or #2 by doing this.
I didn’t show in this example but you can even combine a back to back plot with a split stem and leaf plot if you want! This can be complicated so I will dedicate an article to this in the future.
Do you have anything you would like to see a post on? If so, email me or comment below.
Let’s find the range of the following set:
