2.8: Mean, Median and Mode
Introduction
The Carrot Review
Tania and Alex are continuing to plan for next year’s garden. Today, Tania has decided to complete a harvesting review of carrots. She wants to use the number of carrots that were picked each week to make some conclusions about carrot growth.
Here are the three questions that she is trying to figure out.
- What is the average amount of carrots that were picked overall?
- What number of carrots was harvested the most often?
- What is the middle number of carrots that were picked?
Here is Tania’s data about the number of carrots picked each week over nine weeks of harvest.
2, 8, 8, 14, 9, 12, 14, 20, 19, 14
This is a total of 120 carrots-the number of carrots that we saw from the last section.
Your task is to help Tania. To do this, you will need to learn all about mean, median and mode. Once you have learned about these mathematical ways of analyzing data, you will be ready to help Tania with her carrot review.
What You Will Learn
In this lesson, you will learn how to use the following skills.
- Find the mean of a set of data.
- Find the median of a set of data.
- Find the mode of a set of data.
- Identify the range of a set of data.
- Select the best average to represent given sets of data.
Teaching Time
I. Find the Mean of a Set of Data
The first way of analyzing data that we are going to learn about is called the mean. A more common name for the mean of a set of data is to call it the average. In other words, the mean is the average of the set of data.
An average lets us combine the numbers in the data set into one number that best represents the whole set. First let’s see how to find the mean, and then we’ll learn more about how to use it to interpret data.
There are two steps to finding the mean.
- We add up all of the numbers in the data set.
- We divide the total by the number of numbers in the set.
Let’s look at an example.
Example
10, 7, 3, 8, 2
First, we need to add all the numbers together.
10 + 7 + 3 + 8 + 2 = 30
Now we divide the total, 30, by the number of items in the set.
There are 5 numbers in the set, so we divide 30 by 5.
30 \begin{align*}\div\end{align*}
The mean, or average, of the set is 6.
Next, let’s see how finding the mean helps us interpret data.
Suppose we want to know how tall plants grow when we add a certain nutrient to the water. Here is an example.
Example
The data below shows the height in inches of 10 plants grown with the nutrient-rich water.
9, 10, 7, 3, 11, 9, 8, 11, 7, 10
Let’s find the mean. Add up all of the numbers first.
9 + 10 + 7 + 3 + 11 + 9 + 8 + 11 + 7 + 10 = 85
Now we divide by the number of items in the data set. There are 10 plants, so we get the following answer.
85 \begin{align*}\div\end{align*}
The mean height of the plants is 8.5 inches.
This gives us a nice estimate of how tall a plant might grow with the nutrient-rich water.
Let’s see where the mean falls in relation to the other numbers in the set.
If we reorder the numbers, we get
3, 7, 7, 8, 9, 9, 10, 10, 11, 11
The minimum of the set is 3 and the maximum is 11. Take a good look at all of the numbers in the set.
Here are some conclusions that we can draw from this data.
- Only 3 stands out by itself at one end of the data set. Since it is much smaller than the other numbers, we might assume that this plant didn’t grow very well for some reason.
We can make a prediction based on this.
- Perhaps of the 10 plants it got the least light, or maybe its roots were damaged.
The mean helps even out any unusual results such as the height of this one plant.
Here are a few for you to practice on your own. Find the mean for each set of data. You may
- 3, 4, 5, 6, 2, 5, 6, 12, 2
- 22, 11, 33, 44, 66, 76, 88, 86, 4
- 37, 123, 234, 567, 321, 909, 909, 900
Take a minute and check your answers with a peer.
II. Find the Median of a Set of Data
The median of a set of data is the middle score of the data. Medians are useful whenever we are trying to figure out what the middle of a set of data is. For example, let’s say that we are working to figure out what a median amount of money is or for a runner what a median time is.
Let’s look at an example and the steps to finding the median of a set of data.
Example
2, 5, 6, 2, 8, 11, 13, 14, 15, 21, 22, 25, 27
Here is a set of data. To find the median of a set of data we need to do a couple of things.
1. Write the numbers in order from the smallest to the greatest. Be sure to include repeated numbers in the list.
If we do that with this set, here are our results.
2, 2, 5, 6, 8, 11, 13, 14, 15, 21, 22, 25, 27
2. Next, we find the middle number of the set of data.
In this set, we have an odd number of values in the set. There are thirteen numbers in the set. We can count 6 on one side of the median and six on the other side of the median.
Our answer is 13.
13 is the median.
Take a minute and write these steps down in your notebooks.
This set of data was easy to work with because there was an odd number of values in the set.
What happens when there is an even number of values in the set?
Let’s look at an example.
Example
4, 5, 12, 14, 16, 18
Here we have six values in the data set. They are already written in order from smallest to greatest so we don’t need to rewrite them. Here we have two values in the middle because there are six values.
4, 5, 12, 14, 16, 18
The two middle values are 12 and 14. We need to find the middle value of these two values.
To do this, we take the average of the two scores.
\begin{align*}12 + 14 &= 26\\
26 \div 2 &= 13\end{align*}
The median score is 13.
Here are a few for you to try on your own. Find the median of each set of data.
- 5, 6, 8, 11, 15
- 4, 1, 6, 9, 2, 11
- 23, 78, 34, 56, 89
Take a minute to check in with a neighbor. If your answers don’t match, recheck your work.
III. Find the Mode of a Set of Data
The mode of a set of data is simply the number that occurs most often.
When we put our data in numerical order, it becomes easy to see how often each of them occurs.
Let’s look at the data set below.
61, 54, 60, 59, 54, 51, 60, 53, 54
First, we put the data in numerical order.
51, 53, 54, 54, 54, 59, 60, 60, 61
Now we look for any numbers that repeat.
Both 54 and 60 appear in the data set more than once. Which appears more often?
54 repeats the most times. That is our mode.
Our answer is 54.
What if a data set doesn’t have a repeating number?
If no number occurs more than once, or if numbers appear in the set the same number of times, the set has no mode.
Let’s look at an example.
Example
22, 19, 19, 16, 18, 21, 30, 16, 27
In the set above, both 16 and 19 occur twice.
No number in the set happens the most often, so there is no mode for this set.
How can we use the mode to analyze data?
Because it is the number that occurs most often in a data set, we know that it is the most frequent answer to our question or result of our experiment.
Example
Suppose the data below shows how many people visit the zoo each afternoon.
68, 104, 91, 80, 91, 65, 90, 91, 70, 91
We can see that 91 occurs most often in the set, so we know 91 is the mode.
This number helps us approximate how many people visit the zoo each afternoon because it was the most frequent number.
Here are a few for you to try on your own. Identify the mode of each set of data.
- 2, 4, 4, 4, 6, 7, 8, 8, 10, 10, 11, 12
- 5, 8, 9, 1, 2, 9, 8, 10, 11, 18, 19, 20
- 12, 12, 5, 6, 7, 11, 23, 23, 67, 23, 89, 23
Take a minute to check your answers with a peer.
IV. Identify the Range of a Set of Data
The range of a set of data simply tells where the numbers fall, so that we know if they are close together or spread far apart. A set of data with a small range tells us something different than a set of data with a large range. We’ll discuss this more, but first let’s learn how to find the range.
Here are the steps for finding the range of a set of data.
- What we need to do is put the values in the data set in numerical order. Then we know which is the greatest number in the set (the maximum) and which is the smallest number (the minimum).
- To find the range, we simply subtract the minimum from the maximum.
Take a minute to copy these steps into your notebook.
Take a look at the data set below.
Example
11, 9, 8, 12, 11, 11, 14, 8, 10
First, we arrange the data in numerical order.
8, 8, 9, 10, 11, 11, 11, 12, 14
Now we can see that the minimum is 8 and the maximum is 14. We subtract to find the range.
14 - 8 = 6
The range of the data is 6. That means that all of the numbers in the data set fall within six places of each other.
All of the data results are fairly close together.
How can we use a range to help us answer a question?
Suppose we wanted to know the effect of a special soil on plant growth. The numbers in this data set might represent the height in inches of 9 plants grown in the special soil. We know that the range is 6, so all of the plants heights are within 6 inches of each other.
What if the 9 plants had these heights instead?
6, 11, 4, 12, 18, 9, 25, 16, 22
Let’s reorder the data and find the range.
4, 6, 9, 11, 12, 16, 18, 22, 25
Now we can see that the minimum is 4 and the maximum is 25. Let’s subtract to find the range.
25 - 4 = 21
The range of this data is 21. That means the numbers in the data set can be much farther apart.
What does this mean about plants grown in special soil?
If the first group of plants had a range of only 6, their heights ended up being fairly close together. So they grew about the same in the special soil.
In contrast, the second group of plants had a much greater range of heights. We might not be so quick to assume that the special soil had any effect on the plants, since their heights are so much more varied.
The range has helped us understand the results of the experiment.
Here are a few for you to try on your own. Find the range of the following data sets.
- 4, 5, 6, 9, 12, 19, 20
- 5, 2, 1, 6, 8, 20, 25
- 65, 23, 22, 45, 11, 88, 99, 123, 125
Take a few minutes to check your work with a peer.
V. Select the Best Average to Represent Given Sets of Data
Sometimes when we analyze a set of data we aren’t sure which average is best. We don’t know whether to use the mean, median, range or mode to assist us.
How can we figure out which is the best average to use?
As we have seen, interpreting data sets helps us answer a question or know the result of an experiment. Finding the range, mode, median, and mean allows us to understand a set of data in the context of the question.
The mean in particular helps us approximate a single numerical answer to the question because it points us to the number that is most likely to be a result every time you ask the question or run the test in your experiment.
We call this search for the data that is most likely to occur finding the measure of central tendency.
What are good measures of central tendency?
The mean is usually a good measure of central tendency. For example, if we grow 10 plants and their mean height is 12 inches, we can assume that if we grow 10 more plants, their heights will also be around 12 inches.
Measuring central tendency by finding the mean helps us predict the data we’ll get.
In some cases, however, the mean is not always an accurate predictor of central tendency.
Let’s find the mean of the data set below to see why.
Example
5, 10, 12, 7, 6, 150, 6
First we add the numbers together.
5 + 10 + 12 + 7 + 6 + 150 + 6 = 196
Then we divide by the number of items in the set, which is 7:
196 \begin{align*}\div\end{align*}
The mean for this set of data is 28.
Now let’s look more closely at the data.
Most of the numbers are pretty small. In fact, all but one of them is much less than 28!
150 thrown in there has really thrown off the mean.
It is so much higher than the other numbers that it has pulled the mean far away from the central tendency. If the number were 15, not 150, the mean would be 5 + 10 + 12 + 7 + 6 + 15 + 6 \begin{align*}=\end{align*}
In a set of numbers that fall between 5 and 15, 8.7 are a pretty good indication of the central tendency.
In our first set of data, on the other hand, six of the seven numbers fell well below the mean.
Therefore the mean is not a good predictor of central tendency when there is a particularly high or low number in the data set.
What can we do to find the measure of central tendency in this case?
In these cases, we should use the median to predict central tendency instead.
Let’s look again at the first set of data.
Example
What’s the median?
5, 6, 6, 7, 10, 12, 150
Remember, the median is the middle number in the set. The median of this data set is 7.
If we look at the other numbers in the set, it seems that 7 better represents most of the numbers in the set.
Therefore the median, 7, is a better estimate of future data than 28, the mean, is.
It’s possible that we could get another number as high as 150, but six other numbers in the set indicate that it’s more likely future data will be closer to them.
Let’s practice spotting situations where we should use the mean or the median to measure central tendency.
Example
Which is the better measure of central tendency for the data below: the mean or the median?
43, 58, 61, 47, 52, 7, 55
Look carefully at all of the numbers in the set.
What are the minimum and the maximum?
The minimum is 7 and the maximum is 61.
Now think about where all of the other numbers fall in between these two numbers.
Most of them are much closer to 61 than to 7.
This is probably going to be a case where the mean is skewed by the low number.
Let’s check to make sure by finding the mean and the median.
To find the mean, we add the numbers and then divide by 7 because there are 7 numbers in the set.
\begin{align*}46 + 57 + 60 + 48 + 51 + 7 + 53 &= 322\\
322 \div 7 &= 46\end{align*}
The mean of this data set is 46.
Now let’s find the median by reordering the numbers to find the middle number.
7, 46, 48, 51, 53, 57, 60
The median of the set is 51.
Now let’s look at the distribution of the numbers in the set to see which number is the better measure of central tendency, 46 or 51.
Where does 46 fall in relation to the other numbers?
There is only one number less than it, 7, which happens to be a lot less.
There are five numbers above 46.
This suggests that 46 does not really fall in the middle of the data set.
What about 51? It sits nicely among the numbers 46 – 60, which make up the bulk of the data set. Therefore 51 is a better measure of central tendency.
The mean is too low; it has been pulled down by that stray 7 which doesn’t fit with the rest of the numbers in the data set.
Using the median instead of the mean helps correct this flaw.
Our answer is to use the median.
Real Life Example Completed
The Carrot Review
Now we are ready to help Tania analyze her carrot growth.
Let’s look at the problem one more time.
Tania and Alex are continuing to plan for next year’s garden. Today, Tania has decided to complete a harvesting review of carrots. She wants to use the number of carrots that were picked each week to make some conclusions about carrot growth.
Here are the three questions that she is trying to figure out.
- What is the average amount of carrots that were picked overall?
- Which number of carrots was harvested the most often?
- What is the middle number of carrots that were picked?
Here is Tania’s data about the number of carrots picked each week over nine weeks of harvest.
2, 8, 8, 14, 9, 12, 14, 20, 19, 14
This is a total of 120 carrots-the number of carrots that we saw from the last section.
First, we can underline all of the important information.
Next, let’s answer the first question.
1. What is the average amount of carrots that were picked overall?
To answer this question, we add up the values in the data set and divide by the number of values in the data set.
\begin{align*}2 + 8 + 8 + 14 + 9 + 12 + 14 + 20 + 19 + 14 &= 120\\
120 \div 10 &= 12\end{align*}
The mean or average is 12.
2. Which number of carrots was harvested the most often?
To answer this question, we need to reorder the data to find the mode or the number that occurs the most often.
2, 8, 8, 9, 12, 14, 14, 14, 19, 20
The number 14 occurs the most often, that is the mode of this data set.
3. What is the middle number of carrots that were picked?
This question is asking us to find the median or middle number.
We look at a set of data listed in order.
2, 8, 8, 9, 12, 14, 14, 14, 19, 20
The median is between 12 and 14.
The median number is 13.
Now Tania has an idea of how many carrots were harvested when. If she doubles production next year she will be able to make predictions based on this data. She know that since the average number of carrots collected in one week is 12, then doubling production will mean that the average number of carrots collected in one week will go up to 24.
Tania and Alex are excited about growing vegetables next year.
Vocabulary
Here are the vocabulary words that are found in this lesson.
- Mean
- the average of a set of numbers. The mean gives us a good overall assessment of a set of data.
- Maximum
- the greatest score in a data set
- Minimum
- the smallest score in a data set
- Median
- the middle score in a data set
- Mode
- the number or value that occurs most often in a data set
- Range
- the difference between the smallest value in a data set and the greatest number in a data set
- Measures of Central Tendency
- ways of selecting which value in a data set best expresses the set of data.
Technology Integration
James Sousa, Mean, Median & Mode
Khan Academy Statistics: The Average
Other Videos:
- http://mail.clevelandcountyschools.org/~ahunt/FOV1-0002AEE9/S025ECADF?Plugin=Podcast – This is a song only, but has good content.
- www.teachers.tv/video/1495 – This is a British video on house prices.
Time to Practice
Directions: Find the mean for each set of data. You may round to the nearest tenth when necessary.
1. 4, 5, 4, 5, 3, 3
2. 6, 7, 8, 3, 2, 4
3. 11, 10, 9, 13, 14, 16
4. 21, 23, 25, 22, 22, 27
5. 27, 29, 29, 32, 30, 32, 31
6. 34, 35, 34, 37, 38, 39, 39
7. 43, 44, 43, 46, 39, 50
8. 122, 100, 134, 156, 144, 110
9. 224, 222, 220, 222, 224, 224
10. 540, 542, 544, 550, 548, 547
Directions: Find the median for each pair of numbers.
11. 16 and 19
12. 4 and 5
13. 22 and 29
14. 27 and 32
15. 18 and 24
Directions: Identify the mode for the following sets of data.
16. 2, 3, 3, 3, 2, 2, 2, 5, 6, 7
17. 4, 5, 6, 6, 6, 7, 3, 2
18. 23, 22, 22, 24, 25, 25, 25
19. 123, 120, 121, 120, 121, 125, 121
20. 678, 600, 655, 655, 600, 678, 600, 600