2.18: Mean
Have you ever heard of the word "average" in math? Have you ever tried to figure out the average or mean of a set of numbers?
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. First, she wants to figure out the mean or the average 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 in the last Concept.
Use what you will learn in this Concept to help Tania.
Guidance
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.
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. 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.
Now let's practice. Find the mean for each set of data.
Example A
3, 4, 5, 6, 2, 5, 6, 12, 2
Solution: 5
Example B
22, 11, 33, 44, 66, 76, 88, 86, 4
Solution: 47.7 or round up to 48
Example C
37, 123, 234, 567, 321, 909, 909, 900
Solution: 500
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 find the mean
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.
Vocabulary
Here are the vocabulary words in this Concept.
- 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
Guided Practice
Here is one for you to try on your own.
Jacob has the following quiz scores.
78, 90, 83, 88, 67, 90, 84, 69
Given these scores, what is his average for the quarter?
Answer
To begin, add up all of the scores.
\begin{align*}78 + 90 + 83 + 88 + 67 + 90 + 84 + 69 = 649\end{align*}
Next, we divide by the number of scores.
\begin{align*}649 \div 8 = 81.1\end{align*}
Jacob's average is an 81.
Video Review
Here is a video for review.
Khan Academy Statistics: The Average
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
11. 762, 890, 900, 789, 780, 645, 700
12. 300, 400, 342, 345, 403, 302
13. 200, 199, 203, 255, 245, 230, 211
14. 1009, 1000, 1200, 1209, 1208, 1217
15. 2300, 2456, 2341, 2400, 2541, 2321
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Average
The arithmetic mean is often called the average.Geometric mean
The geometric mean is a method of finding the ‘middle’ value in a set that contains some values that are intrinsically more influential than others.Harmonic mean
A harmonic mean is calculated by dividing the number of values in the set by the sum of the inverses of the values in the set.Maximum
The largest number in a data set.mean
The mean, often called the average, of a numerical set of data is simply the sum of the data values divided by the number of values.measures of central tendency
The mean, median, and mode are known as the measures of central tendency.Minimum
The minimum is the smallest value in a data set.Population Mean
The population mean is the mean of all of the members of an entire population.Sample Mean
A sample mean is the mean only of the members of a sample or subset of a population.weighted
A weighted value or set of values takes into account varying levels of importance among members of the set.weighted average
A weighted average is an average that multiplies each component by a factor representing its frequency or probability.weighted harmonic mean
A weighted harmonic mean is a harmonic mean of values with varying frequencies or weights.Image Attributions
Here you'll learn to find the mean of a set of data.