11.1: Mean
Have you ever heard of the Iditarod?
The students in Mr. Hawkins class are studying about the Iditarod. Some of the students had never heard of the Iditarod before, so Mr. Hawkins started off his class by showing them this video about the race.
The students sat through the video in awe. When it was over, the room was so quiet that you could have heard a pin drop. Marcus was the first one to raise his hand.
“How far is it?” he asked.
“That is a great question,” Mr. Hawkins said. “The race is 1,150 miles from Anchorage Alaska to Nome Alaska. Men and women have finished it and won it too. This year, there were 10 men who finished on top. One finished in 8 days and the rest in 9 days.”
“How fast did they go?” Karen asked from the back of the room. “I mean, you can’t go very fast on a dog sled, right?”
“Well, for you and me it might not seem fast, but for those dogs I am sure that it is. This leads us to a great math problem. Here are the speeds of the top 10 finishers. What is the average speed here?”
Mr. Hawkins wrote these speeds on the board.
4.81 mph, 4.79 mph, 4.76 mph, 4.67 mph, 4.66 mph, 4.64 mph, 4.62 mph, 4.6 mph, 4.58 mph, 4.55 mph.
“Take out a piece of paper and figure this out.”
Marcus took out a piece of paper, but he couldn’t remember how to figure out the average score.
If you remember how to do it, figure it out now. Then go through this Concept to begin learning all about data and statistics. When you are finished this Concept, you can check your work with Marcus’ work and see if you have the correct average speed. Look for this problem again at the end of the Concept.
Guidance
Data is one of those words that we hear all the time, especially in math class. We hear about collecting data, organizing data, analyzing data, etc. But....
What is data?
Data is numerical information collected in a set.
When we look at data in math and science, we look at information that has been gathered over time or that has been gathered to evaluate a topic. Learning to look at data is part of the work of people in math and science. Analyzing data can help scientists predict future events too.
We can analyze numerical data several different ways. We can look for the mean, the median, the mode and the range of a set of data.
Let’s start with the mean.
The mean is sometimes referred to as the average of a set of data. The mean is the sum of the data values divided by the number of data values. You often hear about averages with grades or speeds. What is your average grade in math class? That number determines your final grade or the grade that you received on a test or quiz. We use averages all the time. Let’s look at the steps to figuring out the mean or average through this problem.
Find the mean for the set of data: 47, 56, 51, 45, and 41.
Step 1: Add the data values to determine the sum.
\begin{align*}47 + 56 + 51 + 45 + 41 = 240\end{align*}
Step 2: Divide the sum by the number of data values in the set. In this case, there are five numbers in the data set, therefore divide the sum by five.
\begin{align*}240 \div 5 = 48\end{align*}
The mean is 48.
Take a few minutes and write these steps for finding the mean of a set of values down in your notebook.
You can see that when we follow the steps that we can find the mean or average of a set of numbers quite easily. Let’s look at another one.
The chart below depicts the daily temperature in San Diego for the first seven days in August. Calculate the mean temperature for the first seven days in August.
Date: | Temperature: |
---|---|
Sunday 8/1 | \begin{align*}88^\circ F\end{align*} |
Monday 8/2 | \begin{align*}83^\circ F\end{align*} |
Tuesday 8/3 | \begin{align*}87^\circ F\end{align*} |
Wednesday 8/4 | \begin{align*}89^\circ F\end{align*} |
Thursday 8/5 | \begin{align*}82^\circ F\end{align*} |
Friday 8/6 | \begin{align*}79^\circ F\end{align*} |
Saturday 8/7 | \begin{align*}87^\circ F\end{align*} |
Step 1: Add to determine the sum of the data values.
\begin{align*}88 + 83 + 87 + 89 + 82 + 79 + 87 = 595\end{align*}
Step 2: Divide the sum, 595 by 7 because there are seven numbers in the given data set.
\begin{align*}595 \div 7 = 85\end{align*}
The mean temperature for the first week in August was \begin{align*}85^\circ F\end{align*}.
This was a real life example of how averages help us figure out weather. If you think about the weather forecast, you will often hear data mentioned. The meteorologist will talk about average snowfall or average temperature or average rainfall.
Sometimes, an average will not be an even number. When this happens, you can round to the nearest whole number.
Now it’s your turn to practice. Find the mean for each data set below. You may round when necessary.
Example A
\begin{align*}11, 13, 14, 15, 16, 22, 24, 25, 30, 32\end{align*}
Solution: \begin{align*}20\end{align*}
Example B
\begin{align*}34, 36, 38, 41, 43, 44, 50, 53, 50, 50, 62, 66\end{align*}
Solution: \begin{align*}47\end{align*}
Example C
\begin{align*}8, 16, 24, 32, 40\end{align*}
Solution: \begin{align*}24\end{align*}
Here is the original problem once again. Reread it and then compare your answer for the mean with the given solution.
The students in Mr. Hawkins class are studying about the Iditarod. Some of the students had never heard of the Iditarod before, so Mr. Hawkins started off his class by showing them this video about the race.
The students sat through the video in awe. When it was over, the room was so quiet that you could have heard a pin drop. Marcus was the first one to raise his hand.
“How far is it?” he asked.
“That is a great question,” Mr. Hawkins said. “The race is 1,150 miles from Anchorage Alaska to Nome Alaska. Men and women have finished it and won it too. This year, there were 10 men who finished on top. One finished in 8 days and the rest in 9 days.”
“How fast did they go?” Karen asked from the back of the room. “I mean, you can’t go very fast on a dog sled, right?”
“Well, for you and me it might not seem fast, but for those dogs I am sure that it is. This leads us to a great math problem. Here are the speeds of the top 10 finishers. What is the average speed here?”
Mr. Hawkins wrote these speeds on the board.
4.81 mph, 4.79 mph, 4.76 mph, 4.67 mph, 4.66 mph, 4.64 mph, 4.62 mph, 4.6 mph, 4.58 mph, 4.55 mph.
“Take out a piece of paper and figure this out.”
Marcus took out a piece of paper, but he couldn’t remember how to figure out the average score.
By now you understand that the average is the same thing as the mean. The students have been asked to find the mean speed of the dog sleds on the 2010 Iditarod. They have been given the speeds of the top ten finishers. This is the data that we will use to figure out the mean.
First, add up all of the speeds.
\begin{align*}4.81 + 4.79 + 4.76 + 4.67 + 4.66 + 4.64 + 4.62 + 4.6 + 4.58 + 4.55 = 46.68\end{align*}
Next, we divide this sum by 10 because there were ten dog sleds, so there were 10 different speeds.
\begin{align*}46.68 \div 10 = 4.668\end{align*} rounds to 4.67
The average speed is 4.67 mph.
Vocabulary
Here are the vocabulary words in this Concept.
- Data
- pieces of numerical information collected in a set.
- Mean
- the average value of a set of data.
Guided Practice
Here is one for you to try on your own.
John has the following quiz scores.
78, 90, 83, 88, 67, 90, 84, 69, 56
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 + 56 = 705\end{align*}
Next, we divide by the number of scores.
\begin{align*}705 \div 9 = 78.3\end{align*}
John's average is a 78.
Video Review
Here is a video for review.
- This is a Khan Academy video on mean, median and mode.
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, 6, 7, 8
2. 6, 7, 8, 3, 2, 4, 9, 10, 11, 12
3. 11, 10, 9, 13, 14, 16, 20, 22, 22
4. 21, 23, 25, 22, 22, 27, 18, 20
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, 120, 123, 130
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
Average
The arithmetic mean is often called the average.Data
Data is information that has been collected to represent real life situations, usually in number form.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.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.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.
Concept Nodes:
Average
The arithmetic mean is often called the average.Data
Data is information that has been collected to represent real life situations, usually in number form.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.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.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.