10.1: Measures of Central Tendency and Dispersion
Introduction
The Track Team
“It is time to get ready for competition!” Mr. Watson the track coach said to his track and field team on Monday afternoon.
“What does that mean coach?” Marco asked smiling a huge grin.
The Hawks were very excited that their season had gone so well and now they were ready to prepare for regionals.
“It means that we are all going to figure out where we are in our team standings and then set goals to improve. That way we can have a great showing at regionals,” Mr. Watson explained.
As soon as he heard this, Alfredo began to figure out his standing. Alfredo is a high-jumper at his school. He has 8 teammates whose records are 172 cm, 174 cm, 175 cm, 179 cm, 181 cm, 181 cm, 182 cm, and 185 cm. If Alfredo’s record is 176 cm, how does he compare to the rest of the team?
You can help Alfredo figure this out by understanding statistics and data. Pay close attention during this lesson and you will see this problem again at the end of the lesson.
What You Will Learn
By the end of this lesson, you will be able to complete the following skills.
- Find the mean, median and mode of a set of data.
- Find the deviation of each data point from the mean of a set of data.
- Find the range and the mean absolute deviation of a set of data.
- Find, analyze and compare measures of central tendency and dispersion to describe real – world data.
Teaching Time
I. Find the Mean, Median and Mode of a Set of Data
In the real world, there are many situations in which a large group of data is collected. In order to make sense of the data, we use a number of statistical measures. These measures help us to generalize a group of data, make inferences about it, and compare it with other groups of data. These statistical measures include mean, median, mode, range, deviation from the mean, and absolute deviation from the mean. Depending on the situation, certain measures may be more helpful than others in interpreting data. Let’s look at these statistical measures.
The mean, median, and mode are three common measures of central tendency; they are three mathematical tools frequently used to analyze data. The mean, commonly referred to as the average, is the sum of all the data items divided by the number of data items. The median is the middle number in a set of data that is ordered from lowest to highest. If there is an even number of data, we take the average of the middle two numbers to find the median. Finally, the mode is the number that occurs most often.
Example
A manager at a small movie theater was analyzing the number of people who came to the movies during the week. Over nine days, he found the following data: 81, 89, 92, 85, 93, 62, 85, 105, and 90. Find the mean, median, and mode of the data.
First, let’s find the mean. Remember that the mean is the same as the average.
Mean: add all of the data items and divide by the number of items.
\begin{align*}& = \frac{81+89+92+85+93+62+85+105+90}{9}\\ &= \frac{782}{9}\\ &= 86.8 \end{align*}
The average or mean is 86.8 which could be rounded up to 87.
Next, let’s find the median.
Median: the middle number when the data is ordered from lowest to highest.
First reorder the data from least to greatest:
\begin{align*} & 62, 81, 85, 85, 89, 90, 92, 93, 105 \\ & \qquad \qquad \quad \ \ \uparrow \\ & \text{The middle number, 89, is the median.}\end{align*}
The median is 89.
Finally, let’s find the mode.
The mode is the number that occurs most often. In this case, 85 occurs two times and all of the other number only once. The number 85 is the mode.
The mode is 85.
Now it is important for you to learn these definitions so that you can find the mean, median and mode of any set of data.
II. Find the Deviation of Each Data Points from the Mean of a Set of Data
At a restaurant, the food servers report how much money in tips they earn each night. One Saturday, the food servers reported the following tips: $45, $37.50, $51, $89, $47, and $55. Greg is the newest food server and he reported $51. He wants to know how well he did compared with the others. One way to do this is to find the deviation from the mean. This tells you how far away from the mean, or average, each food server was. In order to find this, find the mean. Then find the difference between each number and the mean by subtracting. Because we want to find “how far away” each person was, it is like finding a distance—we only use positive numbers.
Step 1: Find the mean.
\begin{align*}& = \frac{45+37.5+51+89+47+55}{6}\\ &= \frac{324.5}{6}\\ &= 54.08\end{align*}
The average food server tip was $54.08.
Step 2: Find the difference of each food server tip from the mean. This is the deviation from the mean. Notice that Greg’s value is always used because we are looking for the deviation between the tips of the other waiters and Greg.
Difference from mean | Deviation from the mean |
---|---|
54.08 - 37.5 | 16.58 |
54.08 - 45 | 9.08 |
54.08 - 47 | 7.08 |
54.08 - 51 | 3.08 |
55 - 54.08 | .92 |
89 - 54.08 | 34.92 |
When we subtract, we place the largest of the two numbers first so that the difference is positive.
The deviation from the mean can let each food server know how far he or she was from the average tips that night.
Greg earned $51. The average was $54.08 so he was very close to the average although a little bit lower.
Write these steps down in your notebook.
Remember that when we look for the deviation from the mean, we are looking for the difference between an average and a value.
III. Find the Range and the Mean Absolute Deviation of a Set of Data
An important measure of data deals with how spread out the data is. In other words, two sets of data could have identical means, medians, and modes. This will not tell, however, how high the data went nor how low. The deviation from the mean shows how far an individual item is from the mean but does not say anything about the group as a whole. Two additional statistical measures help to analyze this factor in a set of data.
The range is found by subtracting the smallest number from the largest number. This will give you an idea of the span or the breadth of the data. For example, if you are a new car buyer and have just entered the work force, it may help for you to know that the mean price of new cars is $22,300. This may be too high for you, but the mean doesn’t really supply you with enough information. If you know that new car prices vary from $10,500 to $89,900, then you have a better idea about what a car might cost at the lower end, easier for your income level. The range of car prices, in this example is \begin{align*} \$89,900 - \$10,500 = \$79,400 \end{align*}. It is a broad range that should allow for you to purchase a car.
A second statistical measure that can be useful is the mean absolute deviation. We already found the deviation from the mean, which is how far an individual item is from the mean. Because we only found the positive difference, we actually found the absolute deviation. The mean absolute deviation, then, is the mean of those deviations.
I know that it seems confusing, try to think of it in another way. Let’s suppose that a school teacher gives an exam. The mean in the class is 82%. That seems pretty good, right? Doesn’t that mean that the average student got a B? Not necessarily. If the mean absolute deviation is high, it means that the average student either did much better or much worse than 82%. It indicates a greater range of scores. On the other hand, if the mean absolute deviation is low, it is a stronger indication that the group as a whole is working around the B level. Let’s look at another example.
Example
A city surveyor took elevation measurements around a coastal city that has a reported mean elevation of 35 feet above sea level. He went to various homes and gathered the following data: 152, 316, 26, 64, 20, 506, 210, and 89. Find the range and mean absolute deviation.
Step 1: Find the range. Subtract the smallest number from the largest.
\begin{align*}506 - 20 = 486\end{align*}
Step 2: Find the deviations from the mean (mean of 35 already given).
Difference from mean | Deviation from the mean |
---|---|
35 – 20 | 15 |
35 – 26 | 9 |
64 – 35 | 29 |
89 – 35 | 54 |
152 – 35 | 117 |
210 – 35 | 175 |
316 – 35 | 281 |
506 – 35 | 471 |
Step 3: Find the mean of the deviations from the mean.
\begin{align*}& = \frac{15+9+29+54+117+175+281+471}{8}\\ &= \frac{1151}{8}\\ &= 143.9 \end{align*}
The mean absolute deviation is about 143.9 feet.
You have been learning about all the different types of statistical data measures. Now let’s look at how to apply these in real-life situations.
IV. Find, Analyze and Compare Measures of Central Tendency and Dispersion to Describe Real – World Data
We have seen a variety of measures of central tendency. Mean, median, mode, range, deviation from the mean, and mean absolute deviation are all valuable tools for considering data. The exact tool that is best for a situation depends on what it is that an individual wants to know.
Example
An amusement park is designing a new section for children over 3 years old and under 8 years old. They take a survey of the heights and weights of a thousand children in that age group. Which measure of central tendency should they use to accommodate the greatest number of children on a roller coaster?
In order to attract the more customers, they should accommodate as many children as possible. For this reason, they should use the range which will include from the tallest to the shortest children on the ride.
Now that you have an idea how to do this, you can think about examples that would require the mean or average – think about grades or scores overall. You can also think about the mode if you thought about how many times a number occurs in a set of data. As in our amusement park, for example, how many days did 500 people attend? When using the median, remember to think about questions that ask for a middle score.
Real-Life Example Completed
The Track Team
Here is the problem from the introduction. Reread it and then use what you have learned to help Alfredo figure out his standings on the team.
“It is time to get ready for competition!” Mr. Watson the track coach said to his track and field team on Monday afternoon.
“What does that mean coach?” Marco asked smiling a huge grin.
The Hawks were very excited that their season had gone so well and now they were ready to prepare for regionals.
“It means that we are all going to figure out where we are in our team standings and then set goals to improve. That way we can have a great showing at regionals,” Mr. Watson explained.
As soon as he heard this, Alfredo began to figure out his standing. Alfredo is a high-jumper at his school. He has 8 teammates whose records are 172 cm, 174 cm, 175 cm, 179 cm, 181 cm, 181 cm, 182 cm, and 185 cm. If Alfredo’s record is 176 cm, how does he compare to the rest of the team?
Now solve this problem for Alfredo.
Solution to Real – Life Example
In this case, finding the deviation from the mean would be most useful in answering the question because he wants to compare his individual record to that of the team.
Step 1: Find the mean.
\begin{align*}& = \frac{172+174+175+179+181+181+182+185}{8}\\ &= \frac{1429}{8}\\ &= 178.6 \end{align*}
Step 2: Calculate his deviation from the mean: \begin{align*}178.6 - 176 = 2.6cm\end{align*}.
Alfredo’s record is 2.6cm lower than the mean. He should continue to work on improving his high jump. Now that Alfredo understands his place on the team, he can work with the coach on a plan to improve.
Notice that when comparing to others, we use the deviation from the mean.
Vocabulary
Here are the vocabulary words for this lesson.
- Statistical Measures
- measures used to generalize a set of data, make inferences and compare it with other groups of data.
- Measures of Central Tendency
- Math tools used to analyze data
- Mean
- the average of a set of data.
- Median
- the middle score in a set of data that has been arranged from smallest to largest.
- Mode
- the value that occurs the most times in a data set.
- Deviation from the mean
- how far a value is from the mean or average
- Range
- the breadth of the data, the difference between the largest and smallest values.
- Mean Absolute Deviation
- the mean of the deviations
Time to Practice
Directions: Find the mean, median, mode, range, deviations from the mean for the given data. Round all answers to the nearest tenths place. Notice that each answer has five answers.
- 13, 18, 24, 21, 16, 24, 14, 17, 24
- 116, 137, 120, 75, 98, 139
- 22, 24, 25, 30, 32, 34, 37, 38, 40
Directions: Define each term.
- Mean
- Median
- Mode
- Deviation from the mean
- Range
Directions: Complete the following.
- Invent a data set that has 8 items, a mean of 16, a median of 17, a mode of 10, and a range of 21.
- Two groups of adult female harbor seals were weighed from different parts of the globe, one from the Pacific Ocean and one from the Atlantic Ocean. The Pacific Ocean group had weights of: 126kg, 130kg, 135kg, 136kg, 137kg, 140kg, 148kg, and 150kg. The Atlantic Ocean group had weights of 117kg, 119kg, 122kg, 123kg, 130kg, 131kg, 141kg, 149kg, and 152kg. A marine biologist was concerned about their health and accessibility to food. Use a central tendency to compare the groups. Explain why you chose that central tendency.
- A convenience store supervisor oversees 3 stores in a geographical region. Part of his job is to analyze store performance. Because they are in different parts of town, it is only fair to compare a store to its own historical data. Look at the sales data and determine which store’s July sales is best using a measure of central tendency. Explain your reasoning.
Store # | January | February | March | April | May | June | July |
---|---|---|---|---|---|---|---|
Store 1 | 82,000 | 86,000 | 75,000 | 77,000 | 72,000 | 79,000 | 80,00 |
Store 2 | 110,000 | 115,000 | 116,000 | 112,000 | 105,000 | 107,000 | 103,000 |
Store 3 | 65,000 | 62,000 | 59,000 | 61,000 | 58,000 | 63,000 | 68,000 |
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