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Describing the Center of a Distribution Using the Median




 

Video Solutions to help grade 6 students learn how calculate the median of the given data and estimate the percent of values above and below the median value.

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Common Core For Grade 6

New York State Common Core Math Module 6, Grade 6, Lesson 12



Lesson 12 Student Outcomes

• Given a data set, students calculate the median of the data.
• Students estimate the percent of values above and below the median value.

Lesson 12 Summary

In this lesson, you learned about a summary measure for a set of data called the median. To find a median you first have to order the data. The median is the midpoint of a set of ordered data; it separates the data into two parts with the same number of values below as above that point. For an even number of data values, you find the average of the two middle numbers; for an odd number of data values, you use the middle value. It is important to note that the median might not be a data value and that the median has nothing to do with a measure of distance. Medians are sometimes called a measure of the center of a frequency distribution but do not have to be the middle of the spread or range (maximum-minimum) of the data.

Lesson 12 Classwork

How do we summarize a data distribution? What provides us with a good description of the data? The following exercises help us to understand how a numerical summary answers these questions.

Example 1: The Median – A Typical Number
Suppose a chain restaurant (Restaurant A) advertises that a typical number of french fries in a large bag is 82. The graph shows the number of french fries in selected samples of large bags from Restaurant A.

Exercises 1–3

1. You just bought a large bag of fries from the restaurant. Do you think you have french fries? Why or why not?

2. How many bags were in the sample?

3. Which of the following statements would seem to be true given the data? Explain your reasoning.
a. Half of the bags had more than 82 fries in them.
b. Half of the bags had fewer than 82 fries in them.
c. More than half of the bags had more than 82 fries in them.
d. More than half of the bags had fewer than 82 fries in them.
e. If you got a random bag of fries, you could get as many as 93 fries.




Example 2: The Median

Sometimes it is useful to know what point separates a data distribution into two equal parts, where one part represents the larger “half” of the data values and the other part represents the smaller “half” of the data values. This point is called the median. When the data are arranged in order from smallest to largest, the same number of values will be above the median as are below the median.

Exercises 4–5

4. Suppose you were trying to convince your family that you needed a new pair of tennis shoes. After checking with your friends, you argued that half of them had more than four pairs of tennis shoes, and you only had two pairs. Give another example of when you might want to know that a data value is a half-way point? Explain your thinking.

5. Use the information from the dot plot in Example 1. The median number of fries was 82.
a. What percent of the bags have more fries than the median? Less than the median?
b. Suppose the bag with 93 fries was miscounted and there were only 85 fries. Would the median change? Why or why not?
c. Suppose the bag with 93 fries really only had 80 fries. Would the median change? Why or why not?

Exercises 6–7: A Skewed Distribution

6. The owner of the chain decided to check the number of french fries at another restaurant in the chain. Here is the data for Restaurant B: 82, 83, 83, 79, 85, 82, 78, 76, 76, 75, 78, 74, 70, 60, 82, 82, 83, 83, 83.
a. How many bags of fries were counted?
b. Sallee claims the median is 75 as she sees that 75 is the middle number in the data set listed above. She thinks half of the bags had fewer than 75 fries. Do you think she would change her mind if the data were plotted in a dot plot? Why or why not?
c. Jake said the median was 83. What would you say to Jake?
d. Betse argued that the median was halfway between 60 and 85 or 72.5. Do you think she is right? Why or why not?
e. Chris thought the median was 72. Do you agree? Why or why not?

7. Calculate the mean and compare it to the median. What do you observe about the two values? If the mean and median are both measures of center, why do you think one of them is lower than the other?

Exercises 8–10: Finding Medians from Frequency Tables

8. A third restaurant (Restaurant C) tallied a sample of bags of french fries and found the results below.
a. How many bags of fries did they count?
b. What is the median number of fries for the sample of bags from this restaurant? Describe how you found your answer.

9. Robere decided to divide the data into four parts. He found the median of the whole set.
a. List the 13 values of the bottom half. Find the median of these 13 values.
b. List the 13 values of the top half. Find the median of these 13 values.

10. Which of the three restaurants seems most likely to really have 82 fries in a typical bag? Explain your thinking.


 

Rotate to landscape screen format on a mobile phone or small tablet to use the Mathway widget, a free math problem solver that answers your questions with step-by-step explanations.


You can use the free Mathway widget below to practice Algebra or other math topics. Try the given examples, or type in your own problem and check your answer with the step-by-step explanations.



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