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Variability in a Data Distribution




 

Video Solutions to help grade 6 students learn how to informally evaluate how precise the mean is as an indicator of the typical value of a distribution, based on the variability exhibited in the data.

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New York State Common Core Math Module 6, Grade 6, Lesson 8



Lesson 8 Student Outcomes

• Students interpret the mean of a data set as a “typical” value.
• Students compare and contrast two small data sets that have the same mean but different amounts of variability.
• Students see that a data distribution is not characterized only by its center. Its spread or variability must be considered as well.
• Students informally evaluate how precise the mean is as an indicator of the typical value of a distribution, based on the variability exhibited in the data.
• Students use dot plots to order distributions according to the variability around the mean for each of the data distributions.

Lesson 8 Summary

We can compare distributions based on their means, but variability must also be considered. The mean of a distribution with small variability (not a lot of spread) is considered to be a better indication of a typical value than the mean of a distribution with greater variability (wide spread).

Lesson 8 Classwork

Example 1: Comparing Two Distributions
Robert’s family is planning to move to either New York City or San Francisco. Robert has a cousin in San Francisco and asked her how she likes living in a climate as warm as San Francisco. She replied that it doesn’t get very warm in San Francisco. He was surprised, and since temperature was one of the criteria he was going to use to form his opinion about where to move, he decided to investigate the temperature distributions for New York City and San Francisco. The table below gives average temperatures (in degrees Fahrenheit) for each month for the two cities.

Exercises 1–2
Use the table above to answer the following:
1. Calculate the annual mean monthly temperature for each city.
2. Recall that Robert is trying to decide to which city he wants to move. What is your advice to him based on comparing the overall annual mean monthly temperatures of the two cities?

Example 2: Understanding Variability
In Exercise 2, you found the overall mean monthly temperatures in both the New York City distribution and the San Francisco distribution to be about the same. That didn’t help Robert very much in making a decision between the two cities. Since the mean monthly temperatures are about the same, should Robert just toss a coin to make his decision? Is there anything else Robert could look at in comparing the two distributions?
Variability was introduced in an earlier lesson. Variability is used in statistics to describe how spread out the data in a distribution are from some focal point in the distribution (such as the mean). Maybe Robert should look at how spread out the New York City monthly temperature data are from its mean and how spread out the San Francisco monthly temperature data are from its mean. To compare the variability of monthly temperatures between the two cities, it may be helpful to look at dot plots. The dot plots for the monthly temperature distributions for New York City and San Francisco follow.

Exercises 3–7
Use the dot plots above to answer the following:
3. Mark the location of the mean on each distribution with the balancing ∆ symbol. How do the two distributions compare based on their means?

4. Describe the variability of the New York City monthly temperatures from the mean of the New York City temperatures.

5. Describe the variability of the San Francisco monthly temperatures from the mean of the San Francisco monthly temperatures.

6. Compare the amount of variability in the two distributions. Is the variability about the same, or is it different? If different, which monthly temperature distribution has more variability? Explain.

7. If Robert prefers to choose the city where the temperatures vary the least from month to month, which city should he choose? Explain.




Example 3: Using Mean and Variability in a Data Distribution
The mean is used to describe the “typical” value for the entire distribution. Sabina asks Robert which city he thinks has the better climate? He responds that they both have about the same mean, but that the mean is a better measure or a more precise measure of a typical monthly temperature for San Francisco than it is for New York City. She’s confused and asks him to explain what he means by this statement.
Robert says that the mean of 63 degrees in New York City (64 in San Francisco) can be interpreted as the typical temperature for any month in the distributions. So, 63 or 64 degrees should represent all of the months' temperatures fairly closely. However, the temperatures in New York City in the winter months are in the 40s and in the summer months are in the 80s. The mean of 63 isn’t too close to those temperatures. Therefore, the mean is not a good indicator of typical monthly temperature. The mean is a much better indicator of the typical monthly temperature in San Francisco because the variability of the temperatures there is much smaller.

Exercises 8–14
Consider the following two distributions of times it takes six students to get to school in the morning and to go home from school in the afternoon.
8. To visualize the means and variability, draw dot plots for each of the two distributions.

9. What is the mean time to get from home to school in the morning for these six students?

10. What is the mean time to get from school to home in the afternoon for these six students?

11. For which distribution does the mean give a more precise indicator of a typical value? Explain your answer. Distributions can be ordered according to how much the data values vary around their means.
Consider the following data on the number of green jellybeans in seven bags of jellybeans from each of five different candy manufacturers (AllGood, Best, Delight, Sweet, Yum). The mean in each distribution is 42 green jellybeans.

12. Draw a dot plot of the distribution of number of green jellybeans for each of the five candy makers. Mark the location of the mean on each distribution with the balancing △ symbol.

13. Order the candy manufacturers from the one you think has least variability to the one with most variability. Explain your reasoning for choosing the order.

14. For which company would the mean be considered a better indicator of a typical value (based on least variability)?


 

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