ccm6+7+ unit 12 packet: statistics and data...

CCM6+7+ Unit 12 Packet: Statistics and Data Analysis

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CCM6+7+ Unit 12

Data Collection and Analysis

Big Ideas Page(s) What is data/statistics?

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Measures of Reliability and Variability: Sampling, Box Plots and Dot Plots, Histograms and MAD (and some review)

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8-13 14-17 18-23

Measures of Center: Mean, Median, Mode Shapes of Distributions

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Analyzing and Comparing Data Distributions

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Study Guide/Review 55-58


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Random Samples and Surveys


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Common Core Additional Investigations Grade Seven page 31

Reviewing Good Sampling:


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The histogram is a popular graphing tool. It is used to summarize discrete or continuous data that are

measured on an interval scale. It is often used to illustrate the major features of the distribution of the

data in a convenient form. A histogram divides up the range of possible values in a data set into

classes or groups. For each group, a rectangle is constructed with a base length equal to the range of

values in that specific group, and an area proportional to the number of observations falling into that

group. This means that the rectangles will be drawn of non-uniform height. A histogram has an

appearance similar to a vertical bar graph, but when the variables are continuous, there are no gaps

between the bars. When the variables are discrete, however, gaps should be left between the bars. Figure 1 is a good example of a histogram.

A vertical bar graph and a histogram differ in these ways:

In a histogram, frequency is measured by the area of the column.

In a vertical bar graph, frequency is measured by the height of the bar.

Histogram characteristics

Generally, a histogram will have bars of equal width, although this is not the case when class intervals

vary in size. Choosing the appropriate width of the bars for a histogram is very important. As you can

see in the example above, the histogram consists simply of a set of vertical bars. Values of the variable

being studied are measured on an arithmetic scale along the horizontal x-axis. The bars are of equal

width and correspond to the equal class intervals, while the height of each bar corresponds to the frequency of the class it represents.

The histogram is used for variables whose values are numerical and measured on an interval scale. It

is generally used when dealing with large data sets (greater than 100 observations). A histogram can also help detect any unusual observations (outliers) or any gaps in the data.

http://www.statcan.gc.ca/edu/power-pouvoir/glossary-glossaire/5214842-eng.htm#disvariable

http://www.statcan.gc.ca/edu/power-pouvoir/glossary-glossaire/5214842-eng.htm#convariable


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Histographs

A histograph, or frequency polygon, is a graph formed by joining the midpoints of histogram column

tops. These graphs are used only when depicting data from the continuous variables shown on a histogram.

A histograph smoothes out the abrupt changes that may appear in a histogram, and is useful for

demonstrating continuity of the variable being studied. Figure 2 and 3 are good examples of histographs.

Unlike Figure 2, this histograph has spaces between the bars. By just looking at this illustration, the

reader can immediately tell that the spaces mean the variables are discrete. In this way, histographs make it easier for the readers to determine what type of variables has been used.


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Data Distributions CMP2 Grade 7 Lesson 2.2 pages 32-35

You can think of the MEAN as the balance point in a distribution of data. It acts like the

fulcrum of a see-saw.

Here is a set of data showing the amount of sugar in a serving of each of ten cereals, in

grams:

1, 3, 6, 6, 6, 6, 6, 6, 10, 10

Make a line plot to show the distribution.

What is the mean for these data?

Make one or more changes to the data so that the mean is 7 and the range is:

a) the same as the range of the original data set

b) greater than the range of the original data set

c) less than the range of the original data set

Below is a dot plot showing the sugar per serving in nine cereals. What is indicated by the

arrows on each side of the line marking the mean?


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Graph A and Graph B show two different data distributions. Latoya guesses that each

distribution has a mean of 5 grams of sugar per serving. For each distribution, answer

questions 1-3.

1. Find the difference from Latoya’s guess of 5 for each data value that is greater than 5.

What is their sum?

2. Find the difference from Latoya’s guess of 5 for each data value that is less than the

mean. What is their sum?

3. Is Latoya correct that the mean is 5? Does the distribution “balance?” If so, explain. If

not, change one or more data values to make it balance.


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**A few more MEAN problems!**

5. Send It Quick Mail House mailed five packages with a mean weight of 6.7 pounds.

Suppose the mean weight of four of these packages is 7.2 pounds. What is the weight, in

pounds, of the fifth package?

A) 3.35 B) 4.7 C) 6.95 D) 8.7

6. Del Kenya’s test scores are 100, 83, 88, 96, and 100. His teacher tells the class that they

can choose the measure of center she will use to determine final grades. Which measure

should Del Kenya use?

A) mean B) median C) mode D) range

7. a) A gymnast receives these scores from five judges: 7.6, 8.2, 8.5, 8.2, 8.9

What happens to the mean of the scores when you multiply each data value by 2?

By 2

3? By 0.2?

b) Why do you think the mean changes as it does in each situation?

8. A student gets 40 points out of 100 on a test. Her teacher announces that this test and

next week’s test will be averaged together for her grade. The student wonders if she

can still get a 70% if she gets a 100 on the next test. She reasons, “I think my average

(mean) would be 70 because half of 40 is 20 and half of 100 is 50. That is a 70%

because 20 plus 50 is 70.” Does her method always work? Explain your thinking.


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CMP2 Data Distributions Grade 7 Lesson 2.3 pages 36-39

The MODE is the data value that occurs the MOST frequently in a set of data.

The MEDIAN is the MIDPOINT in an ordered distribution. If the graph of a distribution, data

values are located below, above, or at this midpoint. The graph below shows numerical data

about the numbers of pets each of the 26 students has.


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Using the graph above, answer the following questions.

1. What is the range of this data?

2. What is the mean and where is it located?

3. How does the location of the median compare to the location of the mean?

4. Why do you think this is so?

5. Do the data seem to cluster in some parts of the distribution?

6. Does clustering of the data appear to be related to the locations of the median and the

mean?


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Repeated Values in a Distribution

A. Jorge is ordering pizza for a party. Tamika shows Jorge the graph below. She tells him

to order only thin-crust pizzas because thin-crust is the mode. Do you agree or disagree?

Explain.

B. The data from 70 cereals are shown below. Which option do you suggest using to find

the typical amount of sugar in a serving of cereal? Explain.

Option 1 Use the mode, 3 grams. The typical amount of sugar in a serving of cereal

Is 3 grams.

Option 2 Use the median, 7.5 grams. The typical amount of sugar in a serving of

cereal is 7.5 grams.

Option 3 Use clusters. There are several cereals that have either 3 or 6 grams of

sugar per serving. 40% of the data seem to be evenly spread between 8

and 12 grams of sugar.

Option 4 Use something else. Write your own statement about what you consider

to be the typical amount of sugar in a serving of cereal.


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C. An advertiser wants more people to listen to a phone message. He uses the graph

below.

Which of the options below should the advertiser use to decide how long the message should

be? Explain.

Option 1 Use the mode. The most frequent amount of time spent listening to the

phone advertisement was 3 minutes.

Option 2 Use the mean. Listening times lasted, on average, 1.51 minutes per

person.

Option 3 Use clusters. One third of the people listened less than 1 minute and

more than half listened less than 1.5 minutes. Only 20% listened for 3 minutes.

Option 4 Use something else. Write your own response.


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D.

1. In the plots below, the data for the 70 cereals in Question B are organized by the

cereals’ locations on the shelves in a supermarket. Use means, medians, clusters, or other

strategies to compare the three distributions. Explain your reasoning.

2. Use the information from part 1 to make a prediction about the sugar content of a

cereal based on its shelf location.


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CMP2 Data Distributions Grade 7 Lesson 2.4 pages 40-43

Unlike with categorical data, the mode is not always useful with numerical data. Sometimes

there is no mode and sometimes there is more than one mode.

Sometimes the mean and median of a distribution are located close together. The graph

below shows the distribution of the amount of sugar in cereals located on the bottom shelf in

a supermarket. The mean and the median are marked. The median is 7 grams and the

mean is 6.9 grams.

The overall shape of a distribution is determined by where the data cluster, where there are

repeated values, and how spread out are the data. The shape of a distribution influences

where the median and mean are located. In the next problem, you will experiment with

making changes to distributions. Observe what these changes do to the locations of the

mean and median in a distribution.


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For Questions A-C, predict what will happen. Then do the computation to see whether you

are correct.

A. The graph below shows the distribution of the amount of sugar in 20 cereals found on

the top shelf. The sum of the values in this distribution is 91 grams. Use sticky notes or

blocks to make a copy of the distribution. Note the location of the mean at 4.55 grams of

sugar and the median at 3 grams of sugar.

1. Suppose you remove the three cereals with 6 grams of sugar per serving and add

three new cereals, each with 9 grams of sugar per serving. What happens to

the mean and the median? Why do you think this happens?

2. a) Use the new distribution from part 1. Suppose you remove a cereal with 3

grams of sugar and add a cereal with 8 grams of sugar. How do the mean and

median change?

b) Suppose you remove another cereal with 3 grams of sugar and add another

cereal with 8 grams of sugar. How do the mean and median change?

c) Suppose you remove a third cereal with 3 grams of sugar and add a third cereal

with 8 grams of sugar. How do the mean and median change?


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B. Use the new distribution from Question A, part 2. Experiment with removing data values

and replacing them with new data values.

1. How does replacing smaller data values with larger data values affect the mean

and the median?

2. How does replacing larger data values with smaller data values affect the mean

and the median?

3. How does replacing larger and smaller data values with values that are closer to

the middle of the distribution affect the mean and the median?


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CMP2 Data Distributions Grade 7 Lessons 3.3 and 3.4 pages 60-61


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REACTION TIME CARDS ARE ON THIS AND THE FOLLOWING PAGES…feel free to pull

them out of the packet and cut them apart to help group/categorize/compare them!


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COMPARING DISTRIBUTIONS: Unequal Numbers of Data Values


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1. Based on visual inspection of the dotplots, which group appears to have the larger

average height? Which group appears to have the greater variability in the heights?

2. Compute the mean and mean absolute deviation (MAD) for each group. Do these

values support your answers in #1?

3. How many of the 12 basketball players are shorter than the tallest field hockey player?

4. Imagine that an athlete from one of the two teams told you she needs to go to practice.

You estimate that she is about 65 inches tall. If you had to pick, would you think that

she was a field hockey player or a basketball player? Explain your reasoning.

5. The women on the Maryland field hockey team are not a random sample of all female

college field hockey players. Similarly, the women on the Maryland basketball team

are not a random sample of all female college basketball players. However, for

purposes of this task, suppose that these two groups can be regarded as random

samples of all female college field hockey players and all female college basketball

players, respectively. If these were random samples, would you think that female

college basketball players are typically taller than female college field hockey players?

Explain your decision using answers to questions 1-4 or any other additional analysis.


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This is an old test review from the old pre-algebra curriculum…see how you do!


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ccm6+7+ unit 12 packet: statistics and data...

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