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Objective
We want students to learn to describe the important features of a distribution: shape, center, spread, and unusual features. We also expect students to examine the difference between mean and median and their respective measures of spread, and when to apply them to a particular distribution. Graphical displays such as: histograms and boxplots are also introduced in this activity.
The Activity
Prior to assigning this activity, students should have had an introduction to how quantitative data is displayed and described numerically. The activity involves a set of data dealing with the percentage difference between measure calories and labeled calories of particular items, in two categories, per item and per gram. Students are asked to make stem-and-leaf plots, dotplots, and histograms of the two variables. The students are then asked to describe the distributions of the two variables: per gram and per item and compare the resulting distributions. The students are asked to use the appropriate descriptive statistics to describe the respective distributions. Lastly, the students are asked to make boxplots of the two variables and compare the boxplots to the histograms, noting any outliers that are now apparent. One distribution is unimodal, right skewed and the other is unimodal, symmetric. Following student completion of the activity the instructor can engage the students in the discussion of other distributions and the importance of the outlier rule.
Assessment
The assessment of this assignment will be based mainly on completion of the assignment with good logical explanations of why the certain unusual features were chosen such as: outliers in the skewed distribution and slight skew in the unimodal symmetric distribution. Correct reasoning for using median and IQR with a skewed distribution since mean and standard deviation are sensitive to outliers. Using outlier rule correctly (1.5*IQR below and above first and third quartiles).
Class participation can be counted as well. How active were particular students? Did they ask intuitive and intelligent questions regarding the activity? An example is: what if the distribution was still symmetric, but bimodal, would we still use the mean and standard deviation to describe the distribution?
Formal assessment can include exam questions about particular data sets or homework questions that will reinforce the concepts presented by this activity. A few possible questions include:
Given that you have a right skewed distribution, would the mean or median be higher?
You could show the student three histograms and have them match the histograms to three boxplots of the same data sets.
A histogram could be presented and the student could have to describe the data, noting: shape, center, spread, and unusual features, then choose the appropriate measures of center and spread given the shape.
Many different questions could be asked regarding describing types of data; I have only presented a few of the most common.
Teaching Notes
The estimated time to complete this activity is approximately 50 minutes.
This activity can be done in class or assigned as out-of-class work. Either way I would suggest that students be allowed to work together on the assignment so that they might discuss the issues together.
This activity is not dependent upon any particular piece of technology. I recommend StatCrunch since it is a free piece of statistical software, but students could do the activity using another software package or graphing calculator.
Activity: Displaying Quantitative Data and Describing Distributions (Chapters 4, 5)
Introduction:
Counting calories is not the be all, end all solution to healthy weight loss. However, it certainly has its benefits, but what if the labels indicating the number of calories are incorrect? Counting calories is an idea being implemented more and more in diet programs these days, so if the labels are incorrect many dieters may be easily frustrated.
The Study:
In 1993, Allison, Heshka, Sepulveda, and Heymsfield decided to see if the labels on different kinds of food had the correct number of calories as indicated on the package. They discovered that some labels on food packages understated the calorie content by more than 85%.
The differences between the calorie content reported on labels and the calorie content measured in the lab are examined for 40 foods marketed as "diet," "low-fat," or "health foods." Also included in the data is whether the product is "nationally advertised," "regionally distributed," or "locally produced." taken from the Electronic Encyclopedia of Statistical Examples and Exercises under the story topic Counting Calories.
The Variables:
Per gram = percentage difference between measured calories and labeled calories per gram
(100% x (measured-labeled)/labeled)Per item = percentage difference between measured calories and labeled calories per itemClassification = N if nationally advertised, R if regionally distributed, L if locally prepared
The Data:
Food_1 Per gram_1 Per item_1 Classif._1
noodles and alfredo sauce -2.0 2.0 N
cheese curls -1.0 -25.0 N
green beans -4.5 -6.0 N
mixed fruits 4.0 8.0 N
cereal 5.0 6.0 N
fig bars -12.0 -1.0 N
oatmeal raisin cookie 1.0 10.0 N
crumb cake 9.0 13.0 N
crackers 10.0 15.0 N
blue cheese dressing -6.0 -4.0 N
imperial chicken -6.0 -4.0 N
vegetable soup -17.0 -18.0 N
cheese 7.0 10.0 N
chocolate pudding 4.0 5.0 N
sausage biscuit -4.0 3.0 N
lasagna -11.0 -7.0 N
spread cheese 14.0 3.0 N
lentil soup -1.0 -0.5 N
pasta w/shrimp and tomato sau -10.5 -10.0 N
chocolate mousse 2.0 6.0 N
lemon pound cake -16.0 2.0 R
banana cake 5.0 25.0 R
brownie 11.0 39.0 R
butterscotch bar 7.0 16.5 R
blondie 12.0 17.0 R
oat bran snack bar 21.0 28.0 R
granola bar -13.0 -3.0 R
apricot bar 13.0 14.0 R
chocolate chip cookie 16.0 34.0 R
chinese chicken * 15.0 L
gyoza * 60.0 L
jelly diet candy-reds flavor * 250.0 L
jelly diet candy-fruit flavor * 145.0 L
Florentine manicotti * 6.0 L
egg foo young * 80.0 L
hummus with salad * 95.0 L
baba ghanoush with salad * 3.0 L
A histogram is a common tool for describing the data. Use StatCrunch to create a histogram of the variables per gram and per item.
Describe the distribution of per gram. Make sure to include:
Shape
Center
Spread
Unusual features
Describe the distribution of per item.
Why do you think the distributions of per gram and per item look so different?
Calculate the descriptive statistics, including the mean, standard deviation, and 5-number summary.
Based on your description in #2, what are the appropriate measures of center (median or mean) and spread (standard deviation or IQR) for the distribution of per gram? Explain why you made this choice. What do you notice about the difference between the mean and the median for this variable? Is this what we expect?
Based on your description in #2, what are the appropriate measures of center and spread for the distribution of per item? Explain why you made this choice. What do you notice about the difference between the mean and the median for this variable? Is this what we expect?
Create side-by-side boxplots for both variables.
Do these boxplots reinforce the description you gave in Question 2?
Consider the boxplot for per gram. How many outliers are there for this variable? Are these the same outliers that you chose in #2a? How did StatCrunch determine the number of outliers? (Hint: Remember the fences.)
Consider the boxplot for per item. How many outliers are there for this variable? Are these the same outliers that you chose in #2b? Why did StatCrunch choose these values as outliers? (Hint: Remember the fences.)
Instructors Solution Edition
A histogram is a common tool for describing the data. Use StatCrunch to create a histogram of the variables per gram and per item.
Describe the distribution of per gram. Make sure to include:
Shape
Center
Spread
Unusual features
Describe the distribution of per item.
Why do you think the distributions of per gram and per item look so different?
Answer:
HYPERLINK "http://www.statcrunch.com/grabimage.php?image_id=204969" INCLUDEPICTURE "http://www.statcrunch.com/grabimage.php?image_id=204969" \* MERGEFORMATINET
a.
The shape is approximately unimodal and symmetric. The center seems to be at approximately 2. Most of the data lies from -5 to 10, with no outliers. The range is from approximately from -20 to 20.
HYPERLINK "http://www.statcrunch.com/grabimage.php?image_id=204971" INCLUDEPICTURE "http://www.statcrunch.com/grabimage.php?image_id=204971" \* MERGEFORMATINET
b.
The shape is unimodal right skewed with the center at approximately zero. Most of the data lies from -20 to 90, and there are two outliers one at approximately 160 and one at approximately 240.
c.
Because of the way per gram and per item were calculated. Per gram is the percentage difference between labeled and actual calories per gram of the food. Per item calculates this difference for the entire food item. Since per gram is working on a standardized scale, the values are closer together (i.e. there is less spread). But since the item sizes are different, per item has outliers and values that are vastly different from the others.
Calculate the descriptive statistics, including the mean, standard deviation, and 5-number summary.
Based on your description in #2, what are the appropriate measures of center (median or mean) and spread (standard deviation or IQR) for the distribution of per gram? Explain why you made this choice. What do you notice about the difference between the mean and the median for this variable? Is this what we expect?
Based on your description in #2, what are the appropriate measures of center and spread for the distribution of per item? Explain why you made this choice. What do you notice about the difference between the mean and the median for this variable? Is this what we expect?
Summary Statistics
ColumnnMeanVarianceStd. Dev.Std. Err.MedianRangeMinMaxQ1Q3Pergram_1291.2758621101.7604710.087641.8732276238-1721-69Peritem_13722.4864862525.326250.2526258.261486275-25250-0.517
mean and SD because it is unimodal and symmetric. The mean and median are very close to eachother in this distribution. This is what we expect because the distribution is symmetric without any outliers.
Median and IQR because the distribution is skewed. The mean and median are very different because the the mean is pulled towards the tail by the high outling values. Again this is what we expect.
Create side-by-side boxplots for both variables.
Do these boxplots reinforce the description you gave in Question 2?
Consider the boxplot for per gram. How many outliers are there for this variable? Are these the same outliers that you chose in #2a? How did StatCrunch determine the number of outliers? (Hint: Remember the fences.)
Consider the boxplot for per item. How many outliers are there for this variable? Are these the same outliers that you chose in #2b? Why did StatCrunch choose these values as outliers? (Hint: Remember the fences.)
Answer:
HYPERLINK "http://www.statcrunch.com/grabimage.php?image_id=204972" INCLUDEPICTURE "http://www.statcrunch.com/grabimage.php?image_id=204972" \* MERGEFORMATINET
HYPERLINK "http://www.statcrunch.com/grabimage.php?image_id=204974" INCLUDEPICTURE "http://www.statcrunch.com/grabimage.php?image_id=204974" \* MERGEFORMATINET
HYPERLINK "http://www.statcrunch.com/grabimage.php?image_id=204975" INCLUDEPICTURE "http://www.statcrunch.com/grabimage.php?image_id=204975" \* MERGEFORMATINET
hopefully
Outlier Rule = 1.5*IQR above upper quartile and below lower quartile
1.5*IQR = 1.5*(Q3 Q1) = 1.5*(9.0 (-6.0)) = 22.50
Upper Fence = Q3 + 1.5*IQR = 9.00 + 22.50 = 31.5
Lower Fence = Q1 1.5*IQR = -6.00 22.5 = -28.50
So, since we have no values outside of the fences, we can say that there are no outliers. This should have been the same conclusion from part 2.
c. Outlier Rule = 1.5*IQR above upper quartile and below lower quartile
1.5*IQR = 1.5*(Q3 Q1) = 1.5*(17.00 (-0. 50)) = 26.25
Upper Fence = Q3 + 1.5*IQR = 17.00 + 26.25 = 43.25
Lower Fence = Q1 1.5*IQR = -0. 50 26.25 = -26.75
So, according to our boxplots and the outlier rule, we can clearly see that we have five outliers. In part b it is likely that you noticed only two outliers, since the other three seemed closer to the large group of data.
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