The mathematical purpose of this activity is for students to understand that approximately normal distributions have a similar proportion of area in regions described by the mean and standard deviations. Students are given a data set including the number of words for 200 submitted short stories. Students are asked to determine the proportion of submitted stories that are within certain ranges. 

Monitor for students who use these methods to find the area. The strategies are listed in order from most straightforward to more use of tools:

• Return to the original data set for each question and count the values in each interval.
• Create a frequency table to break the data into natural intervals.
• Create a histogram to visualize the areas in question.
• Use technology to count values in each region.

The problem mentions that a page holds approximately 200 words, to help students identify natural intervals to summarize the data into a frequency table or histogram.

In the digital version of the activity, students use a spreadsheet to find the proportions in the activity. The spreadsheet allows students to count the number of values in an interval more quickly. Use the digital version if students could benefit from extra time to understand the proportions rather than to count values from the data set.

Making statistical technology available gives students an opportunity to choose appropriate tools strategically (MP5).

The goal of this discussion is to get students thinking about histograms as an area model for a distribution.

Invite previously selected groups to share how they computed the area. Sequence the discussion of the strategies in the order listed in the Activity Narrative. If possible, record and display the students’ work for all to see.

Connect the different responses to the learning goals by asking questions, such as:

• “What are the advantages and disadvantages of each strategy?”
• “Which strategy is most efficient with this many values in the data set?”
• “How could you check whether your answer is reasonable using each strategy?”

If possible, display a histogram created by the students. If none of the students created a histogram, display the one here.

Here are some questions for discussion.

• “How is this activity related to the Warm-up?” (In the Warm-up, we figured out the area of a shaded region and divided it by the total area under the curve. In this activity, we can think of the bars as having an area.)
• “Why would a normal distribution be a good model for the data summarized in the histogram?” (The distribution is approximately symmetric and bell-shaped.)
• “Why do you think the area for a bar representing 2,000 to 2,200 words has the same proportion of the whole as the proportion of stories in the same region to the whole?” (The are the same since the area came from , which is equivalent to .)
• “Was it surprising to you that more than 95% of the data was within two standard deviations of the mean or that almost 70% of the data was within one standard deviation of the mean? Explain your reasoning.” (Both were surprising to me. I had never really thought about using the standard deviation to create an interval around the mean, so I had never thought about using the standard deviation to estimate the proportion of the data that was within a specific interval. But since most of the data is near the center, I can understand why it is this way.)
• “Why do you think it could be useful to look at the proportion of values that are within one standard deviation of the mean?” (It can be useful to know that distributions that can be modeled with a normal distribution have certain percentages of the data clustered near the center of the distribution.)

Invite students to share their comparisons.

Students may comment on the fact that the two activities had some questions that are very much alike, and their answers are very much alike. If no students mention this, point it out. For example, each task asked about the proportion of stories or websites that were within 1 standard deviation of a value that was half of a standard deviation below the mean (within 1 standard deviation of 2,400 words or within 1 standard deviation of 2.8 seconds), and they both had about 68% of the values in that region.

Display the images for the normal distributions that model each situation. Tell students that since both questions ask for the proportion of participants within one standard deviation of a value that is the same number of standard deviations from the mean, and we know that these distributions are approximately normal, we would expect the region in each image to include approximately the same proportion of values from the data set. That is what it means to say that a normal distribution is defined only by the mean value and the standard deviation.

Here are some questions for discussion:

• “Describe what is happening in the two graphs.” (Each graph represents a normal curve fit to the data in the problem. The shaded regions represent the proportion of the data that is within one standard deviation of 2,400 words and 2.8 seconds, respectively.)
• “Describe a story that includes 2,400 words, in terms of the mean and standard deviation only. Then do the same for a website that loads in 2.8 seconds.” (Both values are half of a standard deviation below the mean.)
• “How would this graph change if the shaded region was within one standard deviation of the mean?” (The shaded region would be shifted to the right, and it would take up about 68% or 69% of the area under the curve. It would be symmetric around the normal curve’s vertical line of symmetry.)
• “Why do you think it is helpful to model bell-shaped symmetric data using a normal curve?” (It is helpful because then we can compare data sets with different means and different standard deviations.)

If students don’t understand the phrase “within 1 standard deviation of the mean,” consider asking:

• “What is 1 standard deviation in this scenario? What is the mean in this scenario?”

• “Draw a number line that contains the mean value. Where on the number line are values that are 1 standard deviation from that mean?”