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Exploring Statistics: Heart Rate

Have each student calculate their heart rate while sitting down in class and enter their results in a joint spreadsheet. Then have the class jump up and down for one minute. Each student will then find their new heart rate and enter their results in the same joint spreadsheet. Each student’s data is labeled by their seat number in column A: A student observes that after exercising, her heart rate increased. What can she observe about the entire class’s data?

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First, let’s explore this problem using a graphing calculator.

Enter the data into the calculator. Enter the “before” heart rates into column 1 and the “after” heart rates into column 2.

Now draw a histogram of your “before” heart rates.

What can you conclude about the before values? The before values range from 60 bpm to 90 bpm. Each bin tells you how many students’ heart rates were within that range. For example, the first bin says that 5 students had a heart rate between 60 bpm and 67.8 bpm.

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Repeat this process for the “after” heart rates.

What does the histogram of the “after” heart rates tell you? The heart rates after exercise range from 87 bpm to 129 bpm. The largest bin says that 6 students had a heart rate between 103.8 bpm and 112.2 bpm. Change both plots to box and whisker plots. View both at the same time.

What can you conclude about the two sets of values? The “before” values are clearly lower than the “after” values. However, there is some overlap. Some student’s “before” values were higher than other student’s “after” values. Everyone’s heart rate increased after exercise. The range of heart rates was greater in the “after” category than in the “before” category.

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Now let’s get statistics for the two lists. Calculate statistics for the “before” values. What information does this give you?

x̅ = 78.3. This is the mean (average) of the “before” heart rates. Σx is the sum of the “before” heart rates. Σx² is the sum of the squares of “before” heart rates. Sx is the sample standard deviation. σx is population standard deviation. n is the number of “before” heart rates. minX is the minimum value (in this case 60). Q1 is the first quartile. Med is the median. Q3 is the third quartile. maxX is the maximum value (in this case 99). Calculate the statistics for the “after” values.

x̅ = 107.7. This is the mean (average) of the “before” heart rates. Σx is the sum of the “before” heart rates. Σx² is the sum of the squares of “before” heart rates. Sx is the sample standard deviation. σx is population standard deviation. n is the number of “before” heart rates.

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minX is the minimum value (in this case 87). Q1 is the first quartile. Med is the median. Q3 is the third quartile. maxX is the maximum value (in this case 129). What can you conclude from this data? These values tell you a lot about the data collected. The most important ones for this problem seem to be mean, minimum, maximum, and standard deviation. The mean increases after exercise. The minimum heart rate and the maximum heart rate also increase. The standard deviation is about the same for both sets of data. This must be because the same people were measured for before and after. Their heart rates went up in proportion to the before value. Let’s look at the difference between the before and after scores. Subtract column 1 from column 2. What does this new graph look like?

What does this graph tell us about the data? This graph shows us that about half of the students’ heart rates (11 students’) increased between 20.8 bpm and 30.6 bpm. 2 students’ heart rates increased less than that, and 8 students’ heart rates increased more than that. Now let’s test the differences.

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What does this information tell you? It is not likely that the means are going to be the same for before and after, because the p value is 1.4 x 10-10, which is very close to 0. Let’s take another look at our data by graphing a scatter plot. The x values are set as the “before” list and the y values are set as “after” list. This representation shows that, in general, students with lower “before” values also had lower “after” values. Students with higher “before” values also had higher “after” values.

Draw a line of best fit on the scatterplot:

This confirms our observation of a generally upward slope. The higher your “before” heart rate is, the more likely you are to have a higher “after” heart rate.

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Now, let’s explore this same problem using a TI-Nspire.

Enter the data collected in the joint spreadsheet into a spreadsheet in TI-Nspire.

Now insert a new data and statistics page. Put the before values into a dot plot.

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What do you notice? This dot plot lets you see which values are repeated. It also shows that the lowest value is 60 and the highest value is 99. It doesn't show many other interesting details. What does a box plot of this data look like?

By clicking on different regions of the box plot, we discover that the minimum value is 60 and the maximum value is 99. The median is 75. The lower quartile is 66. The upper quartile is 89.

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Now let's add in the "after" values.

Clearly, we can see that the after data set has higher numbers than the before data set. There are some values that "overlap"; The lowest after values are in the same "range" as the highest before values. Let's look at this information in a histogram.

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We can again see that the set of after values is in general higher than the set of before values. Now that we've observed different types of graphs, let's look at the statistics. On the spreadsheet, calculate one variable statistics on both the before and after lists.

Just like we saw using the graphing calculator, the mean of the before values is 77.76. After exercise, the mean increases to 107.71. The minimum increases after exercise from 60 to 87. The maximum increases after exercise from 99 to 129. The median also increases by 20 beats per minute.

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Now let's conduct a 2-sample t test.

This shows us that the p-value is 1.60. Now let's look at the graph of the difference of the data (after-before).

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This dot plot shows that the most common increase in heart rate was 27. Most students' heart rates increased about 20 to about 35 beats per minutes. You can also see that two students' heart rates increased very little. Another student's heart rate increased by 60! There is variety in the data collected. Lastly, let's look at a scatterplot of the before and after values.

As you can see, this class's data ranges quite a bit. A general observation that a lower before value often corresponds with a lower after value can be made. Similarly, higher before values result in higher after values as well.

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After estimating a line of best fit, have TI-Nspire find it for you. The slope is about .5 and the y-intercept is about 65.9.

STUDENT CONCLUSION- In general, one's heart rate increases after exercise. A lower before value may corresponds with a lower after value. Similarly, a higher before value may result in a higher after value as well.

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Exploring Statistics: Heart Rate Teacher Notes

I found that this activity most corresponds to 8th grade mathematics (and Algebra 1, often taught in 8th grade) in the Alabama Course of Study. However, I did find one content standard from 7th grade interesting:

Understand that statistics can be used to gain information about a population by examining a sample of the population; generalizations about a population from a sample are valid only if the sample is representative of that population. Understand that random sampling tends to produce representative samples and support valid inferences. [7-SP1]

Data collected from one class is a random sampling of population. The students in this problem took this data and made an inference about all people in general. This is only valid if the representative samples (one class) are representative about the entire population. Perhaps this lesson could emphasize that the conclusions drawn can be drawn about which population? Other middle school students? All people, everywhere? 8th grade/Algebra 1 content standards addressed in this activity include the following:

Know that straight lines are widely used to model relationships between two quantitative variables. For scatter plots that suggest a linear association, informally fit a straight line, and informally assess the model fit by judging the closeness of the data points to the line. [8-SP2]

Represent data with plots on the real number line (dot plots, histograms, and box plots). [S-ID1]

Use statistics appropriate to the shape of the data distribution to compare center (median, mean) and spread (interquartile range, standard deviation) of two or more different data sets. [S-ID2]

Fit a linear function for a scatter plot that suggests a linear association. [S-ID6c]

Interpret the slope (rate of change) ... of a linear model in the context of the data. [S-ID7]

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This problem is a great example of some things Dan Meyer discussed in his TED talk, “Math Class Needs a Makeover”. I titled this problem “Exploring Statistics” because that is exactly what it had students do- explore. There wasn’t one question to answer or problem to solve. The activity didn’t give the student certain information and ask him to plug it into a formula. Students had to take a simple set of data- everyone’s heart rates- and explore the myriad of ways to represent it. What does each method tell you about the data? What information is important? What do these numbers really mean for me? These are the kinds of questions that will stick with a student, because the student came up with them! The Technology Principle states:

Students can learn more mathematics more deeply with the appropriate and responsible use of technology. They can make and test conjectures. They can work at higher levels of generalization or abstraction.

Exploring Statistics is a engaging example of how a student can make and test conjectures with technology. They calculated numerous tests and made many graphs of the data quickly. This left time and attention span to decide what these figures really meant. They generalized information gathered from this set of data to “people in general”. This allowed the students to gain a deeper level of understanding and appreciation for statistics then simply computing, for example, the mean, median, and mode of a list of numbers (as many textbooks require) would have.