Observing and Analyzing Data
A thermometer reads "36." Is that 36 degrees Celsius, 36 Fahrenheit, or the temperature of a dog with a fever? Without units, a piece of data is useless. That is why science is so demanding about how data is collected, presented, and interpreted.
Data Tables
During recess you count the pigeons in the schoolyard and jot it down on a sheet of paper: Monday 12, Tuesday 9, Wednesday 15... If you have dozens of loose observations, how do you organize them so you can look them up and compare them at a glance?
You are going to investigate whether soil type (sand, clay, or garden soil) affects the growth of lentil plants over 15 days. Design the data table you would use to…
You are going to investigate whether soil type (sand, clay, or garden soil) affects the growth of lentil plants over 15 days. Design the data table you would use to collect the results: state what columns and rows it would have and what units you would use.
Qualitative and Quantitative Data
While examining a rock you write down: "it is gray with shiny flecks, has a rough texture, and weighs $240\text{ g}$." How do observations that describe a quality differ from those that are measured with a number?
Choosing the Type of Graph
You have the temperature recorded every hour during a hike up the Nevado de Toluca and the number of trees of each species you found. How do you turn those lists into a picture that is understood right away?
Reading a graph
On a line graph you can read values using guide lines and estimate in-between values. You can also see where the change is fastest: where the line is steepest.
You want to represent three sets of data: (a) how classroom temperature changes throughout the day, (b) the percentage of students choosing each sport, and (c) the…
You want to represent three sets of data: (a) how classroom temperature changes throughout the day, (b) the percentage of students choosing each sport, and (c) the number of different species observed in each area of the playground. State which type of graph (line graph, circle graph, or bar graph) you would choose in each case and justify your choice.
The graph shows the height reached by some plants depending on the hours of light they receive each day. Describe the relationship between both variables and, using the…
The graph shows the height reached by some plants depending on the hours of light they receive each day. Describe the relationship between both variables and, using the grid lines to help you, estimate the approximate height of a plant receiving 6 hours of light per day.
The temperature of a glass of hot water cooling on a table is measured: at 0 min it reads $60\ ^\circ\text{C}$, at 5 min $52\ ^\circ\text{C}$, at 10 min…
The temperature of a glass of hot water cooling on a table is measured: at 0 min it reads $60\ ^\circ\text{C}$, at 5 min $52\ ^\circ\text{C}$, at 10 min $45\ ^\circ\text{C}$, at 15 min $40\ ^\circ\text{C}$, and at 20 min $36\ ^\circ\text{C}$. What was the temperature at 10 minutes? In which time interval did the temperature drop the fastest?
Calculating with Data: Mean, Rate, and Precision
Three classmates measure the same leaf and get $8.1$, $8.3$, and $8.2\text{ cm}$. Which value do you report? And if a plant grows differently each week, how do you describe its rate?
Accuracy and precision. If you repeat a measurement and get values that are very close to each other, your measurement is precise. If the mean is close to the true value, it is accurate.
Outliers
Sometimes a data point is far from all the others. It is called an outlier. It may be due to an error (pressing the stopwatch late), and then it is best to discard it, stating so in your report; but if it is due to a real phenomenon, it should not simply be erased. The best thing is to repeat the measurement.
A bean plant, which sprouts at the start of the first week with a height of 0 cm, is 2 cm tall at the end of the first week, 6 cm at the end of the second, 11 cm at the…
A bean plant, which sprouts at the start of the first week with a height of 0 cm, is 2 cm tall at the end of the first week, 6 cm at the end of the second, 11 cm at the end of the third, and 17 cm at the end of the fourth. Calculate how much it has grown in total and its mean weekly growth, and determine whether it has grown at the same rate every week.
Two groups measure the length of the same leaf three times. Group A gets $8.1\ \text{cm}$, $8.3\ \text{cm}$, and $8.2\ \text{cm}$; Group B gets $7.5\ \text{cm}$,…
Two groups measure the length of the same leaf three times. Group A gets $8.1\ \text{cm}$, $8.3\ \text{cm}$, and $8.2\ \text{cm}$; Group B gets $7.5\ \text{cm}$, $9.0\ \text{cm}$, and $8.1\ \text{cm}$. Calculate the mean for each group and determine which of the two measured with greater precision.
When measuring the time it takes for a leaf to fall from $2\ \text{m}$, the following values in seconds are obtained: 3.1, 2.9, 3.0, 7.8, 3.2. One of them is clearly…
When measuring the time it takes for a leaf to fall from $2\ \text{m}$, the following values in seconds are obtained: 3.1, 2.9, 3.0, 7.8, 3.2. One of them is clearly anomalous. Identify it, suggest a possible cause for this error, and decide, with reasoning, what to do with this data point when calculating the mean.
Correlation and Cause
In Mexico, when sales of aguas frescas and ice cream go up, visits to the beaches of Mazatlán also go up. Does ice cream make people go to the beach?
Almost always there is a third factor that explains both. In the example, it is the heat: when it is hot, people buy more ice cream and go to the beach more. Ice cream does not lead anyone to the beach.
This matters for health. It has been observed that people who consume a lot of salt, sugar, or saturated fats have more hypertension or cavities. That correlation is a clue, but to state the cause more studies are needed that compare groups and control other factors (age, physical activity, heredity).
It is observed that the months with the highest ice cream sales are also the months with the most bathers on the beach. Does that mean eating ice cream causes people to…
It is observed that the months with the highest ice cream sales are also the months with the most bathers on the beach. Does that mean eating ice cream causes people to go to the beach? Explain the difference between correlation and causation using this example.
A study finds that students who spend more hours in the library get better marks, and a newspaper headline reads: "The library boosts marks". Give at least two…
A study finds that students who spend more hours in the library get better marks, and a newspaper headline reads: "The library boosts marks". Give at least two alternative explanations for this correlation and explain what type of experiment would be needed to state that a cause-and-effect relationship exists.
Back to the beginning: that "36" only makes sense with its unit: $36\text{ °C}$ is a normal temperature for the human body, whereas $36\text{ °F}$ is equal to barely about $2.2\text{ °C}$, like the inside of a refrigerator.