Curved patterns and claims in the news

What you will be able to do

• recognise growth that multiplies, and straighten it with a logarithm;
• use a square root for a quantity that grows like a square;
• turn a prediction on the log scale back into a count;
• read a claim in the news with the right questions.

A growing culture

A made-up school laboratory counts the yeast cells in a warm sugar solution every hour.

The counts curve upwards: each hour adds more than the hour before. A straight line fits badly: $r = 0.8864$, and the residuals run positive, negative, then positive again. The counts roughly double each hour. Growth that multiplies by the same factor in each step is called exponential.

Taking logarithms

Take $\log_{10}$ of each count. Multiplying by $2$ always adds the same amount, $\log_{10} 2 \approx 0.301$, to the logarithm. So the logarithms grow by equal steps, and they lie close to a straight line. Indeed, the correlation between the time and $\log_{10}$ of the count is $0.9995$.

The least-squares line for the logarithms is

What does it mean? Each hour adds $0.2983$ to the logarithm, so it multiplies the count by $10^{0.2983} \approx 1.988$, nearly $2$. The doubling time is $\dfrac{\log_{10} 2}{0.2983} \approx 1.009$ hours.

To predict, work on the log scale and then undo the logarithm. At $3.5$ hours, $\log_{10} y = 0.7127 + 0.2983 \times 3.5$, so $y = 10^{1.757} \approx 57.1$ thousand cells per mL. The usual warning holds: at $10$ hours the line predicts about five million cells per mL, but the sugar runs out long before.

The square root idea

Some quantities grow like a square. The stopping distance of a car is one: double the speed and the braking distance grows about four times. The test-track data of the course on the least-squares line gave $r = 0.9916$ and a U-shaped residual plot. Take the square root of each distance and plot it against the speed: the bend disappears, and $r$ rises to $0.9998$. In words: a log suits growth by equal factors, and a square root suits growth like a square.

Reading claims in the news

Here is a made-up headline: "Children who eat breakfast get better marks." Before you believe that breakfast causes better marks, ask these questions.

1. Was it an experiment or an observational study? Were the children given breakfast at random, or did the reporters only compare families?
2. What lurking variables could there be? Families who serve breakfast may also have more time, more books or more sleep.
3. How strong is the link? Ask for $r$ or the size of the difference, not just "a link".
4. Is the result about the people in the data? A study of one town, or of adults, may not apply to you.
5. Was the pattern a straight line, and was it an extrapolation? "If one hour helps, ten hours help ten times as much" is rarely true.
6. Who paid for the study, and how many studies agree?

Your turn: new numbers

Try it in Python

The program compares $r$ for the counts and for their logarithms, fits a line to the logarithms, and turns its slope into a growth factor and a doubling time.

The usual mistakes

• Forgetting to undo the logarithm. A prediction of $\log_{10} y = 1.757$ means $y = 10^{1.757}$, not $1.757$.
• Straightening a curve that has a real end. Growth stops when food or space runs out.
• Taking the log of zero or a negative number. It is not defined. Logs suit counts and sizes that are always positive.
• Reading a headline as a proven cause. Ask the six questions first.

Remember these

• Growth by an equal factor each step becomes a straight line on a log scale.
• A slope $b$ for $\log_{10} y$ means a factor of $10^{b}$ for each unit of $x$.
• A square root suits a quantity that grows like a square.
• A claim of cause needs an experiment, or a strong case from many studies.

Check yourself

Try each one on your own first. If you are stuck, open a hint. Then check with Show the answer.

1. A count is $3$, $6$, $12$, $24$ at times $0$, $1$, $2$, $3$. By how much does $\log_{10}$ of the count rise each step?

2. A line for a count is $\log_{10} y = 1 + 0.5x$. By what factor does the count grow for each unit of $x$?

3. Why can we not take the logarithm of a count of $0$?

4. A headline says: "People who drink green tea live longer." Name one lurking variable.

5. The area of a square field grows with its side. Which transformation of the area gives a straight line against the side?

References

• OpenStax, Introductory Statistics, Chapter 12 (Linear Regression and Correlation).
• OpenStax, College Algebra, Chapter 6 (Exponential and Logarithmic Functions).