Many fundamental laws of science can be expressed by an equation which relates one variable to some other variables. For example, the Ideal Gas Law, PV = kT, relates pressure, volume and temperature. Even when the value of one variable cannot be predicted exactly from the values of some other variables, a formula that gives a reasonable approximation can be very useful.

In this lesson students will try to determine if relative humidity and air temperature are related, and, if so, in what way.

This lesson can be integrated into a weather unit in earth science classes, into a unit on gas laws in physics classes or into a data analysis or curve fitting unit in mathematics or statistics classes.

Students should be able to use a search engine on the Internet and to access e-mail. (If students do not have e-mail accounts, the teacher may obtain the data through his/her e-mail.) Students should have some knowledge of basic weather terms and, if possible, basic skills using spreadsheets or graphing calculators.

E-mail access. Spreadsheet with regression line calculation or graphing calculator, if possible.

Use an information service to obtain current weather data by e-mail.

Create a scatterplot showing relative humidity vs air temperature.

Form a conclusion that describes whether a relationship exists between air temperature and relative humidity and, if so, what sort of relationship.

Find a best-fitting line to the data.

Write a report describing the data obtained and the conclusions formed.

Either the teacher or students will obtain relative humidity and temperature data over a 10 to 16 hour time period at some location by subscribing to an information service which can e-mail this data on an hourly basis. This information should be available before starting this lesson.

A weather site that e-mails hourly weather information is http://www.infobeat.com

NOTE: Be sure to Unsubscribe or Cancel after you have all your information.

Part II.

Students should make a scatterplot of relative humidity vs temperature and decide if these variables seem to be related in some way. The scatterplot can be done either by hand or by using a graphing calculator or spreadsheet. If graphing calculator or software is available the equation of a best fitting line (also called the regression line) can be calculated and the correlation coefficient r or the coefficient of determination r² computed. The coefficient of determination r² (always between 0 and 1) describes how well the data fits the line. A rule of thumb is:

(essentially no linear relationship; the equation is not useful)

(poor relationship)

(mild relationship)

(strong relationship; the equation describes the relationship well)

An overhead of "Examples of Scatterplots" can be shown and discussed.

Students should write a one to two page report describing their data and conclusions.

After students have completed their reports, they can compare results. Some questions to ask are:

Was there usually a positive or negative trend?

What factors besides temperature can affect relative humidity?

For those students who found a negative trend, how do the slopes compare?

What can account for different slopes?

Is there any relationship between the slopes and the city where the data was reported?

Is the slope the same when you take data from different days but at the same location?

What kind of experiment could be performed in the laboratory to test the hypothesis that warmer air can "hold" more moisture?

How is relative humidity measured?

Student papers should be graded using the following characteristics.

I. a. Neatness.

b. Clarity.

II. a. Organized table of relative humidity and temperature values or other hourly weather data.

b. Scatterplot and graph of best-fitting line.

c. Equation of best-fitting line.

d. Correlation coefficient or coefficient of determination.

III. a. Statement of the problem.

b. Definition of relative humidity and units of temperature and any other variables reported.

c. Discussion of whether there is a clear pattern describing how relative humidity changes with variation in air temperature. If so, what is the pattern? Interpretation of the coefficient of correlation (if computed).

d. Discussion of other factors besides temperature that may affect relative humidity.

Students may try to find out how "heat index" is calculated and what it has to do with relative humidity.

!!!! Warning !!!! The Web sites given in this lesson may have changed! Before using this lesson with your students, be sure to check if the sites are still working or if you must find another site. Sometimes the sites still have the relevant data but you may need to change the directions to access the data.

Since warmer air can hold more moisture, as the temperature rises during the day, the relative humidity should decrease. Hence students should see a negative slope on their scatterplot. If it is a windy day, if there is a storm front moving in, or if there is lots of moisture from the previous day and hence a lot of evaporation, or etc., this negative relationship might not be apparent. Students should understand that if the amount of water in the air remains approximately constant, then there should be a negative relationship between temperature and relative humidity.

If more information about relative humidity or weather, in general, is needed, any Internet search engine will give many good sites to answer your questions. For example, the site http://www.weatherpost.com is very useful.

 If you wish more background about the statistical concepts involved in the lesson, some good sites to check are:

http://davidmlane.com/hyperstat/index.html

http://www.anu.edu.au/nceph/surfstat/surfstat-home/surfstat.html

http://www.math.unb.ca/~maureen/SSCEdCom/basicstats/basicstats.html

http://www.math.unb.ca/~knight/BasicStat/$content.htm

http://www.bbns.org/us/math/ap_stats

http://www.grad.cgs.edu/wise/linksf.shtml

http://www.cvgs.k12.va.us/DIGSTATS

http://www.statsoft.com/textbook/stathome.html

http://www.stats.gla.ac.uk/steps/glossary/index.html

http://www.crpc.rice.edu/CRPC/GT/sboone/Lessons/lptitle.html

http://forum.swarthmore.edu/library/topics/statistics

http://www.psychstat.smsu.edu/introbook/skb00.htm

TI-83 instructions:

http://www.ti.com/calc/docs/act/koehler001.htm

http://www.wku.edu/~neal/manual/ti83.html

The Calculator website at the Mathematics Department of the University of Wisconsin-La Crosse will perform basic statistical calculations. If you do not have access to a simple statistical computer package or calculators with statistics options, your students may access http://www.compute.uwlax.edu/stats_htdocs/newmenu.html to perform statistical computations on-line.

In order to print out just a copy of the student worksheet, highlight this section, then copy and paste it into your word processor. You may then revise the worksheet if you wish.

Press STAT, choose EDIT and enter the temperature data in, say, list L₁ and the relative humidity data in list L₂. Press STAT and choose CALC. Cursor to 4:LinReg(ax+b) and press ENTER. Then type L₂,L₁,Y₁. The y-variable always is first. The regression equation will be stored as Y₁. The variable Y₁ can be accessed from the VARS menu. The slope, y-intercept and coefficients of correlation and determination will be displayed. If these coefficients are not displayed you will need to set DiagnosticsOn. You can find this word in the CATALOG.

To graph the data and the best-fit line use STAT PLOT. Turn ON one of the Plots, choose the first icon showing a scatterplot, enter the temperature list name as the Xlist and the relative humidity list name as the Ylist. Then press ZOOM and choose ZoomStat.

Type in the temperature, relative humidity and any other variables you wish to consider in columns. To find the line of best fit and also the coefficient of determination, position the cursor on an empty cell and highlight the next 3 rows and the next 2 columns. Then type (for example) = LINEST(D3:D14,C3:C14,1,1) where D3:D14 contains the relative humidity values and C3:C14 the temperature values. (Your columns may be different. But the y-variable range always is first.) Then press Ctrl + Shift + Enter.

The first row displayed shows the slope and y-intercept of the best-fit line. The first number in the third row is r², the coefficient of determination.

To show the graph of this data, position the cursor in an empty cell and click the graph icon. Choose XY(Scatter), then Next. Type C3:D14 for the Data Range and click on Columns and then on Next. Type "Temperature" for the Value(X) Axis and "Relative Humidity" for the Value(Y) Axis. Click Next. Click on As Object In for Chart Location and then click Finish.

You may want to change the limits on the Temperature and Relative Humidity scales. To do so double click on the x-values in the graph to display the Format Axis window. Click on Scale and type in the temperature limits. Similarly change the relative humidity limits.

Finally click on Chart on the top bar and choose Add Trendline to display the graph of the best-fit line.

Have students draw a line that seems to follow the data on their graph. They should calculate the slope of this line by measuring its rise and run using the formula slope = rise / run.

Note in these last two examples that the data is exactly the same except for one "outlier" in the left graph. However, the coefficient of determination changes from 0.52 to 0.92 just because one point has been changed.

How Relative Humidity Varies with Changes in Air Temperature

 We wanted to see if air temperature affects relative humidity. Relative humidity is a percent. It is calculated by the taking the ratio of the amount of moisture in the air divided by the amount of moisture the air can hold. The hypothesis is that warm air can hold more moisture than cold air. You can see this when, for example, your pop can "sweats". The cold air near the can cannot hold so much moisture and so the moisture condenses on the can.

We got temperature and relative humidity information from a weather information site on the Internet. We collected this data for one day to try to make sure that other weather conditions were about the same over our 12-hour period. We assumed that the amount of water in the air was approximately constant over this 12-hour period. Hence during the day when it was warmer, the air could hold more moisture and so the relative humidity should be less. Since it was sunny all day and the wind was mild, from 1 to 12 mph, the weather was pretty constant during this 12 hour period. So it seems ok to assume that the amount of moisture in the air did not change very much during this time.

Just by looking at the data you can see that as the temperature was rising, the relative humidity was decreasing. This means there was a negative relationship between temperature and relative humidity.

The equation of the best-fitting line is

The slope is negative, showing that as temperature increases, relative humidity decreases. In fact, for every increase of one degree Fahrenheit in temperature, the relative humidity decreases by about 1.9%. The coefficient of determination is .922162. This indicates that the line describes the data very well. You can also see by the following graph that the data follows the best-fit line very well.