Lesson 1: Using Standard Deviation, part 1

Can the standard deviation pick the most consistent set of numbers?
or Which city's temperatures vary the least, San Diego or San Francisco?

Grade Level : 9-12

Background needed:

The student should have a knowledge of how the mean and standard deviation of a set of numbers is calculated using a scientific calculator. It is prefered that they know how to get the standard deviation without using the statistical mode in the calculator. Their insight into the uses for the standard deviation will be more complete if this is so.

Objectives:

  1. The student will use the Internet to find statistical data for investigations that use the mean, standard deviation, and percentage error
  2. The student will calculate the standard deviation of monthly temperature means.
  3. The student will draw conclusions from the standard deviations and percentage error of these means.

Materials:

  1. Access to the Internet and Netscape software.

  2. Previously set bookmark on the Netscape at the following URL for
    the weather report for the San Francisco Bay Area
    http://www2.mry.noaa.gov/nwspage/nwshome.html
  3. Scientific calculator

Procedure:

Once students have logged on to the Internet and Netscape, have them go to the San Francisco Bay Area branch of the the National Weather Service by snapping on the bookmark called NWS-San Francisco Bay Area. They should scroll down and click on Climatological Data, then scroll and click on Climate Normals and Extremes. When they reach the choice of cities, have them click on San Diego.

At this point students will see a chart of weather statistics containing averages for the 12 months. The numbers we will be using are under the Temperature Means Column; specifically, the Avg column (3rd from the left.)

Have students copy the 12 temperatures down. If they scroll down far enough, they will come to San Francisco-Airport and then to the San Francisco-Mission District that we want to use. Have the students find the same 12 numbers under Avg and copy those down .

The question they will answer with these numbers is:

Which of the 2 cities has the most consistent temperatures?

This is not an easy question because the two cities have very similar temperatures year round. As you might expect, the standard deviation helps us with this answer. Remind them that the standard deviation is the average amount that a set of numbers differ from their mean. Also remind them that the more close a set of numbers are to each other, the more consistent they are. If they truly understand these facts, students should deduce that the most consistent set of numbers is usually ( but not always) the one with the lowest standard deviation.

After students have done the calculations and made their decision , they can check their work by calculating the percentage error. It is used in science to represent the relationship of the standard deviation to the mean, telling scientists how much the mean truly represents the numbers it came from. The formula is:

Percentage Error = (Standard Deviation / Mean) * 100

For example, if set A has a mean of 5 and a standard deviation of 4, its percentage error is 80%. If set B has a mean of 10 and a standard deviation of 4, its percentage error is 40%. Thus we would know that the numbers in set B are closer to their mean and therefore vary the least. Notice that the standard deviation was identical in both sets. It's obvious that the standard deviation by itself can't always tell us whether the mean is a good statistic.

As an extension of this lesson, have students study other cities and other sets of numbers at this location in the Internet.