Measuring Things

Roger W. Ehrich

"Boys and girls, we have told you a lot this year about the way in which things you read and things you hear are not always what they seeem to be. Many times we have to measure things to tell other people what we are seeing, but if we try to make accurate measurements we often find that it is hard to make them correctly. Have you ever tried to make something, only to find that you made a bad measurement so that things wouldn't fit correctly?"

"Scientists always worry about how accurately they can measure things. Do you know what we mean when we say we have measured something accurately? We know that every year has 31,556,925.5 seconds. Do you think that is an accurate measurement of the length of a year? Well, actually, it's like measuring the distance to the moon correctly to within the length of your classroom. But just writing down a number doesn't tell you anything about how accurate it is, does it. Suppose I had told you it was 41,556,925.5 seconds?" That would not be very accurate, but you wouldn't know that from the number alone, would you?"

"Today we are going to measure the length of your classroom. We will work in 12 teams of 2, table by table. Take a 6" ruler, and with your partner, make one measurement of the length of the room. Don't tell anyone else what your answer is...we'll collect the answers when everybody is done. But try to make your measurements as accurate as you can. Please make your measurements in centimeters."

The teacher now helps the students make their measurements, which are complicated by the layout of the room and the short rulers, and probably by the commotion. When the students finish, they report their results, and the teacher writes the numbers on the board in ascending order. All the results are recorded in the students' notebooks.

"Well, now, there seems to be some disagreement about the length of the room? Who is right? Is anybody right? How do we know? Then how do we explain to anyone how long the classroom is? Can we think of a better way to look at these numbers?"

The teacher now draws an expanded scale and plots the values on the scale. With a little luck the numbers will cluster around some value, and there will be some numbers which are clearly off, a result of measurement error. The students sketch this scale in their notebooks.

"Is there any way to guess whether any of these values are clearly wrong? [the outliers]. What would be your best guess about what the length of the room is?"

The teacher now explains what an average is and what the median is. Next the class calculates the average and the median and plots these on the expanded scale. Notice how both seem to be reasonable values to choose for the length of the room. Now show the children that if someone really made a bad measurement, the average would change but the median would not. We conclude that the median might be useful if we suspect that some measurements might be really bad.

Next, the teacher talks about how accurate the measurement might be. It has something to do with the spread of all the measurements. How could you measure that? You could take the difference between the maximum and minimum values. Actually you would use half the spread and say that the measurement was accurate to half the difference between the maximum and minimum values. The teacher could go on quite a bit on this theme and ask the children if that meant that the next measurement which would be made would be guaranteed to fall within these limits? Hmm, definitely not, but most likely it would.

Finally, the teacher asks the children what they would do to make better measurements. One way would be to use a better ruler. Another way would be to make more measurements.

The Lab Report

Save a copy of the text version of the following report on your computer, paste it into an Email message, and answer all the questions carefully. Then mail it to Mrs. Hood and to Dr. Ehrich.

Experiment Title: Measuring Things Accurately

Hypothesis: The length of the classroom can be measured accurately with a short 6" ruler

Materials: A 6" ruler, actually 17.28cm long

Steps:

  1. Make 12 groups of the 2 students from each table.

  2. Each group is to measure the length of the room as accurately as possible. Don't tell anybody what you measure. Count the number of whole ruler lengths and the distance left over as accurately as possible.

    Number of whole rulers _________  x  17.28 cm = _________ centimeters
    
    Remaining length in centimeters                 _________ centimeters
    
    Total length in centimeters                     _________ centimeters
    
  3. Write down all 12 measurements in order from smallest to largest.
    _________________________________________________________
    
    _________________________________________________________
    
    _________________________________________________________
    
Computing the Results:

  1. What is the 6th largest measurement of the 12? This number is called the median...it is the middle number.
    _________________________________________________________
    
  2. Add up all 12 measurements and divide by 12. This number is called the average.
    _________________________________________________________
    
Looking at the Results:

Sometimes if we draw just the right picture we can understand things better. Numbers aren't always easy to understand. Your teacher will show you how to draw a scale from the smallest measurement to the largest, with the measurements marked on the scale with x's. Copy the drawing into your report and put x's where the numbers should be. Mark the median with an m, and mark the average with an a.


______________________________________________________________
Thinking About the Results:

  1. Why are there so many different measurements?
    _________________________________________________________
    
    _________________________________________________________
    
  2. What do you think the length of the room really is? Why? Can anybody really tell?
    _________________________________________________________
    
    _________________________________________________________
    
  3. Do you think any of the measurements are wrong because of a serious mistake? Why? Which ones?
    _________________________________________________________
    
    _________________________________________________________
    
  4. If someone was to make still another measurement, what do you think it would be? Why? Would you be more certain if the measurements were grouped closer together?
    _________________________________________________________
    
    _________________________________________________________
    
  5. Suppose one of the measurements was really far off. Would the median or the average be most affected by this error? Why?
    _________________________________________________________
    
    _________________________________________________________
    
  6. Can you think of any ways to make more accurate measurements of the length of the room?
    _________________________________________________________
    
    _________________________________________________________
    
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