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In this lesson, you will learn that every measurement has some degree of uncertainty and how to work with that uncertainty in scientific investigations.
Every measurement has some degree of uncertainty, indicated by its precision and its accuracy.
Precision is the exactness to which a measurement can be reproduced. Using a ruler, calibrated to millimeters, you may be able to measure to a precision of one millimeter. A vernier caliper, on the other hand, may provide measurements with a precision of hundredths or thousandths of a millimeter. Even the best measuring instruments have a limit to their precision.
Accuracy is the closeness of a measured value to an accepted value. A measurement can be very precise, but if the instrument is not properly calibrated it provides inaccurate measurements. For example, a thermometer that is calibrated to 0.5° has a precision of +/- 0.5°. However, if the thermometer reads 23°C and another thermometer that is known to be accurate reads 25°C, there is an inaccuracy of 2°C. The process of comparing a measuring instrument to a known standard and adjusting it to read accurately is calibration.
Errors can be introduced into an experiment by inaccurate or imprecise measuring instruments. They can also be introduced by using these instruments in the wrong way.
The precision of every measurement depends on the precision of the instrument. The rules for recognizing significant digits are:
The final condition creates a problem for numbers such as 18,300 if one or more of the zeros should actually be considered significant. Scientific notation takes care of the question of the significance of zeros at the end of a number greater than 1. Remember that the value 24,000 could have 2, 3, 4, or 5 significant digits, depending on the precision of the measurement. By using scientific notation, the uncertainty is removed:
2.4 × 10 5 has 2 significant digits
2.400 × 10 5 has 4 significant digits
Which of the following correctly expresses the measured value of 0.00001030 seconds?
The correct answer is C. Scientific notation uses a number between 1 and 10 multiplied by the power of ten. Because the decimal point was moved 5 places to the right, the exponent is -5. Remember that the final zero is significant.
When you work with significant digits, the result of a mathematical operation can never be more precise than the least precise measurement. It is particularly important to remember this when using a calculator to work with measured values, which generally display results in 8–10 digits.
You must also understand that, because it is used to indicate uncertainty, the concept of significant digits applies only to measurements. Numbers used for counting, such as 4 objects, and numbers that are defined as constants, such as 10 millimeters = 1 centimeter, are considered to have an infinite number of significant digits. For example, (34.98 cm) (10 mm/cm) = 349.8 mm.
Review mathematical concepts that are important in science and how to visualize mathematical data in charts and graphs.
Science grows by constructing new hypotheses based on previous results, so sharing knowledge is the final step of an experiment or investigation. Peer reviews and duplication of experiments are necessary to ensure the validity of experimental results. A scientific investigation is not very useful unless you can communicate results and conclusions to someone else. Because data is generally collected in a numerical form, tools for displaying mathematical data need to be able to condense these into a non-numerical figure or diagram, not simply a list of numbers.
There are many ways to present data in a organized form, generally using some kind of graph or table. A simple table such as this relates the dependent variable to the independent variable:
| heating time (minutes) | temperature (&Deg;C) |
|---|---|
| 0 | 45 |
| 1 | 68 |
| 2 | 88 |
| 3 | 100 |
| 4 | 100 |
| 5 | 100 |
Another way to present data is by using a graph — a visual representation of numerical relationships. One familiar type of graph is a line graph, which plots data on x and y axes. This graph below shows the distance from home versus time for a bicyclist traveling at a constant speed.
This graph shows a linear relationship between the dependent variable (distance) and the independent variable (time). This relationship can be expressed as a linear equation in the form:
y = mx + b
where y is the dependent variable, x is the independent variable, m is the slope of the line, and b is the value of y when x is equal to zero. Notice that on this graph, the slope is negative, indicating that the distance decreases as time increases. The bicyclist started at a point 20 miles from home, represented by the y intercept. The equation represented on the graph is therefore:
distance = (-33.3 m/s) (time in seconds) + 20 m
There are many different kinds of graphs used to display different types of data. Depending on your audience, the subject matter, and the level of detail in the data and/or analyses, you need to choose the right way to show exactly what you want to — and get your ideas across in the simplest but most accurate manner possible. For this, we use visual representations of data.