Geog 201

Supplemental Information

Graphing guidelines

It is important to prepare your graphs carefully. The following set of rules should help you to construct graphs that are both visually appealing and intellectually illuminating.

1. The graph must have a title. Put the title clearly above the graph. Make the title as descriptive as possible in order to explain what the graph is showing.

2. The graph must have axes labeled with names and units. Draw the axes and select appropriate scales. Mark the scales at uniform intervals. Make the graph as large as is reasonable, but be sure that the major divisions of the graph paper have a simple relationship to the scales to be marked along each axis. Usually the axes are drawn from the origin (0,0), but it is not necessary to do so if all the data points plot far from the origin. If the graph is to demonstrate that y is directly proportional to x, then the origin should be included. Most graphs plot only positive values of x and y (+x and +y, i.e., the first quadrant), but this is not always the case. Plan ahead!

3. Plot the points clearly. Use a pencil! Use a dot enclosed in a circle or a cross. This allows the data point to be defined precisely. When a more than once curve is to be shown on the same graph, use different symbols for the points (dots in squares, triangles).

4. For non-linear curves where you are trying to express a mathematical relationship between y and x, draw a neat smooth freehand line through, but not necessarily connecting the data points. (NOTE THAT THIS DOES NOT APPLY TO ALL GRAPHS - see 5). Never join these data using a ruler and the dot-to-dot method. Use a pencil and draw lightly at first, as you may want to erase parts or sections of the curve and try again. Try to get the data points evenly spaced about your curve, with roughly equal numbers above and below your line if the data points do not pass through your curve

5. For graphs where there is no mathematical expression for the relationship, such as profiles where you are trying to follow the trend of one variable with the other, draw a neat smooth freehand line passing through each dot. This method is appropriate as you are interested in the point by point variation. Even though the line must pass through each point, it is a smooth curve (not drawn with a ruler).

6. For histograms, usually use a bar-graph style, and choose simple class sizes of approximate size (see lab 1 ). You may plot the raw count data, and/or the percent, on the y-axis.

Significant figures

Every measurement is uncertain to some extent. Suppose, for example, that we wish to measure the mass of an object. If we use a platform balance, we can determine the mass to the nearest 0.1 g.

An analytical balance, on the other hand, is capable of given results correct to the nearest 0.0001 g. The exactness, or precision, of the measurement depends upon the limitations of the measuring device and the skill with which it is used.

The precision of a measurement is indicated by the number of figures used to record it. The digits in a properly recorded measurement are significant figures. These figures include all those that are known with certainty plus one more which is an estimate.

Suppose that a platform balance is used, and the mass of an object is determined to be 12.3 g. The chances are slight that the actual mass of the object is exactly 12.3 g, no more nor less. We are sure of the first two figures (the 1 and the 2); we know that the mass is greater than 12 g. The third figure (the 3), however, is somewhat inexact. At best, it tells us that the true mass lies closer to 12.3 g than to either 12.2 g or 12.4 g. If, for example, the actual mass were 12.28 ...g or 12.33 ...g, the value would be correctly recorded in either case as 12.3 g to three significant figures.

If, in our example, we add a zero to the measurement, we indicate a value containing four significant figures (12.30 g) which is incorrect and misleading. This value indicates that the actual mass is between 12.29 g and 12.31 g. We have, however, no idea of the magnitude of the integer of the second decimal place since we have determined the value only to the nearest 0.1 g. The zero does not indicate that the second decimal place is unknown or undetermined. Rather, it should be interpreted in the same way that any other figure is (see, however, rule 1 that follows). Since the uncertainty in the measurement lies in the 3, this digit should be the last significant figure reported.

The following rules can be used to determine the proper number of significant figures to be recorded for a measurement.

1. Zeros used to locate the decimal point are not significant. Suppose that the distance between two points is measured as 3 cm. This measurement could also be expressed as 0.03 m since 1 cm is 0.01 m. 3 cm = 0.03 m

Both values, however, contain only one significant figure. The zeros in the second value, since they merely serve to locate the decimal point, are not significant. The precision of a measurement cannot be increased by changing units.

Zeros that arise as a part of a measurement are significant. The number 0.0005030 has four significant figures. The zeros after 5 are significant. Those preceding the numeral 5 are not significant since they have been added only to locate the decimal point.

Occasionally, it is difficult to interpret the number of significant figures in a value that contains zeros, such as 600. Are the zeros significant, or do they merely serve to locate the decimal point? This type of problem can be avoided by using scientific notation. The decimal point is located by the power of 10 employed; the first part of the term contains only significant figures. The value 600, therefore, can be expressed in any of the following ways depending upon how precisely the measurement has been made.

6.00 x 10² (three significant figures)

6.0 x 10² (two significant figures)

6 x 10² (one significant figure)

2. Certain values, such as those that arise from the definition of terms, are exact. Far example, by definition, there are exactly 1000 ml in 1 litre. The value 1000 may be considered to have an infinite number of significant figures (zeros) following the decimal point.

3. At times, the answer to a calculation contains more figures than are significant. The following rules should be used to round off such a value to the correct number of digits.

(a) If the figure following the last number to be retained is less than 5, all the unwanted figures are discarded and the last number is left unchanged.

3.6247 is 3.62 to three significant figures.

(b) If the figure following the last number to be retained is greater than 5 or 5 with other digits following it, the last figure is increased by 1 and the unwanted figures are discarded.

7.5647 is 7.565 to four significant figures

6.2501 is 6.3 to two significant figures

(c) If the figure following the last figure to be retained is 5 and there are only zeros following the 5, the 5 is discarded and the last figure is increased by 1 if it is an odd number or left unchanged if it is an even number. In a case of this type, the last figure of the rounded off value is always an even number. Zero is considered to be an even number.

3.250 is 3.2 to two significant figures

7.635 is 7.64 to three significant figures

8.105 is 8.10 to three significant figures

The number of significant figures in the answer to a calculation depends upon the numbers of significant figures in the values used in the calculation. Consider the following problem. If we place 2.38 g of salt in a container that has a mass of 52.2 g, what will be the mass of the container plus salt? Simple addition gives 54.58 g. But we cannot know the mass of the two together any more precisely than we know the mass of one alone. The result must be rounded off to the nearest 0.1 g, which gives 54.6 g.

4. The result of an addition or subtraction should be reported to the same number of decimal places as that of the term with the least number of decimal places. The answer for the addition

161.032 + 5.6 + 32.4524 = 199.0844

should be reported as 199.1 since the number 5.6 has only one digit following the decimal point.

5. The answer to a multiplication or division is rounded off to the same number of significant figures as is possessed by the least precise term used in the calculation. The result of the multiplication

152.06 x 0.24 = 36.4944

should be reported as 36, since the least precise term in the calculation is 0.24 (two significant figures).