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## Three common data patternsThe best way to spot a pattern in your data is tograph it. Graphs can reveal trends in the data that you simply can't see by looking at a long column of numbers.
## ProportionalitySuppose you are trying to find the relationship between the volume of water and the mass of water experimentally. You weigh several different samples of water and determine their volumes as follows:
What is the pattern in this data? Increasing the mass of the sample seems to increase the volume of the sample by roughly the same amount; doubling the mass doubles the volume. If you divide the volume by the mass for each of the four points, you get 1.003 mL/g every time. For this experiment, water has a constant volume per gram. To make the pattern plain, you can graph the volume as a function of mass:
Take a ruler and draw the best straight line through the data points. The slope (steepness) of the line is telling us how fast volume changes for a particular change in mass. The slope happens to be 1.003 mL/g. We can summarize ALL of the data using a simple equation:
V = c m
where The direct proportionality relation between volume and mass is often written in physical sciences this way:
V m
The symbol
means "is directly proportional to" or "is equal to some constant times". This is a useful way to summarize the relationship between the two variables if we don't know (or don't care) what the actual value of the proportionality constant is.
Saying
This last statement allows you to predict the value of any one measurement given the other three.
- Weekly pay (p) is directly proportional to the number of hours you work (h). The proportionality constant is hourly wage.
- Force (F) is directly proportional to acceleration (a). The proportionality constant is mass (m).
- Distance traveled (d) is directly proportional to time (t) when travelling at a constant speed (v).
Exercise 2. Use the proportionality relationships from Exercise 1 to compute the following.
- What is the pay for working 30 hours if the pay for working 40 hours is $440?
- What is the force (in lbs) required to accelerate a body by 100.0 ft/s
^{2}if the force required to accelerate the same body by 32.0 ft/s^{2}is 10.0 lbs? - What distance will be traveled in one day if after two hours, 10 miles have been traveled? Assume constant speed.
## LinearityIt's easy to see that if the mass of water was zero, the volume of water would also have to be zero. But consider the following experiment, in which the volume of a balloon is measured at several different temperatures:
Is the volume of the balloon zero at 0°C? To see, we can
It seems that the volume of the balloon around 0°C will be around 22.5 mL. So it is incorrect to write
V t
or
V = c t
because the volume isn't zero when when the celsius temperature is zero. We can write
V = c t + b
where
As with proportionality relations, ## Inverse ProportionalitySometimes one measurement decreases as the other increases in a simple but nonlinear way. Consider the following experiment, which measures the pressure of a gas trapped in a closed syringe as the plunger is pulled outward:
It looks like doubling the volume cuts the pressure to one half its original value. Multiplying the pressure and volume together for each point, we get the same number (22.4
PV = c
or
P = c (1/V)
or
P (1/V)
When two variables are related this way, they are said to be | |||||||||||||||||||||||||||||||||||||||||||||

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