In a relationship between two variables, one variable is
an independent variable (input variable) and the other one is a dependent
variable (output variable). The value of the dependent variable depends on the
value of the independent variable. On a graph, the independent variable is
shown on the x-axis, while the dependent variable is shown on the y-axis.
A relation between two variables can be linear or
non-linear. The graph of a linear relationship is a straight line and the graph
of a non-linear relationship is a curve. A non-linear relationship can be linearized
using regression analysis to get a line of best fit. The line of best fit can
be used to extrapolate the given data set. The graph of a straight line is based
on its slope and y-intercept. Slope is a measure of steepness of a line () and the y-intercept is the point at
which a graph crosses the y-axis.
The main purpose of most experiments is to derive
meaningful data. Often it is difficult to see the significance in the tabulated
data when the numbers are displayed in a table. By graphing the data sets that
are given, one will notice that it is much easier to see the relationships
between the numbers. The simplest relationship of recorded data occurs when the
independent (x-variable) and dependent (y-variable) variables share a linear
relation. The linear numbers are displayed on the graph as a straight line; thus,
revealing the x and y-axis relationship as:
(where m is the slope and b is the y-intercept).
The coefficient b is only represented when the line doesn’t pass through the
origin of the graph. However, not all data can be assumed as a direct relation.
For instance, relationships such as logarithmic, inverse, or a combination of
the two can be used in order to derive the linear relation instead.
The purpose of this lab is to complete two exercises that
require graphing of data sets and using those specific data sets to interpret
which relationships between the variables produce the most linear line. When
calculating the equation of the line, the regression analysis is important to
know because it is used to determine the strength of linearity. If the
regression value is close to 1, the data is strongly related, or in other words
linear. On the other hand, if the regression coefficient value is close to 0,
then the data has little or no relation.
In exercise 1, one is expected to graph wavelength vs.
frequency (figure 1), wavelength vs. (figure 2) and log of wavelength vs. frequency (figure 3). Based on the
trendline, figure 1 illustrated that it was not linear, however seemed very
close to a linear relationship. Figure 2 demonstrated to have a linear
relationship with a trendline reading 3E8 x 7E-11 and a coefficient of determination
coming out to be 1. This justifies that the regression line perfectly fit the
given data that was provided. Using the equation from figure 2, I plugged in
the given frequency of 1/s for
‘x’ to calculate the wavelength of the red color. The wavelength of the red color came
out to be 6.504E-7. Figure 3 looked as if it was almost linear, but it was not
close enough to validate.
In exercise 2, one is expected to graph a total of four
graphs, two of which are temperature vs. volume, with figure 4 in Celsius and
figure 6 in Kelvin. The other two graphs are temperature vs. , with figure 5 in Celsius and figure
7 in Kelvin. Both figures 4 and 6 showed a linear relationship, with equations
y =1.2167x-272.15 and y = 1.2167x+0.8535. From those two equations, one is able
to distinguish that both figures 4 and 6 share the same slope because both
graphs had the temperature vs. volume relationship and 1 degree Kelvin is 273
degrees larger than 1 degree Celsius consistently throughout the graph. This
indication is also confirmed by simply looking at the graph. By looking at
figures 5 and 7, one can realize that temperature vs. was not linear.