General Chemistry I (CHEM-UA), Lab 2 | Source: Data Analysis Lab Handout (Geggier, NYU)
Tags: graphing, independent variable, dependent variable, trendline, best-fit line, correlation, slope, y-intercept, interpolation, extrapolation, Boyle's law, Logger Pro
Difficulty: Introductory | Prerequisites: Significant Figures study notes (you need SF rules to report trendline parameters).
Tables work fine for a handful of data points, but when a data set grows or when you need to spot a trend, a graph does the job far more effectively. Graphing is a core lab skill that you will use in every subsequent experiment. This topic teaches you how to construct a graph that meets the course's formal requirements, how to add and interpret a trendline, and how to use that trendline to estimate values you did not measure directly.
Plot the dependent variable (what changes) on the y-axis and the independent variable (what you control) on the x-axis. Label axes with variable names and units, give the graph a descriptive title, use data point markers, add a best-fit trendline (not point-to-point lines), and include your name and date. A correlation near 1 means a strong linear fit; slope and y-intercept come from the line equation y = mx + b.
Independent variable
The variable the experimenter controls or deliberately changes (e.g. temperature, pressure, time). Plotted on the x-axis.
Think of it as: the thing you choose to adjust.
Dependent variable
The variable that changes in response to the independent variable (e.g. volume, absorbance). Plotted on the y-axis.
Think of it as: the thing you measure to see what happened.
Ordinate / Abscissa
Ordinate = the y-axis (vertical). Abscissa = the x-axis (horizontal). These terms appear in textbooks and exam questions.
Trendline (best-fit line)
A line (or curve) drawn through the data that best represents the overall relationship. It does not connect individual data points. In Logger Pro you add it via the "Curve Fit" menu.
Correlation
A number between 0 and 1 that indicates how well the data fits the trendline. A correlation of 1 means a perfect fit. A correlation near 0 means no linear relationship.
Slope (m)
The rate of change of y with respect to x. In y = mx + b, the slope is m. It has units of (y-units)/(x-units).
y-intercept (b)
The value of y when x = 0. In y = mx + b, it is b.
Interpolation
Estimating a value within the range of your measured data using the trendline.
In simple terms: reading a value off the line between your data points.
Extrapolation
Estimating a value outside the range of your measured data using the trendline. Less reliable than interpolation because you are assuming the trend continues beyond where you have evidence.
In simple terms: extending the line past your data and reading a value from the extension.
The independent variable goes on the x-axis. The dependent variable goes on the y-axis. You read the graph as "y vs. x" (e.g. "volume of N₂ vs. temperature").
Example: studying how the volume of nitrogen gas depends on temperature. Temperature is independent (you set it); volume is dependent (you measure it). So plot volume on y, temperature on x.
Every graph in this course must satisfy all six:
Axis labels with variable name and unit. "Volume (L)" not just "Volume" and never just "L".
Visible point symbols (markers). Each data point gets a clearly visible dot, circle, or square. Do not cover data points with text boxes.
Appropriate scale. The data should occupy as much of the available space as possible. Use Autoscale in Logger Pro, or manually adjust the min/max values on each axis.
Trendline (best-fit line), not point-to-point connections. A smooth line or curve that best represents the trend. Do not connect the dots. Exceptions are noted in individual lab manuals.
Descriptive title. "Volume vs. temperature" is too vague. "Volume of N₂ vs. temperature" tells the reader what system is being studied.
Name and date in a footer or text box on the graph.
A straight line is described by y = mx + b, where m is the slope and b is the y-intercept.
The correlation (r value) tells you how well the data fits the line. Logger Pro reports this in the linear fit dialog box. A correlation of 0.9988, for example, means the data is very nearly perfectly linear.
When reporting trendline parameters, round slope, y-intercept, and correlation to the same number of SFs as the data used to create the graph.
Interpolation estimates a value within the range of your data. Extrapolation estimates a value outside that range. Logger Pro's Interpolate tool handles both.
The precision of values read from a graph cannot exceed the precision of the data used to create it.
The data shows an inverse relationship between volume and pressure (Boyle's law: V is proportional to 1/P).
P (atm) | V (L) | 1/P (atm⁻¹) |
|---|---|---|
0.500 | 44.7 | 2.00 |
0.600 | 37.5 | 1.67 |
0.700 | 32.0 | 1.43 |
0.800 | 28.1 | 1.25 |
0.900 | 24.7 | 1.11 |
1.00 | 22.6 | 1.00 |
1.10 | 20.4 | 0.909 |
1.20 | 18.9 | 0.833 |
1.30 | 17.0 | 0.769 |
1.40 | 16.2 | 0.714 |
1.50 | 15.0 | 0.667 |
Plotting V vs. P gives a curve (inverse relationship). Plotting V vs. 1/P gives a straight line, which is easier to analyse with a linear trendline.
For the V vs. 1/P plot, the linear trendline gives:
Equation: V = m(1/P) + b
Slope ≈ 22.4 L·atm (this is the proportionality constant, related to nRT in the ideal gas law)
y-intercept ≈ 0 L (as expected from Boyle's law)
Correlation ≈ 0.9998
Interpolation example: to find V at P = 1.04 atm, read off the trendline at 1/P = 0.962 atm⁻¹.
Extrapolation example: to find V at P = 0.400 atm, read off the trendline at 1/P = 2.50 atm⁻¹ (outside the measured range).
"Connect the dots to make a graph." Never connect data points with straight lines. Always use a trendline (best-fit line or curve). The trendline represents the underlying relationship; individual points include random error.
"Volume vs. temperature means temperature on the y-axis." The convention is "y vs. x". The first variable named is the one on the y-axis.
"A good title is just 'y vs. x'." That is descriptive of the axes, not the experiment. The title should identify what system or substance you are graphing.
"Extrapolated values are just as reliable as interpolated ones." Extrapolation assumes the trend continues unchanged beyond your data range. This is an assumption, not a certainty.
⚠️ Worksheet III (Question 6) shows three flawed graphs and asks you to identify the errors. Use the six requirements as your checklist.
⚠️ Worksheet IV requires two graphs: V vs. P (with an inverse curve fit) and V vs. 1/P (with a linear trendline). Both must meet all six requirements.
⚠️ You must be able to read values from a trendline by interpolation and extrapolation, and state those values with the correct number of SFs.
⚠️ Trendline parameters (slope, y-intercept, correlation) must be reported with the correct number of SFs, matching the precision of the data.
Fill in the blank: The independent variable is plotted on the __________-axis. (x)
True or False: Data points on a graph should be connected with straight lines. (False: use a trendline)
Fill in the blank: A correlation of __________ means a perfect linear fit. (1)
True or False: Extrapolation is more reliable than interpolation. (False)
Fill in the blank: In the equation y = mx + b, m is the __________ and b is the __________. (slope; y-intercept)
Q: A student labels a graph "volume vs. temperature" with no further detail. Name two problems with this title.
A: (1) It does not specify what substance's volume is being measured (e.g. "Volume of N₂ vs. temperature" would be better). (2) As written, it is generic enough to describe any experiment, which defeats the purpose of a descriptive title.
Q: Look at graph (a) in Worksheet III. It has axes labelled "Y" and "X" with no units, a non-descriptive title is missing, and the data does not fill the plot area. List the errors.
A: (1) Axis labels lack variable names and units. (2) No descriptive title for the graph. (3) The scale is not optimised to fill the available space. (4) May also be missing point markers or using inappropriate ones depending on the version.
Q: If the argon data gives V = 22.4(1/P) + 0.0 for the V vs. 1/P trendline, what is the expected volume at P = 0.400 atm?
A: 1/P = 1/0.400 = 2.50 atm⁻¹. V = 22.4 x 2.50 + 0.0 = 56.0 L. This is an extrapolation because 0.400 atm is outside the measured range (0.500 to 1.50 atm).
Q: Why does plotting V vs. 1/P produce a straight line when V vs. P does not?
A: Because V is proportional to 1/P (Boyle's law: PV = constant at constant temperature). The relationship between V and P is an inverse (hyperbolic) curve, but V and 1/P have a linear relationship. Linearising the data by plotting 1/P lets you use a straight-line trendline and extract slope and y-intercept directly.
Graphing is used in every lab this semester. The V vs. P data in Worksheet IV is a preview of Boyle's law, which you will study formally later in the gas laws unit. The concept of linearising data (plotting V vs. 1/P instead of V vs. P) is a technique you will use repeatedly: whenever a relationship is not linear, transform one variable so that it becomes linear, then extract the slope and intercept. Beer's law (absorbance vs. concentration), rate laws, and Arrhenius plots all use this approach.
Related Terms / Search Tags
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