Scientific Measurement and Lab Skills

Translating experimental findings from data points to visual representations, like graphs, is essential to determining the relationship between two or more properties. These properties are called variables. When there are two variables, a graph created from the data is called two-dimensional. The graph has two axes. The independent variable is plotted on the x-axis, and the dependent variable is plotted on the y-axis.

Take, for example, this sample data for the temperature and volume of a gas. The volume of the gas depends on the temperature. Thus, we would plot the measured temperature on the x-axis and the volume on the y-axis.

When there are several data points with the same x-value, like if we measured the volume several times at one temperature, we also calculate the standard deviation of those measurements. The standard deviation is a statistical value, which indicates the amount of variation present in a set of values.

Standard deviation is calculated using this formula, where n is the number of data points, x bar is the mean value of the data points, and x_i represents each individual data point. You can compute the standard deviation by hand, or a spreadsheet program can calculate it automatically. The closer the standard deviation is to 0, the closer the data points are to the mean value. If the standard deviation is equal to 0, then all of the entered data points have the same value.

Let's look at the mean values and standard deviations of the volume measurements at each temperature in our data set. We can summarize each set of data points as the mean plus or minus the standard deviation. We determine the significant figures for each mean from the corresponding measurements and round the mean values accordingly.

The standard deviation of each group must have the same number of decimal places as the mean, so we round each standard deviation to the hundredths place. To graphically determine the relationship between two variables, we can fit the data with a best-fit function.

The function is automatically generated by spreadsheet software and can take the form of a linear trend line, a polynomial function, or an exponential or logarithmic function. In the case of our temperature and volume data, the relationship is a linear one. So, the data points are fit by linear least squares regression. Your spreadsheet program will return the equation for the best fit line and an r-squared value. The closer the r-squared value is to 1, the better the fit of the data.

Next, you can use your spreadsheet software to find the standard deviations of the slope, the y-intercept, and the calculated y-value. To determine the significant figures for the values in the equation, we follow a simple rule. The last significant figure of each value corresponds to the first significant decimal place of its standard deviation.

Thus, we round the slope to the thousandths place and the y-intercept to the tenths place, and we round the standard deviations to match. Our slope is 0.167 +/- 0.003 liters per Kelvin, and our y-intercept is -40.6 +/- 1.2 liters. Any calculated y-value will be rounded to the tenths place and will be +/- 0.8 liters. This equation describes the relationship between temperature and the volume of a gas.

In this lab, you will create a data set of dependent and independent variables by measuring the diameter and circumference of various sizes of beakers. You'll then use this data to create a scatter plot and perform a linear regression, keeping in mind the importance of significant figures. You'll also practice lab skills, like filtering and measuring volume using pipettes, paying attention to the uncertainty in the measurements and analysis.