Name: graphs, curve fit, and standard deviation
Uploaded: 2020-03-26T07:22:24.000Z
Duration: 4 min 34 s
Channel: JoVE

K12 Standards

Texas Essential Knowledge and Skills

High School

Chemistry

CHEM.1

CHEM.1.D

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 xi 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&#39;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&#39;ll then use this data to create a scatter plot and perform a linear regression, keeping in mind the importance of significant figures. You&#39;ll also practice lab skills, like filtering and measuring volume using pipettes, paying attention to the uncertainty in the measurements and analysis.

scientific measurement and lab skills

Two-Dimensional Graphs
Some chemistry experiments take the form of directly changing one property of the system that you are studying otherwise known as the independent variable, like temperature, and measuring the effects on another property, otherwise known as the dependent variable, like volume. Once the data have been collected, the interaction between the two parameters must be quantified &#8212; or converted into a form that can be evaluated &#8212; and compared to other relationships.
Two-dimensional graphs can be used to derive certain types of mathematical relationships between two properties or to establish that no such relationship exists between them. The analysis will ultimately determine how the dependent variable changes in response to the independent variable. In the example of adjusting the temperature of a liquid or gas and monitoring changes in its volume, the temperature is the independent variable, and the volume is the dependent variable.
To make a two-dimensional graph, each data point must have a value known as the coordinate for both the dependent and independent variables. The independent variable is plotted on the x-axis, and the dependent variable is plotted on the y-axis. These plots are easily made in spreadsheet software, which can also be used to analyze the plotted data.
Curve Fitting
Once a data set has been plotted on a two-dimensional graph, curve fitting can be used to generate an equation, or function, for the dependent variable in terms of the independent variable. Functions represent a mathematical model that best models the data from which it is derived. Curve fitting is the technique of finding a function that produces a line that is a good match for the pattern of data points. Spreadsheet software has various curve-fitting tools, which is referred to as &#8216;best-fit&#8217;. This is usually a linear least-squares regression analysis, though most software also offers nonlinear least-squares regression.
The precision of the best-fit linear equation can be checked by plugging in the x values for the data points and comparing the &#8216;theoretical&#8217; results of the equation to the actual y values of the data points. Spreadsheet software will typically calculate the coefficient of determination (R2) value for the function, which shows how well the function matches the data points. The closer the R2 value is to 1, the better the fit for a linear regression. Other functions have more specialized methods to determine how well the fit of the function is to the data.
Determining the uncertainty of dependent values calculated from the best-fit function would require complicated &#8220;error propagation&#8221; techniques. However, it is possible to calculate uncertainty within the equation in the form of the standard deviation for both the slope and y-intercept of a best-fit function. This is usually performed with a different tool than the one used to generate a two-dimensional plot.
Standard Deviation
The standard deviation describes the amount of variation present in a set of values. The population standard deviation (&#963;) is used when there is data from each member of a finite population, such as the mass of each marble in a bag of marbles. The sample standard deviation (s) is used for all other cases and is the default standard deviation calculation in spreadsheet software.1 You can assume that &#8216;standard deviation&#8217; refers to the sample standard deviation.
Random measurement error is assumed to follow a roughly &#8216;normal&#8217; distribution, where about 68% of a set of values lie within a range of one standard deviation on either side of the mean, 95% of the values lie within two standard deviations on either side of the mean, and 99.7% of the values lie within three standard deviations on either side of the mean. Thus, the standard deviation is a useful way to describe error and uncertainty.
The equation for the sample standard deviation is:
<img src="/CDNSource/lm/labs/42/42_Concepts_1.jpg" alt="Standard deviation formula, statistical equation, variance calculation, descriptive statistics." data-altupdatedat="1761830439000" />
In this equation, N is the number of values; <img src="/CDNSource/lm/labs/42/30034eq02.jpg" alt="mean value symbol in statistical equation, relevant for data analysis and interpretation" data-altupdatedat="1761830439000" /> is the average (or mean) of the values; and xi represents each individual value. Thus, to calculate the standard deviation by hand, calculate the mean of the set of values, subtract the mean from each value, square each difference, add the squared differences, divide the total sum by one less than the number of values, and take the square root of the quotient. The closer s is to zero, the less variation there is between the values. If the values are entered into spreadsheet software, the standard deviation can be calculated from within the software.
The number of significant figures in a standard deviation depends on what values it is for. When reporting the standard deviation for a group of data points taken under the same conditions, the appropriate number of significant figures in the mean value must first be determined. The standard deviation is then rounded to the same number of decimal places as the mean. For a set of volumes with four significant figures, a mean of 15.361 mL, and a standard deviation of 0.2313, the mean and standard deviation would be reported as 15.36 mL &#177; 0.23 mL.
When reporting the standard deviation of the mean and y-intercept for a best-fit function determined by least-squares analysis, which is the usual method for spreadsheet software, the first decimal place of the standard deviation is the last significant figure of the mean or y-intercept. Thus, the standard deviation should be rounded to one significant decimal place, and the slope or y-intercept should be rounded to the corresponding decimal place. For example, if the slope is 0.1691 L&#183;K-1 and it has a standard deviation of 0.00512, the slope should be reported as 0.169 L&#183;K-1 &#177; 0.005 L&#183;K-1.
If the standard deviation of the slope or y-intercept is so much smaller than its corresponding value that following this rule would give more significant figures to the slope or y-intercept than the original measurement data would allow, then instead determine the significant figures of the slope or y-intercept from the x and y values and round the standard deviation to one significant figure. Thus, for a slope of 0.1691 L&#183;K-1, a standard deviation of 0.0000512, and x and y values with four significant figures, the slope should be reported as 0.1691 L&#183;K-1 &#177; 0.00005 L&#183;K-&#8205;1. In this case, it&#8217;s fine for the calculated value and its standard deviation to have a different number of decimal places. Keep in mind that the least-squares method uses both the x and y values to calculate the slope and y-intercept.
When reporting standard deviation as an uncertainty range, be sure to note specifically that the uncertainty represents one standard deviation. This tells the reader that there is about a 68% chance that the true value of a measurement falls within that range of the mean, assuming a normal distribution. It may often be more suitable to report uncertainty as two standard deviations from the mean, as that increases the probability to about 95%. To do this, simply multiply the standard deviation by two before rounding it to the appropriate number of significant figures.
References
<ol style="font-size:17px">
	<li style="font-size:17px">Harris, D.C. (2015). Quantitative Chemical Analysis. New York, NY: W.H. Freeman and Company.</li>
</ol>

Two-Dimensional Graphs
Some chemistry experiments take the form of directly changing one property of the system that you are studying otherwise known as ...

Education

JoVE Lab Manual

graphs, curve fit, and standard deviation

Two-dimensional graphs help students compare an independent variable with a dependent variable in science data. In a chemistry example, temperature is the independent variable and volume is the dependent variable. After data are collected, the relationship between the two measurements must be quantified so it can be evaluated and compared with other relationships. A graph can also show when no relationship exists between the two variables.

To make a two-dimensional graph, each data point needs a coordinate for both variables. The independent variable is plotted on the x-axis, and the dependent variable is plotted on the y-axis. These plots are often made in spreadsheet software, which can also be used to analyze the data. This helps show how the dependent variable changes in response to the independent variable.

Once the data are plotted, curve fitting can turn the pattern into an equation, or function. A function is a mathematical model that best matches the data it comes from. Curve fitting looks for a line or curve that fits the data points well. In many cases, spreadsheet tools use a best-fit method based on linear least-squares regression, although nonlinear least-squares regression is also available in most software.

The quality of a best-fit line can be checked by comparing the x values in the equation with the actual y values from the data. Spreadsheet software usually reports the coefficient of determination, or R squared, which shows how well the function matches the plotted points. A value closer to 1 means a better fit for linear regression. Finding uncertainty for values calculated from the equation can be more complicated, but standard deviation can be used to describe the uncertainty in the slope and y-intercept of a best-fit line.

Standard deviation shows how much values vary in a set. Population standard deviation, written as sigma, is used when data are available for every member of a finite population, such as the mass of each marble in a bag. Sample standard deviation, written as s, is used in all other cases and is the default in spreadsheet software. For that reason, standard deviation usually means sample standard deviation. Random measurement error is assumed to follow a roughly normal distribution, where about 68% of values fall within one standard deviation of the mean, 95% fall within two, and 99.7% fall within three.

The sample standard deviation can also be calculated by hand. First find the mean. Then subtract the mean from each value, square each difference, add the squared differences, divide by one less than the number of values, and take the square root. The closer s is to zero, the less variation there is in the data. Spreadsheet software can also calculate standard deviation directly.

When reporting standard deviation, the number of significant figures depends on what it describes. For a group of measurements taken under the same conditions, the standard deviation is rounded to the same number of decimal places as the mean. For example, 15.361 mL with a standard deviation of 0.2313 should be reported as 15.36 mL ± 0.23 mL. When reporting the standard deviation of a slope or y-intercept from least-squares analysis, the first decimal place of the standard deviation becomes the last significant figure of the value. So 0.1691 L·K-1 with a standard deviation of 0.00512 is reported as 0.169 L·K-1 ± 0.005 L·K-1.

If the standard deviation is much smaller than the slope or y-intercept, this rule can give more significant figures than the original x and y data allow. In that case, use the significant figures from the data and round the standard deviation to one significant figure. For example, 0.1691 L·K-1 with a standard deviation of 0.0000512 and x and y values with four significant figures is reported as 0.1691 L·K-1 ± 0.00005 L·K-1. It is important to state whether the uncertainty represents one standard deviation. One standard deviation gives about a 68% chance that the true value lies within that range. Two standard deviations may be a better choice when a wider uncertainty range is needed, since that raises the chance to about 95%.

Graphs, Curve Fit, and Standard Deviation

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The most common laboratory emergencies include chemical spills, ...

chemical-storage-categories-hazards-and-compatibilities

Chemical Storage: Categories, Hazards And Compatibilities

Source: Robert M. Rioux &amp; Taslima A. Zaman, Pennsylvania State University, University Park, PA
While the use of various chemicals in experimental ...

Safe Handling of Mineral Acids

Source: Robert M. Rioux &#38; Taslima A. Zaman, Pennsylvania State University, University Park, PA
A mineral acid (or inorganic acid) is defined as a ...

Handling Chemical Spills

Source: Robert M. Rioux &#38; Taslima Zaman, Pennsylvania State University, University Park, PA
Since chemicals are commonly used for laboratory research ...

Proper Use of Autoclaves

Robert M. Rioux &amp; Zhifeng Chen, Pennsylvania State University, University Park, PA
Autoclaving is one of the most commonly used methods in the ...

Fume Hoods and Laminar Flow Cabinets

Robert M. Rioux &amp; William A. Elliott, Pennsylvania State University, University Park, PA
Fume hoods and laminar flow cabinets are engineering controls ...

handling-air-and-water-sensitive-chemicals-using-a-schlenk-line

Handling Air- and Water-Sensitive Chemicals Using a Schlenk Line

Robert M Rioux, Ajay Sathe, Zhifeng Chen, Pennsylvania State University, University Park, PA
The use of reagents sensitive to oxidation or moisture ...

proper-operation-of-vacuum-based-equipment

Proper Operation of Vacuum Based Equipment

Source: Robert M. Rioux, Ajay Sathe &#38; Zhifeng Chen, Pennsylvania State University, University Park, PA
Vacuum is required for a number of laboratory ...

Operating the Glovebox

Robert M Rioux, Ajay Sathe, Zhifeng Chen, Pennsylvania State University, University Park, PA 
A glovebox is an isolated enclosure that is designed to ...

operation-of-high-pressure-reactor-vessels

Operation of High-pressure Reactor Vessels

Robert M Rioux, Pennsylvania State University, University Park, PA
The use of gases in a synthetic chemistry laboratory is essential for carrying out a ...

CHEM.1.C

Lab Techniques

Measurement Accuracy
The ability to repeat an experiment and get the same results, or reproducibility, is essential in scientific research. However, ...

Proper Lab Notebook Keeping

Organic Lab Notebook Keeping
Record keeping is a vital skill in science and organic chemistry labs. Without proper notes, descriptions, quantities, or ...

Basic Organic Chemistry Techniques

Glassware in Organic Chemistry
There are standard pieces of glassware that are used in the organic chemistry lab. Beakers and Erlenmeyer flasks are ...

Measurement: Standard Units

Every measurement provides three kinds of information: the size or magnitude of the measurement (a number), a standard of comparison for the measurement ...

Measurement: Derived Units

The International System of Units or SI system, by international agreement, has fixed measurement units for seven fundamental properties: length, mass, ...

uncertainty-in-measurement-accuracy-and-precision

Uncertainty in Measurement: Accuracy and Precision

Scientists typically make repeated measurements of a quantity to ensure the quality of their findings and to evaluate both the precision and the accuracy ...

uncertainty-in-measurement-reading-instruments

Uncertainty in Measurement: Reading Instruments

Counting is the type of measurement that is free from uncertainty, provided the number of objects being counted does not change during the process. Such ...

uncertainty-in-measurement-significant-figures

Uncertainty in Measurement: Significant Figures

All the digits in a measurement, including the uncertain last digit, are called significant figures or significant digits. Note that zero may be a ...

Dimensional Analysis

Dimensional analysis, also known as the factor label method, is a versatile approach for mathematical operations. The main principle behind this approach ...

CHEM.1.E

Scientific and engineering practices. The student, for at least 40% of instructional time, asks questions, identifies problems, and plans and safely conducts classroom, laboratory, and field investigations to answer questions, explain phenomena, or design solutions using appropriate tools and models. The student is expected to:

Graphs, Curve Fit, and Standard Deviation

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