Chemical Applications of Statistical Analyses

SI units give chemistry measurements a common standard. A measurement has two parts: a number that shows magnitude and a unit that gives the comparison scale. Using the same unit system helps keep results consistent and reduces mistakes in scientific communication.

The Système International d'Unités, or SI units, includes both fundamental units and derived SI units. Fundamental units describe base physical quantities such as mass, distance, temperature, time, electric current, luminous...

Degrees of freedom describe how many values in a statistical calculation are free to vary. They represent the minimum number of independent numbers needed to determine a statistic. In a data set, some values may be independent while others depend on them.

A simple example uses three unknown numbers with a mean of 10. The first two numbers can be chosen freely. The last number cannot be chosen on its own because it must keep the mean at 10. In that case, the data set has two degrees of freedom.

Statistical analysis helps compare repeated measurements and judge how good the results are. When the same sample is measured again and again, small differences can appear. These differences are called errors. Researchers use statistics to describe the data and to decide whether a method is suitable.

The mean and median are common ways to find the central value of a data set. The mean is the sum of all results divided by the total number of results. The median is the middle value after the...

Measurement errors are the differences between an obtained result and the true value or estimated central value. They can be written as absolute error or relative error. Absolute error is the numerical difference from the true or central value. Relative error compares that difference to the true or central value and is often shown as a percentage.

Errors can also be grouped by their source, magnitude, and sign. In many experiments, the main categories are systematic, random, and gross errors.

Systematic errors are bias in measurement, and their sources can often be identified and reduced. In this topic, the main categories are sampling, instrumental, methodological, and personal errors. Each one comes from a different part of the experiment, so each one needs a different fix.

Sampling errors happen when the sampling method is poor or when the wrong population is chosen. These errors can be reduced by improving the sampling strategy. Better sample selection helps the results...

Random error in measurements comes from uncontrollable variables that affect data from one trial to the next. These variables can include changes in the environment, imperfections in an instrument, or the natural variation of the thing being measured. Because the direction and size of the error can change from one measurement to the next, random error is hard to predict, estimate, or describe directly.

Although random error is difficult to remove, its overall effect can be studied in a large...

Standard deviation describes how spread out data are around the mean. It helps show how tightly values cluster near the center and how far they sit from that central value. Many large data sets follow a Gaussian distribution, also called a normal distribution.

A Gaussian distribution has a bell-shaped curve. The mean, or central value, appears in the middle, where the most frequent values are found. As data points move farther from the center, deviation increases and frequency decreases.

The...

z scores show how far a value is from the mean in standard deviation units. A z score, also called a standardized value, tells whether a data point sits above or below the mean, μ. Values larger than the mean have positive z scores, while values smaller than the mean have negative z scores. If x equals the mean, then the z score is zero.

The z score scale is centered on zero. That means the mean of all z scores is zero, and their standard deviation is one. This makes z scores useful for...

Uncertainty is part of analytical chemistry measurement. When repeated measurements are taken, random errors can cause the results to differ in small ways. These repeated values are compared with the estimated or expected value to judge how much uncertainty is present.

Uncertainty is usually written after the estimated or expected value. It should also be reported with the correct number of significant figures, which are the digits needed to show a precise result. The possible variation from...

Uncertainty changes as measurements move through calculations. In a multi-step experiment, each measurement adds some uncertainty. Because the steps happen one after another, the uncertainty from one step carries into the next step and affects the final result.

The way uncertainty is combined depends on the math operation. For addition and subtraction, the final result is reported with absolute uncertainty. Absolute uncertainty is found as the square root of the sum of the absolute...

Atomic mass can vary because different samples contain different isotope ratios. For oxygen, the measured atomic mass is a weighted average of the isotopic masses in that sample. A single sample may not match the true atomic mass of oxygen found across all oxygen molecules on Earth. That creates sampling error, which is a type of systematic error.

This kind of uncertainty is not random. It is tied to the system being measured. Within the uncertainty range, the chance of finding a particular...

Confidence intervals show a range of values around a sample mean that may contain the true mean. They are written as a probability percentage. A 95% confidence interval means the statistician is 95% confident that the true mean lies within that range.

The two ends of the range are called confidence limits. These limits are estimated from the sample mean, the standard deviation, and the statistical factor t, or t-score. The t-score depends on the number of degrees of freedom and the confidence...

Significance testing helps analytical chemists decide whether a difference is meaningful or just caused by random error. It is used to test whether a claim about a parameter is valid. In analytical chemistry, it mainly helps determine whether a change between two values comes from determinate error or indeterminate error.

A determinate error has a clear source. It may come from a change in the measurement protocol, the analyst, or the sample itself. These changes can create a deviation from...

The F-test compares variance in two data sets or compares a sample variance with a population variance. It helps decide whether an indeterminate error can explain a difference in the values. In statistics, variance shows how spread out the data are.

The test assumes that the data set or sets are normally distributed. It also assumes that the data sets are independent of each other. These conditions must be met before the F-test is used.

The F test statistic is found by dividing one variance...

Student's t-test is a statistical tool for checking whether two means are different in a meaningful way. It can compare a sample mean with a population mean, or it can compare the means from two data sets. The test uses the mean, standard deviation, and number of measurements, along with a chosen confidence interval.

The test statistic is then compared with a table of critical values at that confidence level. If the test statistic is smaller than the critical value, the null hypothesis is...

The Q test helps identify outlier data points that sit far from the rest of a data set. In high school lab work, this can help students decide whether an unusual value should stay in the data or be removed. Outliers often come from gross errors, or human mistakes, and may not reflect the real result being measured.

Some points that look like outliers may still reflect a true difference in the phenomenon being studied. In those cases, a statistical test can help judge whether the value should...

A calibration curve links an instrument's signal to known concentrations of a substance. Scientists use it to set response levels with standards. They can also fit an equation to the curve and use it to find the concentration of unknown samples.

When the data make a straight line, the linear least-squares method is the standard way to fit the curve. This method finds the line that best matches the points by minimizing the sum of the squared differences between the predicted and actual values.

Calibration curves use the correlation coefficient, written as r, to show how strongly two variables are related and which direction the relationship takes. In a linear calibration curve, this value helps describe the association between the variables being compared.

The correlation coefficient ranges from -1 to +1. A value of +1 means a perfect positive linear correlation, while -1 means a perfect negative correlation. A value of 0 means there is no correlation between the two variables.

A...

Correlation and regression are used to study the relationship between two variables. Correlation shows how strongly the variables are associated. Regression helps describe that relationship with a line that can be used for prediction.

In linear regression, the relationship is written as a correlation coefficient. This coefficient is shown as r and ranges from -1 to +1. A positive value means the variables move in the same direction. A negative value means they move in opposite directions.

The limit of detection, or LOD, is the smallest amount of analyte that can be distinguished from background noise. In measurement science, background noise is the natural signal in a sample that can interfere with the analyte signal. The LOD helps show whether a substance is present, but it does not usually give a reliable amount.

The LOD is set at the concentration where the analyte signal is three times larger than the standard deviation of the blank signal. A blank is a sample without the...

The Grubbs test is a statistical way to spot a possible outlier in a data set. It is used when the data is assumed to follow a normal distribution. A Grubbs test helps decide whether one unusual numerical value should be treated as a true outlier.

To run a two-tailed Grubbs test, first find the absolute difference between the questionable value and the mean. Then divide that difference by the sample standard deviation. The result is the Grubbs statistic, or G.

Next, compare the calculated G...

ANOVA, or Analysis of Variance, is a statistical test used to compare the means of three or more samples. It was developed by Ronald Fisher in 1918. In high school statistics, ANOVA helps determine whether group averages are equal or whether at least one group is different.

Before ANOVA is used, the samples should meet three key assumptions. The data should come from normally distributed samples. The samples should also be randomly and independently selected. In addition, the populations...