Statistical Data Analysis

Introduction

Throughout this course, you should use statistical measures when analyzing your data and drawing conclusions. You are expected to:

• Record and report the correct number of significant digits.
• Track uncertainties and propagate them in calculations.
• Identify and report sources of error.
• Analyze samples in triplicate and report the mean, standard deviation, and 95% confidence interval of your measurements.
• Use confidence intervals to test if means are significantly different where necessary.
• Perform Grubbs’ tests to detect outliers when necessary.
• Determine and report the limit of detection and/or limit of quantitation for each technique when conducting quantitative work.

Please read Chapter 22 in the Granger textbook (Hardcopy) or Chapter 4 in the Harvey book (online) for a refresher on statistical data analysis.

Types of Error

You will make many mistakes when analyzing your samples, and that’s OK! The important thing is that you can recognize when you’ve made a mistake and figure out how to correct it.

Gross Error

Gross error occurs when the analyst makes a mistake. For example, you may make your standards too concentrated, dilute a sample incorrectly, or record a value incorrectly in your lab notebook. If only one sample is affected, you can use a Grubb’s test to identify the sample as an outlier and exclude it. If the error affects all your samples, you may need to start over.

See Section 22.2 in your textbook for more information.

Systematic Error

Systematic error occurs when a measurement is systematically high or low, usually the result of instrument drift, improper calibration, or habitual misuse of laboratory glassware (for example, filling to the top of the meniscus instead of the bottom). If your systematic error is big enough you may need to redo your analysis.

See Section 22.2 in your textbook for more information.

Random Error

Random error occurs due to unpredictable (statistically random) fluctuations in instrument response, pipette delivery volumes, balance fluctations, etc. A key factor of random error is that it averages out as the sample size increases. Generally, there is little that can be done to eliminate random error other than increasing the averaging time and/or number of samples.

See Section 22.2 in your textbook for more information.

Replicate Measurements

Most unknown samples should be analyzed in triplicate. You should measure the sample three separate times and report the mean, standard deviation, and relative standard deviation of each measurement. See section 22.4 in your textbook for more information. An example results table is shown below.

Table 1: Sulfate concentration in multivitamin pills

In Julia, the mean is:

using Statistics
x = [1 2 3];
avg = mean(x)

The standard deviation is:

x = [1, 2, 3]
sd = std(x)

The relative standard deviation (RSD, in percent) is:

rsd = sd/avg * 100

Limits of Detection and Quantitation

For this course, the Limit of Detection ($LOD_y$) is defined as 3 times the standard deviation of a blank measurement, $y_{blank}$, plus the mean value of the blank ($\bar{y}_{blank}$). This is the 3$\sigma$ detection limit and is represented as:

When a calibration plot (standard curve) is used, this becomes:

where $m$ is the slope of the calibration curve and $LOD_x$ is the LOD in the concentration units of the calibration curve.

The Limit of Quantitation ($LOQ_y$) is defined as 10 times the standard deviation of the blank:

When a calibration plot (standard curve) is used, this becomes:

Comparing Means

If you are comparing two means or comparing a mean to an expected value or (e.g. comparing the concentration of a compound to the concentration listed on the manufacturer’s label) or to each other (e.g. comparing a post-vape to pre-vape sample), you should use the 95% confidence interval to draw conclusions about significance. If the confidence intervals overlap, there is no significant difference; if they do not overlap the difference is significant.

Recall that the confidence interval (CI) is calculated as:

where $t$ is obtained from a t-table, $s$ is the sample standard deviation, and $n$ is the sample number. Thus, the mean is reported as:

Outlier Detection

You should never throw away data unless you have a good reason to do so. Generally, this means that you’ve identified the data point as an outlier using a Grubbs’ test. See section 22.4 in your textbook for more information.