Statistics Guide For VEN 123/L

Workflow Overview

This flow chart outlines the standard analytical path from experiment design through final evaluation. Each stage feeds directly into the next: raw data must be summarized before outliers can be flagged, and outliers must be resolved before descriptive statistics are finalized and carried into statistical treatment. Use this as a checklist to confirm no step has been skipped or reordered.

1  Experiment Design
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2  Conduct Experiment & Collect Raw Data
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3  Preliminary Descriptive Statistics
mean, SD, median — inputs for outlier testing
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4  Outlier Detection & Removal
Grubbs', Z-score, or IQR method, applied per protocol
if flagged points require manual inspection, return to raw data (step 2) before proceeding
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5  Finalized Descriptive Statistics
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6  Statistical Treatment
test selection based on distribution, variance, and design
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7  Evaluate Results
against hypothesis, QC thresholds, and reproducibility across replicates
Introduction

Statistical analyses turn a set of measurements into a result or conclusion. A mean value reported without a measure of variability is just one number, and a comparison between samples without a statistics test is an opinion. Statistics allows you to make general conclusions from limited data.

Statistical analyses fall into two categories. Descriptive statistics summarize what you measured: the central value, variability, and sample size. Inferential statistics build on those summaries to test hypotheses — whether two wines differ, or whether a treatment had a significant effect. Descriptive statistics always come first, because every inferential test depends on them.

Software — All statistics in this course may be performed in Microsoft Excel or GraphPad Prism. Students enrolled in VEN 123/L receive a free GraphPad Prism license for the duration of the quarter.
Symbols & Abbreviations

Greek letters, Roman symbols, and abbreviations used throughout this guide.

Greek letters
Symbol Name Pronounced Meaning
μ mu "myoo" Population mean — the true mean of all possible measurements from a defined population. In the context of measurement error (IUPAC), μ represents the limiting mean: the value approached as the number of measurements approaches infinity. μ and the limiting mean are the same quantity — the IUPAC term emphasizes what happens operationally as n → ∞
σ sigma "sig-mah" Population standard deviation — theoretical spread of an infinite set of measurements; appears in the Gaussian distribution (±1σ, ±2σ, ±3σ). Distinguished from s, which is calculated from your data
α alpha "al-fah" Significance level — the probability that the true mean falls outside the confidence interval; α = 1 − CL
τ tau "taw" True value IUPAC
Δ delta
(uppercase)		 "del-tah" Bias — the difference between the limiting mean and the true value IUPAC Note: uppercase Δ and lowercase δ are both pronounced "delta" but mean different things
δ delta
(lowercase)		 "del-tah" Random error — the difference between an observed value and the limiting mean IUPAC Some statistics and econometrics textbooks use ε (epsilon, "ep-si-lon") for the same quantity. In analytical chemistry, the IUPAC notation δ is standard.
Roman symbols
Symbol Meaning
x̄ Sample mean — the arithmetic average calculated from your data; distinguished from μ, which is the theoretical population mean
x̂ Measurement result — the value attributed to a measurand IUPAC
s Sample standard deviation — calculated from your data using n−1 in the denominator; what you compute and report in VEN 123L. Distinguished from σ, the theoretical population SD
s² Variance — the square of the sample standard deviation, in squared units of the data
n Sample size — the number of measurements in your sample. In VEN 123L reports, n refers to biological replicates. Distinguished from N, the population size
N Population size — the total number of measurements in the entire population. Rarely known in practice; distinguished from n, the sample size
E Absolute error — the difference between a measured value and the true or accepted value
Er Relative error — absolute error expressed as a percentage of the true value
Abbreviations
Abbrev. Meaning
SDStandard deviation
RSDRelative standard deviation — same as CV, expressed as a percentage
CVCoefficient of variation — relative standard deviation as a percentage (CV = s/x̄ × 100)
CIConfidence interval
CLConfidence level
dfDegrees of freedom — defined as n − 1 in confidence interval calculations
SRMStandard Reference Material — a certified material used to validate analytical methods
Key Terms

The following terms appear throughout this guide and carry specific meanings in the context of VEN 123L. Click any card to flip it and reveal the definition, example, and equation where applicable. Click again to flip back.

25 terms
Data
Data
Values collected during experimental measurements. Can be qualitative (chemical identity) or quantitative (numerical amounts). Raw data is collected directly; calculated data is derived from raw values.
Example
Measuring absorbance at 520 nm = raw data. Converting that absorbance to mg/L anthocyanins = calculated data.
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Population
Population
The complete collection of all possible measurements of interest defined by the experimenter.
Example
Every possible ethanol measurement from every Cabernet Sauvignon bottle produced in Napa Valley constitutes the population.
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Sample
Sample
Has two meanings: (1) a statistical subset of measurements drawn from a population; (2) a physical portion of matter analyzed in the lab.
Example
Six bottles selected from a 2023 vintage to measure SO₂ are your statistical sample. Each bottle is also an analytical sample.
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Sample size (n)
Sample size (n)
The number of independent measurements in a sample. In VEN 123L reports, n refers to biological replicates, not technical replicates.
Example
Three separate fermentations of the same juice give n = 3, even if each fermentation is measured three times by HPLC.
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Replicate
Replicate
A single independent measurement within a sample. Technical replicates repeat measurements on the same material; biological replicates are independent samples from the same population.
Example
Three HPLC injections from one wine = technical replicates. Three separate fermentation batches = biological replicates.
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Measurement error
Measurement error
The difference between a measured value and the true value. Has two components: systematic error (bias, affects accuracy) and random error (affects precision).
Example
A pH meter reading 0.1 units too high = systematic error. Small fluctuations between injections = random error.
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Systematic error
Systematic error
Error with a consistent direction and magnitude across replicate measurements. Affects accuracy. Sources include instrument calibration errors, method errors, and personal errors.
Example
An uncalibrated refractometer that consistently reads approximately 0.5 °Brix too high introduces systematic error across all measurements.
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Random error
Random error
Indeterminate error from uncontrollable variables. Affects precision, cannot be fully eliminated, and follows a Gaussian distribution for most quantitative analyses.
Example
Small temperature fluctuations causing slight variation between replicate SO₂ titrations represent random error.
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True value (τ)
True value (τ)
The actual, error-free quantity being sought through measurement. Typically unknown; approached by the population mean (µ) when systematic error is zero.
Example
The exact tartaric acid concentration in a wine sample is the true value — we can only approximate it through repeated measurement.
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Accepted value
Accepted value
A highly reliable surrogate for the true value, typically derived from Standard Reference Materials (SRMs) or expert determination.
Example
A certified wine SRM with stated ethanol content of 12.4% ± 0.1% provides the accepted value for validating your method.
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Accuracy
Accuracy
How close a measurement (or mean) is to the true or accepted value.
\(E = x_i - x_t\)     \(E_r = \dfrac{x_i - x_t}{x_t} \times 100\)
Example
Certified standard: 12.4% ethanol. Your reading: 12.1%. Absolute error = −0.3%, relative error = −2.4%.
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Precision
Precision
The reproducibility of measurements obtained under identical conditions.
\(s = \sqrt{\dfrac{\sum(x_i-\bar{x})^2}{n-1}}\)    \(CV = \dfrac{s}{\bar{x}} \times 100\)
Example
pH readings of 3.51, 3.52, 3.51 show high precision. Readings of 3.41, 3.55, 3.63 show low precision.
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Gaussian distribution
Gaussian distribution
The bell-shaped distribution assumed for random errors. 68.3% of results fall within ±1σ, 95.4% within ±2σ, and 99.7% within ±3σ of the mean.
Example
Wine pH measured 100 times with mean 3.50 and σ = 0.02 — about 68 readings will fall between 3.48 and 3.52.
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Standard deviation (s)
Standard deviation (s)
Measures the scatter of replicate results around the mean. Has the same units as the data.
\[ s = \sqrt{\frac{\sum(x_i - \bar{x})^2}{n-1}} \]
Example
Five TA measurements: 6.2, 6.4, 6.3, 6.5, 6.3 g/L. Mean = 6.34, s = 0.11 g/L.
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Mean (x̄)
Mean (x̄)
The arithmetic average of measurements (sum divided by n). Most common measure of central tendency for Gaussian-distributed data.
\[ \bar{x} = \frac{\sum x_i}{n} \]
Example
Ethanol readings of 12.2, 12.5, 12.3% → mean = (12.2 + 12.5 + 12.3) / 3 = 12.33%.
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Coefficient of variation (CV)
Coefficient of variation (CV)
The relative standard deviation expressed as a percentage. Useful for comparing variability across datasets with different units.
\[ CV = \frac{s}{\bar{x}} \times 100 \]
Example
pH: s = 0.02, x̄ = 3.50, CV = 0.57%. Ethanol: s = 0.15, x̄ = 12.3%, CV = 1.2%. Directly comparable despite different units.
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Confidence interval (CI)
Confidence interval (CI)
A range around the sample mean within which the true population mean (µ) is expected to fall with a specified probability.
\[ \mu = \bar{x} \pm \frac{t \cdot s}{\sqrt{n}} \]
Example
Wine pH = 3.50 ± 0.04 at 95% CL means you are 95% confident the true pH falls between 3.46 and 3.54.
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Confidence level (CL)
Confidence level (CL)
The probability (typically 95% or 99%) that the population mean falls within the confidence interval.
Example
At 95% CL, if you repeated the experiment 100 times, approximately 95 of those CIs would contain the true mean.
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Significance level (α)
Significance level (α)
The probability that the true mean falls outside the confidence interval. Defined as α = 1 − CL.
Example
At 95% CL, α = 0.05 — a 5% chance the CI does not capture the true mean.
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Degrees of freedom (df)
Degrees of freedom (df)
Defined as n − 1 in CI calculations. Represents the number of independent values free to vary when estimating a parameter.
Example
With n = 4 measurements, df = 3. Look up the t-value at df = 3 and your chosen confidence level to calculate the CI.
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Outlier
Outlier
A data point lying far from the rest of a sample. Removal is only acceptable when justified by a statistical test (Grubbs or ROUT) applied before analysis begins.
Example
SO₂ readings: 42, 44, 43, 91 mg/L. The value 91 is suspected — run a Grubbs test before deciding to exclude it.
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Variance (s²)
Variance (s²)
The square of the standard deviation, expressed in squared units of the data. Used directly in many inferential statistical tests.
\[ s^2 = \frac{\sum(x_i - \bar{x})^2}{n-1} \]
Example
If s = 0.11 g/L for a TA dataset, s² = 0.012 (g/L)². Variance is used directly in ANOVA calculations.
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Range
Range
The difference between the maximum and minimum values in a dataset. Simple but sensitive to extreme values.
Example
pH values: 3.41, 3.50, 3.52, 3.55 → range = 3.55 − 3.41 = 0.14. One outlier can make this misleadingly large.
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Significant figures
Significant figures
All certain digits in a measured number plus the first uncertain digit. Determine how precisely a result should be reported.
Example
A pH reading of 3.52 has 3 significant figures. Reporting it as 3.5200 implies a precision your instrument cannot support.
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Measurement result
Measurement result
IUPAC Gold Book
Also: Bias · Expectation value · Expected value · Limiting mean
The outcome of an analytical measurement attributed to a measurand. Total error has two components: bias (Δ) and random error (δ).
\[ \hat{x} = \tau + e = \tau + \Delta + \delta = \mu + \delta \]
Example
If μ = 3.52 but the true pH = 3.50, bias Δ = +0.02. The scatter of readings around μ is the random error δ.
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Error and Uncertainty

Every measurement contains error — the difference between what is measured and the true value. Understanding the nature of that error is the first step toward interpreting your results correctly. Error in analytical chemistry falls into two fundamentally different categories, each with distinct causes, behaviors, and consequences for your data.

Types of measurement error
Systematic error
Produces a consistent bias in one direction. Affects accuracy — how close your result is to the true value. Cannot be reduced by taking more measurements.
  • Instrument calibration errors
  • Method errors (nonideal chemical behavior)
  • Personal errors (number bias, color blindness)
Random error
Produces unpredictable scatter around the mean. Affects precision — the reproducibility of your results. Can be reduced by averaging more measurements.
  • Instrumental noise
  • Environmental fluctuations (temperature, vibration)
  • Analyst inconsistency (reading a burette)
A measurement can be precise without being accurate (consistent results that are all wrong), accurate without being precise (correct on average but scattered), or both — which is the goal.
Accuracy

Accuracy describes how close a measured value is to the true or accepted value. It is quantified by absolute error or relative error.

Absolute error
\[ E = x_i - x_t \]
where \(x_i\) is the measured value and \(x_t\) is the true or accepted value
Relative error (%)
\[ E_r = \frac{x_i - x_t}{x_t} \times 100 \]
Example: A certified standard contains 12.4% ethanol. Your measurement gives 12.1%. Absolute error = 12.1 − 12.4 = −0.3%. Relative error = (−0.3 / 12.4) × 100 = −2.4%.
Precision

Precision describes the reproducibility of measurements made under identical conditions. It is quantified by standard deviation (s) and coefficient of variation (CV).

Standard deviation
\[ s = \sqrt{\frac{\sum (x_i - \bar{x})^2}{n-1}} \]
Coefficient of variation (%)
\[ CV = \frac{s}{\bar{x}} \times 100 \]
The Gaussian distribution

Random errors in most quantitative analyses follow a Gaussian (normal) distribution — a symmetric bell curve centered on the mean.

↓  Continue below: The next section explores four key concepts that follow from the Gaussian distribution — the 68–95–99.7 rule, confidence intervals, effect of sample size, and accuracy vs precision — through interactive diagrams.
The Gaussian Distribution

Random errors in most quantitative analyses follow a Gaussian (normal) distribution — a symmetric bell curve centered on the mean. The four interactive diagrams below illustrate the key statistical concepts that follow from this distribution.

1 — The 68–95–99.7 rule

For any Gaussian distribution, fixed percentages of measurements fall within each standard deviation band. These are constants — they hold regardless of the mean or SD values of your specific dataset.

±1σ
68.3% of measurements
±2σ
95.4% of measurements
±3σ
99.7% of measurements
2 — Confidence interval and confidence level

A confidence interval (CI) is the range within which the true mean is expected to fall at a given probability. Select a confidence level to see how the CI width and critical value change. A higher confidence level requires a wider interval.

Confidence level z = 1.960
CI half-width
±1.960σ
Significance level (α)
0.05
Area excluded
5.0%
3 — Effect of sample size (n) on precision

Increasing n narrows the distribution of the sample mean, not the distribution of individual measurements. The standard error of the mean (SEM = s/√n) decreases as n increases, producing a tighter estimate of the true mean. The gray curve shows individual measurement spread; the navy curve shows how precisely the mean is estimated.

Sample size (n) n = 5
SEM = s/√n
0.447
Reduction vs n = 1
55% narrower
4 — Accuracy vs precision

Accuracy and precision are independent properties. A method can be precise but inaccurate (all results cluster away from the true value), or accurate on average but imprecise (results scatter around the true value). The goal is both.

Describing Your Data

Descriptive statistics summarize what you measured — the central value, how much it varies, and how many measurements you took. Every inferential test depends on these summaries, so they always come first.

Mean

The mean (x̄) is the arithmetic average of your measurements — the sum of all values divided by n. It is the most common measure of central tendency and is appropriate for Gaussian-distributed data.

Mean
\[ \bar{x} = \frac{\sum x_i}{n} \]
where \(x_i\) are the individual measurements and \(n\) is the sample size
Example: Ethanol readings of 12.2, 12.5, 12.3% → x̄ = (12.2 + 12.5 + 12.3) / 3 = 12.33%
Excel syntax
Assumes your data values are in cells A1 to A10. Adjust the range to match your spreadsheet.
Mean: =AVERAGE(A1:A10)
Sample size: =COUNT(A1:A10)
Confidence interval

A confidence interval (CI) is a numerical range established around a sample mean (x̄) within which the true population mean (µ) is expected to fall with a specified probability. It is the primary way to express the uncertainty of a mean in VEN 123L reports.

Confidence interval for the mean
\[ \mu = \bar{x} \pm \frac{t \cdot s}{\sqrt{n}} \]
where \(\bar{x}\) is the sample mean, \(t\) is the critical t-value at \(n-1\) degrees of freedom, \(s\) is the sample standard deviation, and \(n\) is the sample size
Example: Wine pH measured three times: x̄ = 3.50, s = 0.02, n = 3. At 95% CL, t = 4.303 (df = 2). CI = 3.50 ± (4.303 × 0.02) / √3 = 3.50 ± 0.05. The true pH is expected to fall between 3.45 and 3.55 with 95% confidence.
Excel syntax
Assumes your data values are in cells A1 to A10. Adjust the range to match your spreadsheet.
1. Sample mean (x̄): =AVERAGE(A1:A10)
2. Sample standard deviation (s): =STDEV(A1:A10)
3. Sample size (n): =COUNT(A1:A10)
4. Degrees of freedom (df = n − 1): =COUNT(A1:A10)-1
5. Critical t-value at 95% CL: =T.INV.2T(0.05, df) — replace df with your value from step 4
6. CI half-width (t · s / √n): =t * s / SQRT(n) — replace t, s, and n with your values from steps 5, 2, and 3
7. Report as: x̄ ± CI half-width
Key terms for confidence intervals
Confidence level (CL)
The probability, expressed as a percentage (typically 95% or 99%), that the population mean falls within the CI. A 95% CL means that if the experiment were repeated 100 times, approximately 95 of those CIs would contain the true mean.
Significance level (α)
α = 1 − CL
The probability that the true mean falls outside the CI. At 95% CL, α = 0.05 — a 5% chance the interval does not capture the true mean.
Critical value (t)
The statistical multiplier that sets the width of the CI. In VEN 123L, t is always used (not z) because the population standard deviation σ is never known — only the sample standard deviation s is available. The t-value is looked up from a t-table at the chosen CL and df = n − 1.
Degrees of freedom (df)
df = n − 1
The number of independent values free to vary when estimating a parameter. For CI calculations, df = n − 1, where n is the number of biological replicates.
Why t and not z? The z critical value assumes σ (population SD) is known exactly. In practice, σ is never known — you always estimate it from your data as s. The t-distribution accounts for this additional uncertainty and is wider than the normal distribution, especially at small n. As n → ∞, t approaches z.
Significant figures

Significant figures are all certain digits in a measured number plus the first uncertain digit. They determine how precisely a result should be reported and must be consistent with the precision of the measurement.

A pH reading of 3.52 has 3 significant figures. Reporting it as 3.5200 implies a precision your instrument cannot support. When reporting a mean and CI, match the significant figures of the CI to those of the mean: 3.50 ± 0.05, not 3.4997 ± 0.0498.