Statistical Concepts and Error Propagation

Statistical Concepts

• Variance (s²) is calculated using the formula ( s^2 = \frac{\sum_{i=1}^N (x_i - \bar{x})^2}{N-1} ), where ( N ) is the number of observations.
• Coefficient of Variation (CV) is a measure of relative variability, calculated as ( CV = \frac{s}{\bar{x}} \times 100% ).
• Relative Standard Deviation (RSD) is given by ( RSD = \frac{s}{\bar{x}} ), reflecting the standard deviation relative to the mean.
• Spread (Range, (\omega)) is defined as ( \omega = x_{\text{max}} - x_{\text{min}} ).

Error Propagation & Significant Figures

• When standard deviations in results are known, propagation of error methods can be applied to assess uncertainty in calculations.
• Significant figures define the precision of a measurement:
  • Nonzero digits are always significant.
  • Zeroes between significant figures are significant.
  • Leading zeroes are not significant.
  • Trailing zeroes after a decimal point are significant.

Confidence Intervals (CI)

• CI for a population mean with known standard deviation:
  • ( 95% , CI = \bar{x} \pm \frac{z \cdot \sigma}{\sqrt{N}} )
• Example calculation for glucose measurements with ( \bar{x} = 1108 ) mg/L and ( s = 19 ):
  • ( 95% , CI = 1108 \pm 14.1 ) mg/L
• CI for small sample sizes uses the t-distribution, such as ( N = 3 ) for blood alcohol content, resulting in ( 95% , CI = 0.084 \pm 0.012% ).

Hypothesis Testing

• Null Hypothesis ((H_0)) asserts no effect or no difference among groups.
• Alternative Hypothesis ((H_a)) posits a difference or effect exists.
• Common tests include comparing sample means or standard deviations across different groups.

Paired t-test Methodology

• A paired t-test compares means between two related groups, using the difference calculated as ( \bar{d} ).
• T statistic is computed as ( t = \frac{\bar{d} - \Delta_0}{s_d/\sqrt{N}} ) where ( \Delta_0 ) is the difference under the null hypothesis.
• In an example, with ( t_{crit} = 2.57 ) at ( 95% ) CI, a t-stats of ( 4.628 ) leads to the rejection of ( H_0 ).

F-test for Variance Comparison

• F-test compares variances of two data sets to determine equivalence.
• Test statistic is calculated as ( F = \frac{s_1^2}{s_2^2} ).
• Null hypothesis is ( H_0: \sigma_1^2 = \sigma_2^2 ).

Gravimetric Analysis Techniques

• Precipitation Gravimetry: Involves forming and weighing a precipitate to quantify an analyte.
• Volatilization Gravimetry: Analyte is volatilized and mass loss indicates concentration.
• Electrogravimetry: Analyte deposited on an electrode is weighed to determine concentration.
• Gravimetric Titrimetry: Mass of a reagent is used to determine analyte concentration.
• Atomic Mass Spectrometry: Measures ion concentration from a sample.

Calculations for %SO₃

• Determine %SO₃ using dimensional analysis based on the relationship between reagent weights.
• Examples presented show calculations for sulfate samples, confirming consistent theoretical %SO₃ values through trials, averaging 56.37%.

This quiz covers key statistical concepts including variance, coefficient of variation, relative standard deviation, and the spread. Additionally, it explores error propagation methods and the rules governing significant figures. Test your understanding of these fundamental statistical principles.