chemistry PDF on statistical analysis

Groundwater Sample Analysis

Sample Collection

• A student collects twelve groundwater samples from a well.

• Measures dissolved oxygen concentration in six samples:

  • Values (mg/L): 8.8, 3.1, 4.2, 6.2, 7.6, 3.6

• Sample mean calculation:

  • ( x = \frac{8.8 + 3.1 + 4.2 + 6.2 + 7.6 + 3.6}{6} = 5.6 \text{ mg/L} )

Standard Deviation Calculation

• Formula: ( SD = \sqrt{\frac{\sum (x - \bar{x})^2}{n-1}} )

• Calculation Steps:

  • Differences:

    • 8.8 - 5.6 = 3.2 (9.84)

    • 3.1 - 5.6 = -2.5 (6.25)

    • 4.2 - 5.6 = -1.4 (1.96)

    • 6.2 - 5.6 = 0.6 (0.36)

    • 7.6 - 5.6 = 2.0 (4.00)

    • 3.6 - 5.6 = -2.0 (4.00)

  • Sum of squares: 26.81

  • ( SD = \sqrt{\frac{26.81}{5}} = 2.3 \text{ mg/L} )

Standard Error Mean (SEM)

• ( SEM = \frac{SD}{\sqrt{n}} = \frac{2.3}{\sqrt{6}} = 0.95 \text{ mg/L} )

Additional Observations from Remaining Samples

Second Group of Samples

• Sample concentration values: 5.2, 8.6, 6.3, 1.8, 6.8, 3.9

• Grand average calculation of all twelve samples:

  • Mean: ( x = \frac{8.8 + 3.1 + 4.2 + 5.2 + 8.6 + 6.3 + 1.8 + 6.8 + 3.9}{12} = 5.5 )

Combined Standard Deviation

• Standard deviation for combined twelve observations:

  • Calculation of deviations and their squares:

    • Repeat similar steps to find deviations from mean for each sample.

  • Final SD: 2.2 mg/L

Confidence Interval Calculation

• For 95% confidence:

  • Degrees of freedom: 11

  • Using t-distribution with 0.025 (two-tailed, 95%): ( t_{0.025} = 2.201 )

  • Confidence interval: ( 5.5 \pm (2.201 \times 0.65) )

    • Yielding limits: 4.1 to 6.9 mg/L

Precision & Reliability in Measurements

Measurement Improvements

• Suggests increasing sample size reduces error

Q-Test Application

• Used to determine if outliers should be rejected from datasets:

  • Example values: 4.3, 4.1, 4.0, 3.2

  • Calculation of Q for 3.2:

    • ( Q = \frac{3.2 - 4.0}{4.3 - 3.2} = 0.727 )

  • Compare with Q critical values to determine acceptance

Significance Testing using F-Test

• Compares variances of two sample sets

  • Example standard deviations: ( S_A = 0.210, S_B = 0.641 )

  • F statistic calculation: 9.4

  • Comparison against critical thresholds for significance

Confidence Intervals and Regression Analysis

Importance of Confidence Limits

• Higher sample quantity decreases confidence intervals

• Assessment of reliability and accuracy in measurements

Linear Regression Analysis

• Important for data modeling and prediction in spectroscopic analysis

• Relationship of dependent and independent variables analyzed through regression equation.