ANALYTICAL CHEMISTRY CHEM 3811 CHAPTER 4 DR. AUGUSTINE OFORI AGYEMAN Assistant professor of chemistry Department of natural sciences Clayton state university

CHAPTER 4 STATISTICS

STATISTICS - Random errors are always associated with measurements - No conclusion can be drawn with complete certainty - Scientists use statistics to accept conclusions that have high probability of being correct and to reject conclusions that have low probability of being correct - Statistics deals with only random errors - Systematic errors should be detected and eliminated

THE GAUSSIAN DISTRIBUTION - Symmetric bell shaped curve representing the distribution of experimenal data - Characterized by mean and standard deviation - The Gaussian function is - a is the height of the curve’s peak, - µ is the position of the center of the peak (the mean) - σ is a measure of the width of the curve (standard deviation)

THE GAUSSIAN DISTRIBUTION f(x) -3σ -2σ -σ μ σ 2σ 3σ x

THE GAUSSIAN DISTRIBUTION Arithmetic Mean - Also known as the average - Is the sum of the measured values divided by the number of measurements ∑ = sigma, symbol for the sum xi = a measured value n = number of measurements

Standard deviation has the same number of decimal places as the mean THE GAUSSIAN DISTRIBUTION Standard Deviation - A measure of the width of the distribution - Small standard deviation gives narrow distribution curve xi = a measured value n = number of measurements n-1 = the degrees of freedom Standard deviation has the same number of decimal places as the mean

THE GAUSSIAN DISTRIBUTION Relative Standard Deviation - The sample mean and sample standard deviation (x and s) apply to finite set of measurements - For infinite set of data use is made of the true mean or population mean (designated µ) and the true standard deviation (designated σ)

- Is the square of the standard deviation THE GAUSSIAN DISTRIBUTION Variance - Is the square of the standard deviation - Variance = σ2

THE GAUSSIAN DISTRIBUTION Median - The middle number in a series of measurements arranged in increasing order - The average of the two middle numbers if the number of measurements is even

THE GAUSSIAN DISTRIBUTION Mode - The value that occurs the most frequently Range - The difference between the highest and the lowest values All but the range should be expressed with one extra digit beyond the last significant digit

THE GAUSSIAN DISTRIBUTION Probability - Range of measurements for ideal Gaussian distribution - The percentage of measurements lying within the given range (one, two, or three standard deviation on either side of the mean) Range µ ± 1σ µ ± 2σ µ ± 3σ Gaussian Distribution (%) 68.3 95.5 99.7

CONFIDENCE INTERVAL - Range of values within which there is a specified probability of finding the true mean (µ) - The t-test is used to express confidence intervals - Also used to compare results from different experiments

CONFIDENCE INTERVAL - That is, the range of confidence interval is – ts/√n below the mean and + ts/√n above the mean - For better precision reduce confidence interval by increasing number of measurements Refer to table 4-2 on page 87 for t-test values

To test for comparison of Means CONFIDENCE INTERVAL To test for comparison of Means - Calculate the pooled standard deviation (spooled) - Calculate t - Compare the calculated t to the value of t from the table - The two results are significantly different if the calculated t is greater than the tabulated t at 95% confidence level

For two sets of data with CONFIDENCE INTERVAL For two sets of data with - n1 and n2 measurements - averages of x1 and x2 - standard deviations of s1 and s2 Degrees of freedom = n1 + n2 - 2

GRUBBS TEST FOR AN OUTLIER - An outlier is a datum that is far from other points - Grubbs test is used to determine whether an outlier should be rejected or retained

GRUBBS TEST FOR AN OUTLIER - Calculate mean, standard deviation, and then G - Compare G calculated to G tabulated (Table 4-4 on page 92) - The questionable datum is rejected if G calculated is greater than G tabulated - The questionable datum is retained if G calculated is smaller than

(METHOD OF LEAST SQUARES) BEST STRAIGHT LINE (METHOD OF LEAST SQUARES) The equation of a straight line y = mx + b m is the slope (y/x) b is the y-intercept (where the line crosses the y-axis)

BEST STRAIGHT LINE (METHOD OF LEAST SQUARES) The method of least squares - finds the best straight line - adjusts the line to minimize the vertical deviations Only vertical deviations are adjusted because - experimental uncertainties in y values > in x values - calculations for minimizing vertical deviations are easier

BEST STRAIGHT LINE (METHOD OF LEAST SQUARES) - n is the number of data points Knowing m and b, the equation of the best straight line can determined and the best straight line can be constructed

(METHOD OF LEAST SQUARES) BEST STRAIGHT LINE (METHOD OF LEAST SQUARES) xiyi ∑(xiyi) = xi2 ∑xi2 = xi ∑xi = yi ∑yi =