Analytical Chemistry - Statistical Treatment of Analystical Data

ANALYTICAL CHEMISTRY EXAMPLE QUANTITATIVE ANALYSIS: STATISTICAL TREATMENT OF ANALYSTICAL DATA • From the data set in the table below, find the (a) mean, (b) median and (c) mode. Coin number mass coin, g 1 5.0305 2 5.0383 3 5.1118 4 5.0827 5 5.1123 6 5.0098 7 5.0476 8 5.1118 9 5.0825 10 5.1118 • • Total number of measurement (N) = 10 Solution: • (b) Arrange the data according to increasing or decreasing value. The data below is arranged according to increasing value. INTRODUCTION EXPERIMENTAL MEASUREMENT • • • Experimental measurements have associated random errors with it and conclusions cannot be drawn with certainty. Statistics give us a tool to accept conclusions with a high probability of being correct and to reject conclusions that do not. Modern analytical chemistry is concerned with the detection, identification, and measurement of the chemical composition of substances present in a sample of matter. It is a quantitative science which means results are numeric. Quantitative results are obtained using devices or instruments that allow us to determine the concentration of a chemical in a sample-based on an observable signal. Because we are using instruments, there are always variations in that signal over time due to noise and/or drift within the instrument. In addition, we need to calibrate the response as a function of analyte concentration to acquire meaningful quantitative data. In analytical chemistry, statistics is a tool used to establish the uncertainty associated with the measurement. LESSON 1 - MEAN, MEDIAN AND MODE o • MEAN • • • It is the most widely used measure of central value. It is commonly referred to as the average or arithmetic mean. Below is the formula for the mean: LESSON 2 - RELATIVE AND ABSOLUTE ERROR ABSOLUTE ERROR • • where xi represents the individual values of x and N is the set of replicate measurements. • MEDIAN • • • the middle value of a data set when the data are arranged according to increasing or decreasing value odd-numbered data set: locate the middle result even-numbered data set: take the mean of the middle pair MODE • Since the number of measurements is even, the median is the average of the two-middle pair. o Median = 5.0825 + 5.0827 = 5.0826 2 (c) mode = 5.1118 ( most frequently occurring measurement) The absolute error of a measurement is the difference between the measured value (xi) and the true value (x--t) The absolute error (E) in the measurement of a quantity x is given by the equation: E = xi - xt (Note: We keep the sign in stating the error. A negative sign means the measurement result is low; a positive sign means the measurement value is high.) RELATIVE ERROR • • • The relative error is a more useful quantity than the absolute error. The relative error (Er)of measurement is the absolute error divided by the true value. Er = xi - xt = absolute error (g) xt xt It is the value that occurs most frequently. ANACHEM • 1 • • • • • It may be expressed as percent relative error, parts per thousand (ppt), or parts per million (ppm) depending on the magnitude of the result. The percent relative error is given by the expression: Er Er = xi - xt x 100% xt Example: A loss of 0.4 mg of Zn occurs during an analysis of that element. Calculate the (a) absolute error and (b) percent relative error due to this loss if the mass of Zn in the sample is 150.0 mg. o Solution: o Due to loss of 0.4 mg, the mass of Zn in the analysis is 149.6 mg o (a) Er = xi - xt - 149.6 - 150.0 = -0.4 (the negative sign shows that the experimental result is smaller than the accepted value) o (b) Er = -0.4 x 100 = -0.27 = -0.3 150.0 • ANACHEM • 2