ERT 207-ANALYTICAL CHEMISTRY SIGNIFICANT FIGURES AND STANDARD DEVIATION DR. SALEHA SHAMSUDIN

What is a "significant figure"? The number of significant figures in a result is simply the number of figures that are known with some degree of reliability. The number 13.2 is said to have 3 significant figures. The number 13.20 is said to have 4 significant figures.

Rules for Counting Significant Figures - Overview 1. Nonzero integers 2. Zeros  leading zeros  captive zeros  trailing zeros 3. Exact numbers

Rules for Counting Significant Figures - Details Nonzero integers always count as significant figures. 3456 has 4 sig figs.

Rules for Counting Significant Figures - Details Zeros  Leading zeros do not count as significant figures. 0.0486 has 3 sig figs.

Rules for Counting Significant Figures - Details Zeros  Captive zeros always count as significant figures. 16.07 has 4 sig figs.

Rules for Counting Significant Figures - Details Zeros  Trailing zeros are significant only if the number contains a decimal point. 9.300 has 4 sig figs.

Rules for Counting Significant Figures - Details Exact numbers have an infinite number of significant figures. Independent of measuring device: 1 apple, 10 students, 5 cars…. 2πr The 2 is exact and 4/3 π r2 the 4 and 3 are exact From Definition: 1 inch = 2.54 cm exactly The 1 and 2.54 do not limit the significant figures

100. has 3 sig. fig. = 1.00 x 102 100 has 1 sig. fig. = 1 x 102

Rules For Rounding In a series of calculations, carry the extra digits through to the final result, then round. If the digit to be removed: Is less than 5, then no change e.g. 1.33 rounded to 2 sig. fig = 1.3 Is equal or greater than 5, the preceding digit increase by 1 e.g. 1.36 rounded to 2 sig. fig = 1.4

3. If the last digit is 5 and the second last digit is an even number, thus the second last digit does not change. Example, 73.285 73.28 4. If the last digit is 5 and the second last digit is an odd number, thus add one to the last digit. Example, 63.275 63.28

Rules for Significant Figures in Mathematical Operations Multiplication and Division: # sig figs in the result equals the number in the least precise measurement used in the calculation. 6.38  2.0 = 12.76  13 (2 sig figs)

Example : Give the correct answer for the following operation to the maximum number of significant figures. 1.0923 x 2.07

Solution: 1.0923 x 2.07 = 2.261061 2.26 The correct answer is therefore 2.26 based on the key number (2.07).

Rules for Significant Figures in Mathematical Operations Addition and Subtraction: # decimal places in the result equals the number of decimal places in the least precise measurement. 6.8 + 11.934 = 18.734  18.7 (3 sig figs)

Example : Give the answer for the following operation to the maximum number of significant figures: 43.7+ 4.941 + 13.13

43.7 4.941 + 13.13 61.771 61.8 answer is therefore 61.8 based on the key number (43.7).

Rules Exponential The exponential can be written as follows. Example, 0.000250 2.50 x 10-4

Rules for logarithms and antilogarithms mantissa characteristics The number of significant figures on the right of the decimal point of the log result is the sum of the significant figures in mantissa and characteristic

Rules for Counting Significant Figures.

Example 1 List the proper number of significant figures in the following numbers and indicate which zeros are significant 0.216 ; 90.7 ; 800.0; 0.0670 Solution: 0.216 3 sig fig 90.7 3 sig fig; zero is significant 800.0 4 sig fig; all zeroes are significant 0.0670 3 sig fig; 0nly the last zero is significant

Standard Deviation The standard deviation is calculated to indicate the level of precision within a set of data. Abbreviations include sdev, stdev, s and s. is called the population standard deviation – used for “large” data sets. s is the sample standard deviation – used for “small” data sets.

STANDARD DEVIATION The most important statictics STANDARD DEVIATION The most important statictics. Recall back: Sample standard deviation. For N (number of measurement) < 30

For N > 30,

The Mean Value The “average” ( ) Generally the most appropriate value to report for replicate measurements when the errors are random and small.

The standard deviation of the mean:

The standard deviation of the mean is sometimes referred to as the standard error The standard deviation is sometimes expressed as the relative standard deviation (rsd) which is just the standard deviation expressed as a fraction of the mean; usually it is given as the percentage of the mean (% rsd), which is often called the coefficient of variation

Relative Standard Deviation (rsd) Dimensionless, but expressed in the same units as the value Coefficient of variation (or variance), CV, is the percent rsd: or

Example 1 Calculate the mean and the standard deviation of the following set of the analytical results: 15.67g, 15.69g, and 16.03g Answer: 0.20g

Example 2 The following replicate weighing were obtained: 29.8, 30.2, 28.6, and 29.7mg. Calculate the standard deviation of the individual values and the standard deviation of the mean. Express these as absolute (units of the measurement) and relative (% of the measurement) values.

Median and Range Median is the middle value of a data set If there is an even number of data, then it is the average of the two central values Useful if there is a large scatter in the data set – reduces the effect of outliers Use if one or more points differ greatly from the central value Range is the difference between the highest and lowest point in the data set. Mean and Median Range Median Mean Range

Median and Range For example: 6.021, 5.969, 6.062, 6.034, 6.028, 5.992 Rearrange: 5.969, 5.992, 6.021, 6.028, 6.034, 6.062 Median = (6.021+6.028)/2 = 6.0245 = 6.024 Range = 6.062-5.969 = 0.093