ERT 207-ANALYTICAL CHEMISTRY

SIGNIFICANT FIGURES AND STANDARD DEVIATION

DR. SALEHA SHAMSUDIN

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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

11. Nonzero integers2. Zeros leading zeros captive zeros trailing zeros3. Exact numbers

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Rules for Counting Significant Figures - Details

Nonzero integers always count as significant figures.

3456 has 4 sig figs.

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Rules for Counting Significant Figures - Details

ZerosZerosLeading zerosLeading zeros do not count as do not count as

significant figures.significant figures.

0.04860.0486 has has33 sig figs. sig figs.

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Rules for Counting Significant Figures - Details

Zeros Captive zeros always count as

significant figures.

16.07 has4 sig figs.

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Rules for Counting Significant Figures - Details

Zeros Trailing zeros are significant only

if the number contains a decimal point.

9.300 has4 sig figs.

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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 exactlyThe 1 and 2.54 do not limit the significant figures

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

100 has 1 sig. fig. = 1 x 102

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Rules For Rounding

1. In a series of calculations, carry the extra digits through to the final result, then round.

2. If the digit to be removed:A. Is less than 5, then no change e.g. 1.33 rounded

to 2 sig. fig = 1.3B. Is equal or greater than 5, the preceding digit

increase by 1 e.g. 1.36 rounded to 2 sig. fig = 1.4

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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 73.284. 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

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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 6.38 2.0 = 2.0 =12.76 12.76 13 (2 sig figs)13 (2 sig figs)

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

1.0923 x 2.07

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Solution:1.0923 x 2.07 = 2.261061 2.26The correct answer is therefore 2.26 based on

the key number (2.07).

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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)

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

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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

Log (3.1201)

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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

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Rules for Counting Significant Figures.

Example 1List the proper number of significant figures in the

following numbers and indicate which zeros are significant

0.216 ; 90.7 ; 800.0; 0.0670Solution:0.216 3 sig fig90.7 3 sig fig; zero is significant800.0 4 sig fig; all zeroes are significant0.0670 3 sig fig; 0nly the last zero is significant

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Standard Deviation

The standard deviation is calculated to indicate the level of precision within a set of data. Abbreviations include sdev, stdev, and s.

is called the population standard deviation – used for “large” data sets.

s is the sample standard deviation – used for “small” data sets.

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STANDARD DEVIATIONThe 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.

n

xxxx n

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x

The standard deviation of the mean:

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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

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Relative Standard Deviation (rsd)

Dimensionless, but expressed in the same units as the value

Coefficient of variation (or variance), CV, is the percent rsd:

x

srsd

x

xrsd

%100x

sCV

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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

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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.

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Median and RangeMedian 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.

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Mean and Median

Median

Mean

Range

Range

Median and RangeFor 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

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