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After the end of the experiment, you should be able to:
1. differentiate the three expressions of central value particularly on how these are
computed and used appropriately to describe a set of data;
2. compute absolute error, relative error, and percent error;
3. compute mean deviation and standard deviations of the sample and the
population using Excel.
Activity 4 Statistical Testing for Treatment of Data Introduction Why is experiment necessary to subjects like chemistry? What activities were carried out when you performed your experiment? What did you obtain when you performed your experiment? How did you manipulate the results of your experiment? What did you do then with these manipulated results? You are all made aware that being an experimental science, chemistry involves laboratory activities designed to explain scientific theories. In your general chemistry laboratory, you did not only measure the value of the scientific property but you also gathered data as part of the experimentation. Perhaps the bigger challenge that you encountered was how you were going to manipulate or treat the data systematically in order to find the value of the property that was being measured. Interestingly, statistics will teach you how to manipulate or treat experimental data systematically. As a scientific study, statistics is not only used in treating experimental data as it is also utilized extensively in almost all fields of disciplines. One common situation that every one of you might have already encountered is finding the simple mean or the average of the values. This is a situation that we usually experience not only in the laboratory particularly in treatment of data but also in every day activity.
Example 1- 1
Find the median of the five values 20.4, 20.6, 20.1, 20.7, and 20.0.
Rules:
1) Rearrange the values from the lowest to the highest.
2) Identify the value that is physically located in the middle of the set of data.
1.3.1 Central Value Central value is expressed in three expressions and the use of each depends on how one wants to describe and interpret the data. 1.3.1a Arithmetic mean / Mean The most common central value used by chemists is the arithmetic mean ,๐ฅฬ
, which is obtained by dividing the sum of the individual values by the number of values. Mathematically, ๐ฬ
= ๐๐+ ๐๐+ ๐๐+... + ๐๐ ๐
โ (^) ๐๐ ๐ where x 1 , x 2 , x 3 ,... , xn are the individual values, n is the number of values, and โ^ ๐ฅ๐ the sum of values of x. 1.3.1b Median Another central value which is less commonly used is the median. It is the middle numerical value in a set of values. When the set of data contains an even number of values, the median is the average of the two numerical values.
LESSON PROPER
median =
- 37 + 6. 41 2
For data set B, ๐ฅฬ
=
- 37 + 6. 33 + 6. 41 + 6. 93 4
- 04 4
median =
- 37 + 6. 41 2
1.3.4 Precision and Accuracy The terms precision and accuracy are often used when dealing with the uncertainties of measured values. Precision is a measure of how closely individual measurements agree with one another while accuracy refers to how closely individual measurements agree with the correct, or โtrue,โ value. The dart analogy in Figure 1-1illustrates the differencebetween these two concepts. Figure 1-3. Precision and accuracy (Source: Brown, T. L., et. al., 2012)
1.3.4a Expressions of accuracy As defined, accuracy describes the nearness of an experimental value, xi , or a mean,๐ฅฬ
, to the true value, ฮผ. It is expressed as error. The following are various expressions of error based on how it is computed. Absolute error is calculated this way, error = xi - ฮผ or ๐ฅฬ
- ฮผ Note: error carries the units of xi , and ฮผ. Relative error is used when comparing errors at different quantities. It is calculated by dividing the absolute error by the true value as follows: Relative error = ๐๐๐๐๐ ฮผ Relative error is often expressed by statisticians as pph or ppt, as defined below: Parts per hundred (pph) or percent error is the relative error multiplied by 100. Parts per thousand (ppt) error is the relative error multiplied by 1000, and so on. Notes from the author: pph and ppt used as expressions of relative error should not be confused with or used interchangeably as the percent concentration (mass %, volume %, and mole %) and the ppt. Although the mathematical thought and the procedure of calculating them may be the same in some respect, percent concentration and ppt are some of the expressions of concentration. Example 1- 5 Calculate the absolute error, percent error, and parts per thousand error for the mean of the following data set. xi (mg) 8.33 8.29 8.28 8.34 8. ฮผ (mg) 8.
Example 1- 6
Quantitative analysis of student obtained the following results for the determination of
isooctane in gasoline using Gas Chromatography.
n = number of observations ว ๐ฅ๐ โ ๐ฅฬ
ว = absolute value of the difference between an observation and the mean (inside the absolute value sign is always a positive number) โ ว ๐ฅ๐ โ ๐ฅฬ
ว = sum of the absolute values of the differences between an observation and the mean ๐ฬ
= average deviation between the experimental values and the mean Similar to accuracy, precision measurement such as average deviation can be expressed as an absolute error or as a relative error (% or pph, ppt, etc.) 1.3.6 Standard deviation of the sample The standard deviation, s , or root-mean-square deviation as it is sometimes called, is the preferred measure of precision and is calculated from the equation s =โ โ(๐ฅ๐ โ ๐ฅฬ
)^2 ๐ โ 1 where ๐ฅ๐ = observation ๐ฅฬ
= mean of the observations n = number of observations n โ 1 = degree of freedom (๐ฅ๐ โ ๐ฅฬ
)^2 = square of the differences between an observation and the mean โ(๐ฅ๐ โ ๐ฅฬ
)^2 = sum of the squares of the differences between an observation and the mean s = standard deviation or the measure of the spread of observations The most common mistake made by students using this equation is they โsquare the sum of the deviationsโ rather than โsum the squares of the deviationsโ. Look carefully at the following example to make sure that you learn to use the formula correctly.
Percent Determination
Number isooctane
Calculate the standard deviation from the mean.
You can calculate the standard deviation of the given data set using the formula
s =โ
โ(๐ฅ๐ โ ๐ฅฬ
)^2
. The formula indicates that the mean is necessary. So
you need to calculate first the mean.
=
๐ฅ๐ ว ๐ฅ๐ โ ๐ฅฬ
ว (๐ฅ๐ โ ๐ฅฬ
)^2
0.0128 =โ(๐ฅ๐ โ ๐ฅฬ
)^2
Substituting, you have
s =โ
Both average and standard deviations can also be expressed in relative terms to facilitate comparison between data sets: relative average deviation = ๐ฬ
๐ฅฬ
relative standard deviation = ๐ ๐ฅฬ
Relative standard deviation (RSD) is also referred to as the coefficient of variation.
You can now assess yourselves if you understand the mathematical and statistical procedures that you just read. References Brown, T. L., et al. (2012). Chemistry the central science. 12th^ ed. Illinois: Pearson Education, Inc. Chang, R. (1994). Chemistry. 5thed. New York: Brooks Publishing Hargis, L.G. (1988). Analytical chemistry principles and techniques. New Jersey: Prentice-Hall, Inc. Holler, F. J. and Crouch, S. R. (2014). Skoog and Westโs fundamentals of analytical chemistry. 9 thed. USA: Brooks/Cole CENGAGE Learning Inc. Mann, P. S. (2011). Introductory statistics. New Jersey: John Wiley and Sons, Inc. https://www.chem.tamu.edu/class/fyp/keeney/stddev.pdf
