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# How to Calculate Standard Deviation in Chemistry

Precision in chemistry is vital. However, being humans our ability to take accurate measurements and find the ‘true value’ is limited.

We, therefore, rely on values that are close as possible to the ‘true value’.

However, it is impossible to know the ‘true value’ as there is no such thing as a perfect measurement in chemistry.

But we can ensure that our values are as accurate as possible.

Chemists use the mean to find the average of the values they have obtained in an experiment.

The mean of a set of data points, however, does not provide information on how accurate the data is.

Standard deviation is used in Chemistry to determine the precision of experiments.

Standard deviation is a measurement of the dispersion and is used to measure how widely the data obtained from an experiment is spread.

In chemistry, an experiment is said to have high precision if the data obtained has a low standard deviation. This means that data points obtained from the experiment lie close to the mean value.

If a high number of the values fall far from the mean, the standard deviation of the data set is large. Experiments of low precision have large standard deviations.

Standard deviation is often denoted by the symbol σ.

How to Calculate Standard Deviation

In order to calculate standard deviation, you must first determine the mean (X) of your data set.

This is obtained by summing up all your values and dividing the sum by the number of values in the data set i.e.

Where xi represents each individual measurement obtained in the experiment.

The standard deviation, σ, can then be obtained using the formula below:

N is the number of values obtained in the experiment.

The residual value, which is obtained from (xi – X) shows the deviation of each measurement from the mean of all the measurements obtained in the experiment.

The ‘degrees of freedom’ for the data set is obtained by (N – 1). This represents the number of measurements that have been repeated in the data set.

This formula for standard deviation may seem complex. However, it actually makes a lot of sense when you break it down to understand the individual steps represented:

Step 1: Find the mean of the set of measurements you have obtained in your experiment (X)

Step 2: Find the residual value or deviation of each measurement from the mean (xi– X) and square it.

Step 3: Find the sum of all the squared values from Step 2.

Step 4: Divide the resulting value by the degrees of freedom of the data set (N – 1)

Step 5: Find the square root of the value from Step 4.

Example of Calculating Standard Deviation in Chemistry

Scientists want to determine the level of mercury in fish in a certain river. The data obtained from repeated testing was as shown below:

N (the number of trials) = 8

The standard deviation for the data can be obtained as follows:

Step 1: Find the mean for the mercury measurements in the fish.

Step 2: Find the square of the variation of each measurement from the mean.

(5.0 – 4.8)2 = 0.22 = 0.04

(2.5 – 4.8)2 = -2.32 = 5.3

(5.5 – 4.8)2 = 0.72 = 0.5

(4.2 – 4.8)2 = -0.62 = 0.4

(5.2 – 4.8)2 = 0.42 = 0.2

(5.7 – 4.8)2 = 0.92 = 0.8

(3.7 – 4.8)2 = -1.12 = 1.2

(8.0 – 4.8)2 = 3.22 = 10.2

Step 3: Find the sum of the squared values in Step 2

0.04 + 5.3 + 0.5 + 0.4 + 0.2 + 0.8 +1.2 + 10.2 = 18.64

Step 4: Divide the value from Step 4 by the degrees of freedom (N – 1)

Step 5: Find the square root of the value in Step 4

The standard deviation for the concentration of mercury obtained in the experiment is 1.63.

This means that measurements obtained in the experiment were up to 1.63 units above and below the mean.

The measurement of the concentration of mercury in the fish can therefore be represented as 4.8 + 1.63 ppb (Hg).

Chemists report the error of the data obtained in their experiments using standard deviation. In this case, the actual concentration of mercury in the flesh of the fish in the river could lie anywhere between 5.43 and 3.17 ppb (Hg).

Closing Thoughts

Most quality control programs consider a standard deviation of up to +2 as acceptable.

Measurements with a standard deviation below + 2 are considered to be close to the true value.

The experiment in the example above would therefore be considered to have been accurate and the values would be accepted as close to the true value. The Boffins Portal Team
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