# Suppose you weigh three samples of AgCl precipitate using the exact mass weighing method. The...

## Question:

Suppose you weigh three samples of AgCl precipitate using the exact mass weighing method. The sample weight is 0.2500 grams. However, due to the existence of random error, your three samples' weights are actually 0.2496 grams, 0.2502 grams, and 0.2498 grams.

a. Calculate the standard deviation of chloride concentration of your measurement results.
b. Calculate the relative standard deviation (RSD) of your chloride concentration measurement results.

## Information Derived from Standard Deviation:

Standard deviation is a value that shows the difference between the samples and their average. This also shows how wide the range of the measurands and how much variation there is in the values of the measurements. Moreover, the standard deviation can be used to determine whether the measurements are accurate enough or not.

### Part (a.)

First, we have to calculate the average ({eq}\rm \chi {/eq}) of the measurements by adding up all the measurements (x) and dividing the sum by the number of samples (n).

Average of Measurements:

{eq}\rm \chi = \dfrac {x_1 + x_2 + x_3}{n}\\ \chi = \dfrac {0.2496~g + 0.2502~g + 0.2498~g}{3}\\ \chi = \dfrac {0.7496~g}{3}\\ \chi = 0.2499~g {/eq}

The standard deviation (SD) of a measurement depends on the measurements, the average, and the number of samples in the population.

Standard Deviation:

{eq}\rm SD = \sqrt{ \dfrac {\Sigma (x - \chi)^2}{n - 1}}\\ SD = \sqrt{ \dfrac {(0.2496 - 0.2499)^2 + (0.2502 - 0.2499)^2 + (0.2498 - 0.2499)^2}{3 - 1}}\\ SD = \sqrt{ \dfrac {(-0.0003)^2 + (0.0003)^2 + (-0.0001)}{2}}\\ SD = \sqrt{ \dfrac {9 \times 10^{-8} + 9 \times 10^{-8} + 1 \times 10^{-8}}{2}}\\ SD = \sqrt{ \dfrac {19 \times 10^{-8}}{2}}\\ \boxed{\mathbf{ SD = 9.5 \times 10^{-8}}} {/eq}

### Part (b.)

Relative standard deviation (RSD) is the product of dividing the standard deviation by the mean multiplied by 100%.

Relative Standard Deviation:

{eq}\rm RSD = \dfrac {SD}{\chi} \cdot 100\%\\ RSD = \dfrac {9.5 \times 10^{-8}}{0.2499} \cdot 100\%\\ RSD = 3.8 \times 10^{-7} \cdot 100\%\\ \boxed{\mathbf{ RSD = 3.8 \times 10^{-5}\%}} {/eq}