Calculating Uncertainty for Radioactive Isotope Measurements

[dimensionless]

Let's say that I have a long lived radio active isotope and I make 10 one second measurements of the disintegration. My measurements are as follows:

3, 0, 2, 1, 2, 4, 0, 1, 2, 5

How many one second measurements would I have to make to get an uncertainty of 1%?

The answer is 5,000, but I have no idea how to get it.

 

[jim mcnamara]

I'm not sure what you mean but the margin_of_error which sometimes is called the uncertainty can be calculated knowing only the number of observations. 1 divided by the square root of the number of observations.

for 5000 it is about 1.4% which is 1 / sqrt( 5000 )

 

[Architeuthis Dux]

I can provide some insight into how to calculate uncertainty for radioactive isotope measurements. Uncertainty in this context refers to the range of possible values for a measurement, taking into account factors such as random errors and limitations of the measurement equipment.

To calculate uncertainty in radioactive isotope measurements, we can use the formula:

Uncertainty = Standard Deviation / Average * 100%

In this case, we have 10 measurements with values of 3, 0, 2, 1, 2, 4, 0, 1, 2, 5. The average of these measurements is 2.

To calculate the standard deviation, we first need to find the sum of the squared differences between each measurement and the average:

(3-2)^2 + (0-2)^2 + (2-2)^2 + (1-2)^2 + (2-2)^2 + (4-2)^2 + (0-2)^2 + (1-2)^2 + (2-2)^2 + (5-2)^2 = 20

Next, we divide this sum by the number of measurements (10) and take the square root:

√(20/10) = 2

The standard deviation in this case is 2.

Plugging this into the formula, we get:

Uncertainty = 2/2 * 100% = 100%

This means that the uncertainty for these 10 measurements is 100%. In order to get an uncertainty of 1%, we need to reduce the standard deviation to 1. This can be achieved by making more measurements.

To calculate how many measurements we need to make, we can use the formula:

n = (z*s/d)^2

Where:
n = number of measurements
z = z-score for desired level of confidence (in this case, we will use 2 for a 95% confidence level)
s = standard deviation
d = desired uncertainty level (in this case, 1%)

Plugging in the values, we get:

n = (2*2/1)^2 = 4^2 = 16

Therefore, in order to achieve an uncertainty of 1%, we would need to make at least 16 measurements. However, to be more confident in our results, it is recommended to make at least 30 measurements.

In conclusion, to achieve an uncertainty of