Question

The following data was obtained by an analyst upon taking replicate measurements of sample X: 12.2, 12.4, 12.6, 11.8 and 12.1. Determine the standard error associated with the measurements

Answer 1

The standard error of the sample mean for replicate measurements of sample X is calculated by first determining the sample mean and standard deviation, and then using the formula SEM = s/sqrt(n).

To calculate the standard error associated with the measurements of sample X (12.2, 12.4, 12.6, 11.8, and 12.1), we first need to find the sample mean and sample standard deviation. The sample mean (\( \overline{x} \)) is the sum of all the measurements divided by the number of measurements. Then, we calculate the sample standard deviation (s), which measures the amount of variation or dispersion from the sample mean. The formula for standard deviation is:

\[ s = \sqrt{\frac{\sum{(x_i - \overline{x})^2}}{n - 1}} \]

where \(x_i\) is each individual measurement and \(n\) is the number of measurements. Once the sample standard deviation is calculated, the standard error can be computed using the formula for the standard error of the mean (SEM):

\[ SEM = \frac{s}{\sqrt{n}} \]

In this case, \(n = 5\). With the calculated mean and standard deviation, we would insert these values into the SEM formula to find the standard error. It is essential to consider that the smaller the standard error, the more precise the estimate of the mean is.

The standard error gives an indication of how much the sample mean might vary from the true population mean, and thus it's a vital element in data analysis and interpretation.

Answer 2

To determine the standard error associated with the measurements, we can use the formula:

Standard Error = Standard Deviation / √(n)

First, let's calculate the mean of the measurements:

Mean = (12.2 + 12.4 + 12.6 + 11.8 + 12.1) / 5

Next, let's calculate the deviations from the mean for each measurement:

Deviation = Measurement - Mean

Then, let's square each deviation and calculate the sum of the squared deviations:

Sum of Squared Deviations = (Deviation₁² + Deviation₂² + Deviation₃² + Deviation₄² + Deviation₅²)

Now, let's calculate the variance:

Variance = Sum of Squared Deviations / (n - 1)

Finally, we can calculate the standard deviation:

Standard Deviation = √Variance

And the standard error:

Standard Error = Standard Deviation / √(n)

Now, let's plug in the values and calculate the standard error.