Question on the error of the estimated mean.

In summary: Can you please clarify?In summary, the conversation discusses a method for estimating the mean and error of a series of measurements. By taking the square root of the sum of the squares of the measurement errors, the estimated error in the mean can be calculated. If all measurements have the same error, the equation becomes Δ(x_est) =Δx/√N. However, if each measurement has its own uncertainty, the equation remains Δ(x_est) = Δx/N. The conversation also clarifies that in the case of all measurements having the same error, the standard deviation is the square root of the variance.
  • #1
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So I took a series of measurements, where the reading error for the measurements are either ±0.01 mm or ±0.02 mm or ±0.03 mm or ±0.04 mm. Now, I can calculate the estimated mean of the measurements by x_est=Ʃ x_i / N, where x_i is the i th measurement. Then I can propagate the errors in the x_i quantities by Δx = √((Δx_1)^2 +...+(Δx_n)^2)
Then the error is estimated mean would be Δ(x_est) = Δx/N.
So, if I did everything correctly would the Δ(x_est) be the standard error?
And would each individual measurement X_i have to have the same reading error if the same device was used to measure them (ruler), then the above equation would become Δ(x_est) =Δx/√N?
 
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  • #2
Your presentation is a little confusing. The variance involves the error squares and the standard deviation is its square root. When combining independent measurement errors, the divisor is N for the variance and √N for the standard deviation.
 
  • #3
mathman, OP has a situation where each measurement can have its own uncertainty. This happens if you use different instruments for measurements, for example. Say, estimating velocity by taking measurements from GPS and accelerometers. In the simple example where each measurement is a direct measurment of a quantity and you estimate the mean, OP's equation is exactly right. Estimated error is the square root of the sum of the squares.

In case where all [itex]\Delta x[/itex] are the same, you get [itex]\Delta x/\sqrt{N}[/itex] for the answer, as you'd expect.
 
  • #4
Then the error is estimated mean would be Δ(x_est) = Δx/N.
then the above equation would become Δ(x_est) =Δx/√N?

The above statements look confusing.
 
  • #5


I can say that your approach to calculating the estimated mean and propagating the errors is correct. The estimated mean is the best estimate of the true mean based on your measurements, and the error in the estimated mean is calculated by dividing the propagated error by the number of measurements.

Regarding your question about the standard error, it depends on how you define it. The standard error is typically used to describe the variability of sample means from the true population mean. In your case, the estimated mean is your best estimate of the true mean, so the error in the estimated mean can be considered the standard error.

As for the individual measurements having the same reading error, it is ideal for all measurements to have the same reading error if the same device is used. However, in some cases, there may be slight variations in the reading error due to factors such as human error or instrument calibration. In those cases, it would be more accurate to use the equation Δ(x_est) = Δx/√N, as you mentioned.
 

1. What is the estimated mean?

The estimated mean is a statistical measure that represents the average value of a set of data. It is calculated by dividing the sum of all the data points by the total number of data points.

2. How is the estimated mean used in scientific research?

The estimated mean is often used as a point estimate to represent the overall average of a population. It can also be used to compare different groups or to track changes over time in a single group.

3. What is the margin of error for the estimated mean?

The margin of error for the estimated mean is the range of values within which the true population mean is likely to fall. It is calculated by taking into account the sample size and the variability of the data.

4. What factors can affect the accuracy of the estimated mean?

The accuracy of the estimated mean can be affected by the sample size, the variability of the data, and the presence of outliers or extreme values in the data set. Other factors such as measurement errors and sampling bias can also impact the accuracy of the estimated mean.

5. How can the error of the estimated mean be reduced?

The error of the estimated mean can be reduced by increasing the sample size, using more precise measurement methods, and identifying and removing any outliers or extreme values in the data. It is also important to use a representative sample that accurately reflects the population being studied.

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