# Standard Errors and Margin of Error

• Richard_R
In summary, the conversation discusses the concept of standard errors and margin of error in the context of a sample mean. The formula for calculating margin of error is provided and it is clarified that the confidence interval is an estimate of the unknown population mean. The purpose of confidence intervals is to provide a range of values that is likely to contain the true population mean, but it cannot be assumed that individual measurements or sample means will always fall within this range.
Richard_R
Hello All,

I am having to brush up on my stats for work and it's been a long time (>10yrs) since I've had to even think about this stuff. I could do with some help clarifying a specific point about standard errors and margin of error.

An example I am looking at from my old notes is this: the manager of an ice-cream shop wants to know the average amount of ice-cream his staff put into each ice-cream cone. If 50 samples are taken with an average (mean) of 10.3 ounces and a standard deviation of 0.6 ounces then what is the margin of error (MOE) at the 95% confidence level?

Now I think I've worked out the answer correctly:

MOE @ 95% CL = 1.96 x 0.6/SQRT50 = 0.17 ounces

I.e. MOE = 10.3 oz +/- 0.17 oz at 95% CL

However I am not totally sure how to interpet this result. Does this mean that if I redid the experiment again and again that 95% of the individual results in each sample would be within that range (10.3 +/- 0.17) or that the mean from each sample would be within that range, or both?

I think it's the sample means as standard errors and margins of error have to do with sample means (or proportions) but am not 100% sure...

Thanks!
-Rob

You got it; it's the mean. Put another way, 95% of all 95% confidence intervals contain the true population mean.

"Does this mean that if I redid the experiment again and again that 95% of the individual results in each sample would be within that range (10.3 +/- 0.17) or that the mean from each sample would be within that range, or both?"

Neither. The 95% confidence interval is an estimate, given as a range of values, of the unknown population mean. From sample to sample we would expect the CIs to show some overlap (that is the point of Mapes' post), but we can't say that in general the individual measurements will do so, or that the sample mean from a new sample will fall within a current confidence interval.

One more note: in your calculations the margin of error is 0.17, the confidence interval is the interval from 10.3 - .17 to 10.3 + .17.

...we can't say that in general the individual measurements will do so, or that the sample mean from a new sample will fall within a current confidence interval.
Mmm I am not sure what the point of confidence intervals is then...

In reality, in most circumstances I am only going to have one sample from my population. I can create, say, a 95% confidence interval from this sample but I always thought this meant that there was a 95% chance that the true (i.e. population) mean would be somewhere within that confidence interval.

From what you're saying it sounds as though this isn't true. I'm not sure what the point of confidence intervals are then if we can't infer the likelihood of the true population mean being within the confidence interval. A confidence interval derived from a single sample doesn't really tell you anything about where the population mean lies?

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I think my main problem is that I only have data for one sample from the population. I was given this data and there's no way we can get any more so I have to deal with this one sample.

From what you were saying it sounds as though confidence intervals don't really apply here - I could work out a confidence interval using this one sample (using the sample standard deviation instead of the population standard deviation, which I don't know) but the results would be meaningless since I can't sample from the population many times?

I see confidence intervals associated with single sample results all the time but it sounds as though this doesn't actually tell you anything (I originally thought it meant there was, say, a 95% probability that the true population mean lies within the confidence interval associated with that single sample).

"In reality, in most circumstances I am only going to have one sample from my population. I can create, say, a 95% confidence interval from this sample but I always thought this meant that there was a 95% chance that the true (i.e. population) mean would be somewhere within that confidence interval."

That's what I said - the CI is an estimate of the population mean: we can be reasonably sure the CI falls around the true value of $\mu$. So, you can be reasonably sure (95% confident) that the value of the mean is between the endpoints you calculated. You just can't say that any sample means, or individual values, will always be in the interval.

Ah okay I get it now.

Thanks!

## What is a standard error?

A standard error is a measure of the variability or spread of a sampling distribution. It represents the average amount of error or difference between a sample statistic (such as the mean) and the population parameter (such as the true population mean).

## What is the formula for calculating standard error?

The formula for calculating standard error is: standard error = standard deviation / square root of sample size. In other words, it is the standard deviation of a sample divided by the square root of the number of individuals in that sample.

## What is the margin of error?

The margin of error is a measure of the precision or accuracy of a sample statistic. It represents the range of values within which the true population parameter is likely to fall. It is typically expressed as a plus or minus value, such as ± 3%, and is influenced by the size of the sample and the level of confidence.

## How is margin of error related to standard error?

The margin of error and standard error are closely related. The margin of error is calculated by multiplying the standard error by a critical value based on the desired level of confidence. In general, a larger margin of error indicates a less precise estimate, while a smaller margin of error indicates a more precise estimate.

## Why is it important to consider standard errors and margin of error?

Standard errors and margin of error are important because they provide crucial information about the accuracy and reliability of sample statistics. They allow us to make inferences and draw conclusions about a population based on a smaller sample. Without considering these measures, our conclusions may be inaccurate or unreliable.

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