The assignment for this response was to discuss “…under what circumstances the Finite Population Correction Factor (FPCF) is necessary and why is it necessary? How does the adjusted z-value vary quantitatively (bigger, smaller, much larger, no change) from the normal value?” This was the first written response for my Statistics II class (actually called “Quantitative Tools for Management” in the course catalog, but it is roughly equivalent to a Stats II course) during the current semester (Spring 2007) at the University of Massachusetts at Amherst’s online program; I earned a 5/5 for my response below:

When a sample is greater than 5% of the population from which it is being selected and the sample is chosen without replacement, the finite population correction factor should be used. The adjusted z-value would be larger than the normal z-value, meaning that the value is more standard deviations from the middle than in a non-adjusted z-value.

This factor adjusts the z-value to show the extra precision obtained from the sample size being a greater fraction of the population size than normal. Since the standard deviation becomes smaller as the sample size increases, the FPCF shows that a value in a large sample size not at or near the mean is a greater number of standards deviations from the mean than in a small sample size. In other words it’s rarer for a value in a large sample size to be far away from the mean compared to a small sample size.

Thanks for taking the time to write this article. It’s been a great help. You have managed to simplify the concept in that the z value increases as n gets larger and approaches N, meaning that we can afford greater confidence in our sample or we can lower the sample size and still achive the same level of accuracy.

Brad.

Thanks Brad, I too found this theorem a bit tough to understand upon first glance but using simple language helped emphasize exactly why the FPCF is used.

You have to be careful lowering the sample size while still assuming the same level of accuracy, especially if the lower sample size dips below 5% of the total population. The z value in the FPCF situation is more accurate simply because it encapsulates enough of the total population to be statistically more representative of the total population.

Glad my words were helpful.

I still don’t understand the concept of the fpc ,can you explain further

What in particular would you want me to expand on Owusu? Feel free to send me an e-mail or instant message if you wish (there’s links to both in the about section).

I’ve been looking around on the web trying to find an answer to a question that might relate to the finite correction factor. Let’s say the goal is to estimate a mean of a population of 50. I draw a sample of 25 respondents, and get an average of 100 with a standard dev of 10. If I’m trying to report on the average of the entire population, do I use the FPCF in forming my confidence interval (I get an upper bound of 102.9)? Or do I have another choice to say I’m certain about the 25 being an average of 100, and I have a certain confidence interval about the other 25, so I can calculate an upper bound to my overall confidence intervall by combining the two estimates (25 at 100, and the other 25 at 108 (2 standard errors), for an upper bound of 104?

more explanation is needed. wil u discuss further pls.

Aran,

First, sorry it took me so long to respond. Your question got me a bit off guard and I had to dust off some cobwebs in regards to FPCF before answering it.

You use the FPCF because a) you’re sampling more than 5% of the population (in your case, 50% of the population is sampled) and b) you’re not replacing respondents after they are chosen.

What confidence interval are you using? Using a 95%, so a z of 1.96, the upper bound would be 103.92. It looks like you used a confidence interval of 92.65% with a z of 1.45 to get 102.9?

hi.. i want to ask, why we use fcf when sample is equal or more than 5% from the population? [please send the answer to my mail.. thx..]

Hi Yudi,

I deleted your comment listing your email address for your protection (spammers like to go around the internet find email addresses, you should avoid listing it publicly if possible).

On to your question, we utilize the FPCF to basically boost probability. When the sample is equal or great than 5%, we have a far greater chance that the sample’s data will apply to the population as a whole. In other words, we’re more confident that our data is correct and to show this extra confidence statistically we add the FPCF.

That’s the very general explanation. For a more detailed explanation, please re-read the last paragraph in my post above.

Hope that helps!

Hi, I got a question from a stats hw. It asks, if an initial survey used 1000 subjects for a population of 304 million, how many subjects would we need for a population of 1.3 billion while retaining the same accuracy? I know this relates to the Finite Sample Correction, but how? please explain!

Thanks..

Fiona

Hi Fiona,

I’m unfamiliar with the Finite Sample Correction, is it simply another term for the Finite Population Correction? I’ll assume it is as a google search seems to indicate so.

As 1000 is a lot less than 5% of 304 million, we will not need to use the Finite Population Correction Factor.

Accuracy is a reflection of the proportion of the sample size to the population. For instance, for a population of 1000, a sample size of 100 would theoretically be more accurate than a sample size of 10.

Back to your homework question, to retain the same level of accuracy, the proportion of the population captured in the sample would have to remain the same.

First, we find the proportion of sample size to total population in the initial survey:

1,000 / 304,000,000 = .0000032895.

Then, we multiply the new total population by this proportion to determine the new sample size required:

.0000032895 * 1,300,000,000 = 4276.32.

So the new sample size should be either 4276 or 4277 (depending if you round up or down) in order to maintain (roughly) the same accuracy.

I have seen different formulae for Finite Population Correction Factor (FPCF), such as:

1. FPCF = (N ā n)/N

2. FPCF = square root of (N ā n) / (N -1)

3. FPCF = 1 – n/N, which is the same as #1

4. nā = n / (1+ n/N), which means the FPCF = N / (N + n). nā is the sample size after taking into account of FPCF.

Which one should I use?

Your explanation is very helpful – thank you!

One other question – when is the FPCF ignored?

Thanks