# Formula for confidence intervals for small samples and unknown population standard deviation

I have consulted two texts on how to calculate to calculate confidence intervals when N is small and the population standard deviation is unknown. There are some differences in the formulas they give and the end result varies depending on which text I follow (although not by a large amount). Text one says:

1. Calculate the mean
2. Calculate the standard deviation using the formula: s= √ ((∑ X(squared)/N)–X-bar)
3. Calculate the standard error of the mean using the formula: s/√ N-1
4. Determine the value of T from the t-table
5. Obtain the margin of error by multiplying the standard error of the mean by multiplying it by the value obtained in step 4.
6. Add and subtract this product from the sample mean to obtain the C.I.

Steps, 1,4,5& 6 are exactly the same in the second text. However it gives different formulas for steps 2 & 3. It says:

1. Calculate the standard deviation using the formula: s= √ ((∑ X(squared) /N-1) –X-bar). The difference is that they reduce N by one.
2. Calculate the standard error of the mean using the formula: s/√ N The difference is that N is not reduced by 1.

Can anyone explain why the different formulas are used and why?

Thanks. Anne S

• Can you tell us which two texts gave you these sets of instructions? I'd be very curious to know. – Chris Simokat Jun 11 '11 at 21:24
• Chris, Yes, One is from Fox & Levin and the other is from Haan. Can you assist? – Anne Jun 11 '11 at 21:53
• Is Fox and Levin "Elementary Statistics in Social Research" and is Haan "Practical Statistics for Business" and is the first set of instructions from the first text and the other from the second? I will post an answer momentarily. – Chris Simokat Jun 11 '11 at 22:07
• @Chris Yes you have the correct Fox and Levin. The Haan is: An Introduction to Statistics. I struggle with writing the formula for the standard deviation on a website. I don't know how to use the correct symbols so I had to write it in words. Hope it is still clear. I am looking forward to your response. – Anne Jun 11 '11 at 22:37
• This site uses a system called $\LaTeX$ to display formulas if you're interested you can read more about it here knowing some can help you get the help you need. LyX is an editor program that can help you create the formatting. – Chris Simokat Jun 11 '11 at 22:56

Here are some good notes on standard deviation and the standard error of the mean here.

The Wackerly et al text computes small sample confidence intervals in section 8.8 (page 430) you can see their formula here.

Confidence interval: $\bar{Y} \pm t_{\alpha/2} * \frac{S}{\sqrt{n}}$

Where $\bar{y}$ = $\frac{1}{n}$$\sum{\textstyle y_{i}}$ (the sample mean)

and $S = \sqrt{\frac{1}{n-1}\sum(y_{i}-\bar{y})^2}$ (the sample standard deviation)

t$_{\alpha/2}$ is the critical value for a given value of $\alpha$ (e.g., .1, .05, etc.) and has n-1 degrees of freedom, where n is the sample size, that you'd find in a table.

Now if your sample is a large proportion of a known finite population size there is something called a population correction factor, but for basic needs you probably don't have to worry about this.

• @Chris, Thanks for your reply but I don't understand. Can you comment on the formulas I posted? – Anne Jun 11 '11 at 23:00
• What is it you do not understand exactly? From what you've posted I would say the second set of instructions are the ones I would recommend, but the way you wrote the expressions is unusual. such as s= √ ((∑ X(squared) /N-1) –X-bar) seems like you meant squaring the observation value and dividing it by n - 1. Instead of sum the differences between the quantity of the observation value minus the mean end quantity squared divided by n-1. So assuming what you typed was meant to be that formula then yes that second set of instructions is what you want. If its not what you meant please let me know. – Chris Simokat Jun 11 '11 at 23:07
• Thanks Chris, I meant I do not follow the formulas from Wackerly et al. By s= √ ((∑ X(squared) /N-1) –X-bar) I meant square all observations and then sum them, then divide this by N-1, and then subtract the mean(squared). – Anne Jun 11 '11 at 23:13
• Can you explain why Fox and Levin subtract 1 from N in the formula for standard error of the mean and Hann does not and why Haan subtracts 1 from N in the formula for the standard deviation and does not subtract 1 from N in the formula for standard error of the mean. I read the notes from Murdoch. What I think I get from these is if you subtract 1 from N in the formula for standard deviation then you do not need to do this in later formulas. Is this correct? Many thanks for your time! – Anne Jun 11 '11 at 23:22
• Okay so the S in Fox and Levin is the standard deviation of the sample. The one used in Haan is the sample standard deviation which is an estimate of the population's standard deviation. Yes like you suspect the difference in the instructions is to create an unbiased confidence interval given their two different starting points for the standard deviations. – Chris Simokat Jun 11 '11 at 23:48