Probability and Statistics > Slovin’s Formula

If you take a population sample, you must use a formula to figure out what sample size you need to take. Sometimes you know something about a population, which can help you **determine a sample size**. For example, it’s well known that IQ scores follow a normal distribution pattern. But what about if you know nothing about your population at all? That’s when you can use Slovin’s formula to figure out what sample size you need to take, which is written as n = N / (1 + Ne^{2}) where n = Number of samples, N = Total population and e = Error tolerance

Sample question: Use **Slovin’s formula** to find out what sample of a population of 1,000 people you need to take for a survey on their soda preferences.

Step 1: Figure out what you want your** confidence level** to be. For example, you might want a confidence level of 95 percent (which will give you a margin of error of 0.05), or you might need better accuracy at the 98 percent confidence level (which produces a margin of error of 0.02).

Step 2. **Plug your data into the formula.** In this example, we’ll use a 95 percent confidence level with a population size of 1,000.

n = N / (1 + N e^{2}) =

1,000 / (1 + 1000 * 0.05 ^{2}) = 285.714286

Step 3: **Round your answer to a whole number** (because you can’t sample a fraction of a person or thing!)

285.714286 = 286

Like the explanation? Check out the Practically Cheating Statistics Handbook, which has hundreds more step-by-step explanations, just like this one!

## What is Slovin’s Formula?

Slovins’s formula is used to calculate an appropriate sample size from a population.

**About sampling**

Statistics is a way of looking at a population’s behavior by taking a sample. It’s usually impossible to survey every member of a population because of money or time. For example, let’s say you wanted to know how many people in the USA were vegetarians. Think about how long it would take you to call over 300 million people; Assuming they all had phones and could speak!. The problems with surveying entire populations are why researchers survey just a fraction of the population: a sample.

The problem with taking a sample of the population is sample size. Obviously, if you asked just one person in the population if they were vegetarian then their answer wouldn’t be representative of everyone. But would 100 people be sufficient? 1000? Ten thousand? How you figure out a big enough sample size involves applying a formula. While there are many formulas to calculate sample sizes, most of them require you to know something about the population, like the mean. But what if you knew nothing about your population? That’s where **Slovin’s formula** comes in.

**When Slovin’s formula is used**

If you have **no idea about a population’s behavior,** use Slovin’s formula to find the sample size.The formula (sometimes written as Sloven’s formula) was formulated by Slovin in 1960.

The error tolerance, e, can be given to you (for example, in a question). If you’re a researcher you might want to figure out your own margin of error; Just subtract your confidence level from 1. For example, if you wanted to be 98 percent confident that your data was going to be reflective of the entire population then:

1-0.98 = 0.02.

e=0.02.

## Who Invented Slovin’s Formula?

I love a challenge. Out of curiosity I Googled “Who Invented Slovin’s Formula?” today. I remembered waaayyy back when I first learned about Slovin’s formula, it was attributed to “Michael Slovin” but I was looking for a little more information on him. The top search result was Yahoo! Answers with this response as the Best Answer:

I’m sorry, I couldn’t find any information on the net about the origins of Slovin’s Formula or who developed it. Judging by the lack of answers, it looks like not many people of YA know either. Really sorry I couldn’t help. Xxx :)”

Surely it can’t be that hard to figure out where the formula came from…could it? A search for “Slovin’s Formula” just brings up sites (like this one) describing how to use the formula, but not where it came from. Oddly enough, Wikipedia — the site that has a page for everything (Michigan left, anyone?) doesn’t have one for** Slovin’s Formula.** It doesn’t even have one for “Slovin.” The plot thickens…

A somewhat hilarious Google search for the person who invented “Slovin’s Formula” revealed why you shouldn’t trust everything you read on the web. Several authoritative posts on Ask.com, Wiki Answers and other “Answer” sites gave the following answers to the question “Who invented Slovin’s Formula”:

- Mark Slovin
- Michael Slovin
- Kulkol Slovin

There’s also some chat over at Wikimedia Talk, on the topic of even if there should be a Wikipedia page on Slovin’s formula at all!

“…the formula itself seems clearly notable as you get quite a number of hits under Google books ([1]). Slovin publication of the formula is however dated 1960 not 1843, but it might have known to others earlier.–Kmhkmh (talk) 09:05, 1 April 2013 (UTC)++”

“Slovin’s formula I find no evidence of these formulas that doesn’t seem to trace back to the same handbooks. There is no author in MathSciNet with the name “Slovin”, and the only published article I could find for a person named “Slovin” in 1960 is an unrelated patent.”

This mention of “Sloven’s formula” in the 2003 book “Elementary Statistics: A Modern Approach” by Altares et. al might provide a clue (note the spelling):

And Guilford, J.P. and Frucher. B; (1973), Fundamental Statistics in Psychology and Education, New York: MC Graw-Hill does cite Slovin (1960). Now, if I could get my hands on that book, I might be able to solve this mystery!

## Taro Yamane’s Formula

Taro Yamane is often credited with an identical formula. However, his formula was published several years after Slovin’s (in 1967).

**References**

Yamane, Taro. (1967). Statistics: An Introductory Analysis, 2nd Edition, New York: Harper and Row.

the problem I have with this formula is that a 5% MOE at 100,000 is 398 while a 5% MOE at 10,000,000 is 399.

Mike,

When it gets to very large populations, the sample size is often the same as a smaller population. It’s hard to explain that in a comment here, but basically a small sample is often enough to get good results, whether you are surveying 100,000 people or 10 million. (Please note, I haven’t done the math here for 100,000 or 10 million but I am taking your word that the math is correct :)

Best,

Stephanie

how to get or compute the margin of error?

Can I use the slovin’s formula if i intend to conduct a spot-check? I already have a sample population derived out of the total population, but I could not figure out the sample size I would use to spotcheck certain number of respondents to save on cost, money and time.

Aki, you have to figure out what confidence level you are willing to accept. In the example about, a CI of 95 percent will give you a ME or .05.

please i didn’t get adequate answer about at what time or under what condition that i have to apply any formula to determine a sample size

Please indicate a sample on how to use Slovin’s Formula in a categorical way.

Tsegaye,

Sorry — time constraints don’t give me enough time to answer every stats question in the comments. Please post on our forum (click the tab above) and one of our moderators will be glad to help :)

Stephanie

Mailyn,

Time constraints prevent me from answering stats questions in the comments…but post on our forums and our mod will be happy to help :) (In fact, I think a poster recently asked this question…)

Stephanie

Hi there, do you have articles that support the use of Slovin’s formula? Thank you.

Hiram,

Not yet — but they may be coming in the near future.

Thanks for stopping by :)

Stephanie

Hi! I just checked on a site saying that: Not proper to call it Slovin’s Formula since he did NOT derive it! The term Slovin’s Formula originated in the Philippines. Based on the formula, by setting P=0.5, we are getting the largest possible sample, which may be good or bad. It is valid only under simple random sampling and any other design that is theoretically more efficient than simple random sampling (e.g., one-stage stratified sampling). Lastly, the setting up of the value of e varies from one purpose to another. Of course, the smaller, the better.

don’t know if it is legit though but UP Diliman is one of the best universities in the Philippines

Very useful for the research work of young anaesthsesiologists

Very useful for young research workers

Please, I’d like to know if there is another name for Slovin’s fomular.

Not that I know of.

Should we still use Slovin’s in purposive sampling?

No. As purposive sampling is non-random, how large the sample is depends on the researcher. As you can’t run statistics on your sample (due to the non random process), then there’s no point in using Slovin’s. You

coulduse Slovin’s if you wanted to…but there really wouldn’t be any point.what are some of the advantages and limitation of slovin’s formular

i used this formular for my research but everyone told how brightly i chose my sample size

Well you can use it if you know nothing about the population. But it’s a guesstimate and other methods are probably more accurate.

Additional information: On the Misuse of Slovin’s Formula https://www.google.com.ph/url?sa=t&rct=j&q=&esrc=s&source=web&cd=1&cad=rja&uact=8&ved=0ahUKEwiZ07y1qrbOAhXBPCYKHaUEDJMQFggaMAA&url=http%3A%2F%2Fwww.philstat.org.ph%2Ffiles%2Fimages%2F2012_611_9_On_the_Misuse_of_Slovin_s_Formula.pdf&usg=AFQjCNFNpHi0BQrrCLgYDP5iOtDj7Ytjsw

This formula is called Yemane’s formula in Ethiopia

https://www.researchgate.net/post/How_do_I_calculate_the_needed_sample_size_in_the_absence_of_a_known_mean_and_standard_deviation

That’s very interesting! Thanks for letting me know :)

HOW CAN I FIND TOTAL NUMBER OF POPULATION

It depends. For example, for US population data you can check with the Census bureau. What are you trying to find?