Tag Archives: Population

OpenBook Cochran Calculator

OpenBook Cochran Calculator 

OpenBook Cochran Calculator (OC Calculator, 2022) is a free online sample size calculator developed to help research students or other researchers from various fields worldwide, having problem in calculation using Cochran’s Sample Size Formula manually, determine sample size accurately.

Cochran’s Sample Size Formula is written as:

Where:

  • n = Number of Samples,
  • N = Total Population,
  • e = Error Tolerance (Level) or Margin of Error, 0.05,
  • p = Sample Proportion, 0.5,
  • z = z-value found in Z-score Table, 1.96.

The table below is Z-score Table for most use confidence level or confidence interval.

Confidence Level Confidence Interval Area between zero and z-score Z-score
90% 0.10 90/2%=0.4500 1.65
95% 0.05 95/2%=0.4750 1.96
99% 0.01 99/2%=0.4950 2.58

 

Example Question: Use Cochran’s Sample Size Formula to find out what sample of a population of 17,552,940 people in Lagos State, Nigeria, you need to take for a survey on their Techno Phone preferences.

Step 1: Figure out what you want your confidence level to be. For example, you might want a confidence level of 95 percent (giving you an alpha level or confidence interval or margin error of 0.05).

Step 2. Plug your data into the formula. In this example, we’ll use a 95 percent confidence level with a population size of 17,552,940.

  • n = Nz2(p)(1-p)/Ne2 + z2(p)(1-p) =
  • 17552940*(1.962)(0.5)(1-0.5)/17552940*(0.052)+(1.962)(0.5)(1-0.5) 

          = 384.1515925

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

  • 384.1515925 ≅ 384

To use OpenBook Cochran Calculator (OC Calculator, 2022), click here to get it from the homepage.

OpenBook Yamane Calculator

OpenBook Yamane Calculator 

OpenBook Yamane Calculator (OY Calculator, 2022) is a free online sample size calculator developed to help research students or other researchers from various fields worldwide, having problem in calculation using Taro Yamane Formula manually, determine sample size accurately.

Taro Yamane Formula is written as

n = N / (1 + Ne2)

Where:

  • n = Number of samples,
  • N = Total population and
  • e = Error tolerance (level) – (0.05)

The following is the table for most use confidence level or confidence interval.

Confidence Level Confidence Interval
90% 0.10
95% 0.05
99% 0.01

 

Example Question: Use Taro Yamane Formula to find out what sample of a population of 17,552,940 people in Lagos State, Nigeria, you need to take for a survey on their Techno Phone preferences.

Step 1: Figure out what you want your confidence level to be. For example, you might want a confidence level of 95 percent (giving you an alpha level or confidence interval or margin error of 0.05), or you might need better accuracy at the 98 percent confidence level (alpha level or confidence interval or margin 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 17,552,940.

  • n = N / (1 + N e2) =
  • 17552940 / (1 + 17552940 * 0.05 2) = 399.990885

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

  • 399.990885 ≅ 400

To use OpenBook Yamane Calculator (OY Calculator, 2022), click here to get it from the homepage.

 

Slovin’s Formula Credited to Taro Yamane

Slovin’s Formula: What is it and When do I use it?

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 + Ne2)

Where:

  • n = Number of samples,
  • N = Total population and
  • e = Error tolerance (level).

Example 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 (giving you an alpha level of 0.05), or you might need better accuracy at the 98 percent confidence level (alpha level 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 e2) =
  • 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

To use OpenBook Yamane Calculator (OY Calculator, 2022), click here to get it from the homepage.

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 error tolerance; 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.

Problems with Slovin’s Formula

Slovin’s formula gives you a ballpark figure to work with. However, this non-parametric formula lacks mathematical rigor (Ryan, 2013). For example, there is no way to calculate statistical power (which tells you how likely your study distinguishes an actual effect from one of chance). It’s unclear from any reference texts exactly what the “error tolerance” is (a mean, or perhaps a proportion?).

Some texts call the error tolerance a “tolerance margin of error” (e.g. Ariola, 2006), although it seems to be unrelated to the margin of error used in traditional hypothesis tests. The Margin of Error in that sense is the error associated with a result (for example, you could say 62% of people voted for so and so with a 3% margin of error). From the context, it’s almost certainly another name for the alpha level.

The lack of precision with wording is yet another reason the formula has a poor reputation in academia. But perhaps the biggest reason that the formula isn’t widely accepted is that is seems to have materialized out of nowhere. In fact, no one seems to even know who Slovin is, or even if he existed at all.

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 (*cough*) posts on Ask.com, Wiki Answers and other “Answer” sites gave the following answers to the question “Who invented Slovin’s Formula”:

  1. Mark Slovin
  2. Michael Slovin
  3. 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
Ryan, T. (2013). Sample Size Determination and Power. John Wiley and Sons.
Yamane, Taro. (1967). Statistics: An Introductory Analysis, 2nd Edition, New York: Harper and Row.

CITE THIS AS:
Stephanie Glen. “Slovin’s Formula: What is it and When do I use it?” From StatisticsHowTo.com: Elementary Statistics for the rest of us! https://www.statisticshowto.com/probability-and-statistics/how-to-use-slovins-formula/

Sample Size and Calculation

What is ‘Sample Size’?

‘Sample size’ is a market research term used for defining the number of individuals included to conduct research. Researchers choose their sample based on demographics, such as age, gender, or physical location.

Samples can be vague or specific. For example, you may want to know what people within the 18-25 age range think of your product. Or, you may only require your sample to live in Nigeria, which gives you a wide range of the population. The total number of individuals in a particular sample is the sample size.

Why do you need to determine sample size?

Let’s say you are a market researcher in Nigeria and want to send out a survey or questionnaire. The purpose of the survey is to understand your audience’s feelings toward a new cell phone you are about to launch. You want to know what people in Nigeria think about the new product to predict the phone’s success or failure before launch.

Hypothetically, you choose the population of Lagos, which is 17,552,940. You use a sample size determination formula to select a sample of 400 individuals that fit into the consumer panel requirement. You can use the responses to help you determine how your audience will react to the new product.

However, knowing how to determine a sample size requires more than just throwing your survey at as many people as you can. If your sample size is too big, it could waste resources, time, and money. A sample size that’s too small doesn’t allow you to gain maximum insights, leading to inconclusive results.

What are the terms used around the sample size?

Before we jump into sample size determination, let’s take a look at the terms you should know:

  1. Population Size: Population size is how many people fit your demographic. For example, you want to get information on doctors residing in South West, Nigeria. Your population size is the total number of doctors in South West, Nigeria. Don’t worry! Your population size doesn’t always have to be that big. Smaller population sizes can still give you accurate results as long as you know who you’re trying to represent.
  2. Confidence Level: Confidence level tells you how sure you can be that your data is accurate. It is expressed as a percentage and aligned to the confidence interval. For example, if your confidence level is 95%, your results will most likely be 95% accurate.
  3. The Margin of Error (Confidence Interval): When it comes to surveys, there’s no way to be 100% accurate. Confidence intervals tell you how far off from the population means you’re willing to allow your data to fall. A margin of error describes how close you can reasonably expect a survey result to fall relative to the real population value. 
  4. The Sample Proportion: It is an estimated proportion of an attribute that is present in a population. It varies from sample to sample in a way that cannot be predicted with certainty. A proportion of 50% indicates a greater level of variability than that of 20% or 80%. This is because 20% and 80% indicate that a large majority do not or do, respectively, have the attribute of interest. By default, sample proportion denoted by p is considered as 0.5 because it indicates the maximum variability in a population. It is often used in determining a more conservative sample size.

OpenBook Yamane Calculator 

OpenBook Yamane Calculator (OY Calculator, 2022) is a free online calculator developed to help research students or other researchers from various fields, having problem in calculation using Taro Yamane Formula manually, determine sample size accurately. To use the calculator, click here.