Cochran sample size calculation formula pdf

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Cochran sample size calculation formula pdf

Sample Size Formula (Table of Contents) Sample Size Formula The sample size is the most important term used in statistics. It is a part or percentage you choose out of a population for a survey or experiment or opinions or behavior you concern about. It is important to choose the most appropriate sample size because very less sample size will give you only inappropriate results and very larger sample size leads to wastage of time, money, resources, etc. And when you have a larger or smaller population, on which basis one can carry out the survey. For this, the survey is done for a set of a random sample. Cochran's formula is the most appropriate formula for finding the sample size manually. To use this formula, the desired level of precision, the population size should be known. The formula for the sample size can be written mathematically as follows: When you want to identify the sample size for a larger population one can use the following formula. S = (Z2 * P * Q) / E2 When you want to identify the sample size for a smaller population, the above formula can be modified like below. Ssmall = S / (1 + ((S ? 1) / N)) Let's take an example to understand the calculation of Sample Size in a better manner. Sample Size Formula ? Example #1 Assume GRE score is out for a Brand X coaching center for the 1000 students. The Score achieved is 3002 and the mean is found to be 1480. It has a standard deviation of 480. You expect the Margin of Error to be 80%. The proportion is set to be 0.8. Calculate Sample Size using the information: Solution: Z ? Score is calculated using the formula given below Z = (X ? M) / Z ? Score = (3002 ? 1480) / 480 Z ? Score = 3.17 Sample Size is calculated using the formula given below S = (Z2 * P * Q) / E2 Sample Size = (3.172 * 0.8 * 0.2) / (80%)2 Sample Size = 2.51 For this data set the appropriate Sample size is 2.51 Sample Size Formula ? Example #2 Assume a hill station X has a total number of 52 hotels. We need to find how many hotels provide breakfast in X. Half of the hotel may render breakfast service for the customers hence let us take P as 0.5. The Confidence level is 95% and the Margin of Error also consider as 85%. Calculate Sample Size using the information: Assuming this is the normal distribution let us find the Z value from the Z table. For 95 % of the confidence value, the Z value will be 1.96 per the normal table. Z = 1.96. Solution: For Large Population Sample Size is calculated using the formula given below S = (Z2 * P * Q) / E2 Sample Size =(1.962 * 0.5 * 0.5) / (85%)2 Sample Size = 1.33 For Small Population Sample Size is calculated using the formula given below Ssmall = S / (1 + ((S ? 1) / N)) Sample Size = 1.33 / (1 + ((1.33 ? 1) / 52)) Sample Size = 1.32 For this data set the appropriate Sample size is 1.32 Explanation Step 1: Note down value. Z value can be called a Z score or Standard Score value. It is the number of the standard deviation a mean data point of a population has. That is, say you have a particular population size and it has some mean which is a data point. So Z score is the total number of standard deviations it has before and after that mean data point. Generally, you can note this value from the Z table. The Z score has some basic formula too. Z = (X ? M) / Here X is the total number of population and M is the mean of the population and is the standard deviation. Assume you have a normally distributed data set of 80 and mean of the data set is 50 and a standard deviation of 15. Now, Z = (80- 50)/15 = 2. This Z score tells you the number of standard deviation your data set has above from the mean data point. Here it has 2 standard deviations above its mean. Step 2: Note down the value of P. P is nothing but the Proportion of the population. Step 3: Note down the value of E. E is Margin of Error which is a % value that tells how much you can wait for your results for the reflection of the end results or opinions from the overall population. The smaller the E value, the appropriate Sample size one can yield out of this formula. Step 4: Find out the value of Q. Q = 1 ? P. Step 5: Finally note down the value of N. This is the overall population size or the number of people on what you want to do your research. Step 6: Now if you have a larger population you can apply the noted values in the given formula. S = (Z2 * P * Q) / E2 Step 7: Now if you have a smaller population you can apply the noted values in the below formula. Ssmall is simply the sample size for the small size of the population. Ssmall = S / (1 + ((S ? 1) / N)) Relevance and Use of Sample Size Formula Any business field you take, how it goes on live and how much response it gets from the customers and how good or bad it is compared to the other similar things in the market everything should be estimated often in order to improve the performance of any business and to increase its capital and revenue. In that case, when one wants to perform any surveys or research, not the whole amount of data can be tested. Say, for example, a survey for millions of people at a time is time consuming and money wastage. Taking 1 out of millions will not yield you correct result too hence leading to negative results which is a Type II error. Hence for a chosen percentage amount of the whole population, the survey will be carried out. This part of the population will be taken as a random sample. Sample Size Formula Calculator You can use the following Sample Size Calculator S = Sample Size Formula in Excel (With Excel Template) Here we will do the example of the Sample Size Formula. It is very easy and simple. Below are the two different sets of data. Calculate the sample size using the below information. In the excel template, for 2 different sets of data, we have found the sample size. For the first set, manually we found the Z value since the total value, mean value and standard deviation are given. For the second set, a directly Z score is given for 85 % of confidence level. Since the total population size is small, Ssmall is also found for the appropriate sample size value. For Large Population Sample Size is calculated using the formula given below S = (Z2 * P * Q) / E2 For Set 1 Sample Size = (3.232 * 0.7 * 0.3) / (95%)2 Sample Size = 2.43 For Set 2 Sample Size = (1.962 * 0.6 * 0.4) / (88%)2 Sample Size = 1.19 For Small Population Sample Size is calculated using the formula given below Ssmall = S / (1 + ((S ? 1) / N)) For Set 2 Sample Size = 1.19 / (1 + ((1.19 ? 1) / 38)) Sample Size = 1.185 Recommended Articles This has been a guide to Sample Size Formula. Here we discuss how to calculate Sample Size along with practical examples. We also provide a Sample Size calculator with a downloadable excel template. You may also look at the following articles to learn more ? Contents: What is a "Sample Size"? A sample size is a part of the population chosen for a survey or experiment. For example, you might take a survey of dog owner's brand preferences. You won't want to survey all the millions of dog owners in the country (either because it's too expensive or time consuming), so you take a sample size. That may be several thousand owners. The sample size is a representation of all dog owner's brand preferences. If you choose your sample wisely, it will be a good representation. When Error can Creep in When you only survey a small sample of the population, uncertainty creeps in to your statistics. If you can only survey a certain percentage of the true population, you can never be 100% sure that your statistics are a complete and accurate representation of the population. This uncertainty is called sampling error and is usually measured by a confidence interval. For example, you might state that your results are at a 90% confidence level. That means if you were to repeat your survey over and over, 90% of the time your would get the same results. A census is where every member of a population is surveyed, not just a sample. Back to top How to Find a Sample Size in Statistics See also: Determining a sample size in one picture. Watch the video or read on below (also check out the live tutors at ; Your first 30 minutes are free!): Please accept statistics, marketing cookies to watch this video. Links mentioned in the video: 95% CI Table . A sample is a percentage of the total population in statistics. You can use the data from a sample to make inferences about a population as a whole. For example, the standard deviation of a sample can be used to approximate the standard deviation of a population. Finding a sample size can be one of the most challenging tasks in statistics and depends upon many factors including the size of your original population. General Tips Step 1: Conduct a census if you have a small population. A "small" population will depend on your budget and time constraints. For example, it may take a day to take a census of a student body at a small private university of 1,000 students but you may not have the time to survey 10,000 students at a large state university. Step 2: Use a sample size from a similar study. Chances are, your type of study has already been undertaken by someone else. You'll need access to academic databases to search for a study (usually your school or college will have access). A pitfall: you'll be relying on someone else correctly calculating the sample size. Any errors they have made in their calculations will transfer over to your study. Step 3: Use a table to find your sample size. If you have a fairly generic study, then there is probably a table for it. For example, if you have a clinical study, you may be able to use a table published in Machin et. al's Sample Size Tables for Clinical Studies, Third Edition. Step 4: Use a sample size calculator. Various calculators are available online, some simple, some more complex and specialized. For example, this calculator is for group- or cluster-randomized trials (GRTs). Step 5: Use a formula. There are many different formulas you can use, depending on what you know (or don't know) about your population. If you know some parameters about your population (like a known standard deviation), you can use the techniques below. If you don't know much about your population, use Slovin's formula.. Back to top Cochran's Sample Size Formula The Cochran formula allows you to calculate an ideal sample size given a desired level of precision, desired confidence level, and the estimated proportion of the attribute present in the population. Cochran's formula is considered especially appropriate in situations with large populations. A sample of any given size provides more information about a smaller population than a larger one, so there's a `correction' through which the number given by Cochran's formula can be reduced if the whole population is relatively small. The Cochran formula is: Where: e is the desired level of precision (i.e. the margin of error), p is the (estimated) proportion of the population which has the attribute in question, q is 1 ? p. The zvalue is found in a Z table. Cochran's Formula Example Suppose we are doing a study on the inhabitants of a large town, and want to find out how many households serve breakfast in the mornings. We don't have much information on the subject to begin with, so we're going to assume that half of the families serve breakfast: this gives us maximum variability. So p = 0.5. Now let's say we want 95% confidence, and at least 5 percent--plus or minus--precision. A 95 % confidence level gives us Z values of 1.96, per the normal tables, so we get ((1.96)2 (0.5) (0.5)) / (0.05)2 = 385. So a random sample of 385 households in our target population should be enough to give us the confidence levels we need. Modification for the Cochran Formula for Sample Size Calculation In Smaller Populations If the population we're studying is small, we can modify the sample size we calculated in the above formula by using this equation: Here n0 is Cochran's sample size recommendation, N is the population size, and n is the new, adjusted sample size. In our earlier example, if there were just 1000 households in the target population, we would calculate 385 / (1+( 384 / 1000 )) = 278 So for this smaller population, all we need are 278 households in our sample; a substantially smaller sample size. Back to top How to Find a Sample Size Given a Confidence Interval and Width (unknown population standard deviation) Part two shows you how to find a sample size for a given confidence interval and width (e.g. 95% interval, 6% wide) for an unknown population standard deviation. Example question: 41% of Jacksonville residents said that they had been in a hurricane. How many adults should be surveyed to estimate the true proportion of adults who have been in a hurricane, with a 95% confidence interval 6% wide? Step 1: Using the data given in the question, figure out the following variables: za/2: Divide the confidence interval by two, and look that area up in the z-table: .95 / 2 = 0.475 The closest z-score for 0.475 is 1.96. E (margin of error): Divide the given width by 2. 6% / 2 = 0.06 / 2 = 0.03 : use the given percentage. 41% = 0.41. If you aren't given phat, use 50%. : subtract from 1. 1 ? 0.41 = 0.59 Step 2:Multiply by . Set this number aside for a moment. 0.41 ? 0.59 = 0.2419 Step 3: Divide Za/2 by E. 1.96 / .03 = 65.3333333 Step 4: Square Step 3: 65.3333333 ? 65.3333333 = 4268.44444 Step 5: Multiply Step 2 by Step 4: 0.2419 ? 4268.44444 = 1,032.53671 = 1,033 people to survey. Back to top How to Find a Sample Size Given a Confidence Interval and Width (known population standard deviation) Part 3 shows you how to determine the appropriate sample size for a given confidence interval and width, given that you know the population standard deviation. Sample question: Suppose we want to know the average age of an Florida State College student, plus or minus 0.5 years. We'd like to be 99% confident about our result. From a previous study, we know that the standard deviation for the population is 2.9. Step 1: Find z a/2 by dividing the confidence interval by two, and looking that area up in the z-table: .99/2 = 0.495. The closest z-score for 0.495 is 2.58. Step 2: Multiply step 1 by the standard deviation. 2.58 * 2.9 = 7.482 Step 3: Divide Step 2 by the margin of error. Our margin of error (from the question), is 0.5. 7.482/0.5 = 14.96 Step 4: Square Step 3. 14.96 * 14.96 = 223.8016 That's it! Like the explanation? Check out our statistics how-to book, with a how-to for every elementary statistics problem type. Back to top How to find a sample size in Excel. Watch the video or read the steps below: Please accept statistics, marketing cookies to watch this video. How to use Excel Sampling to find a Sample If you have a set of data and you know your sample size, you can use Excel's Data Analysis toolpak to select either a periodic sample or a random sample. A random sample is just that -- randomly selected from your data set. A periodic sample (also called a systematic sample) is where Excel chooses the nth data item to include in your sample. For example, if you wanted to choose every 5th number from the following list: 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, Excel would return 8 and 13 (the 5th and 10th numbers in order). If you don't know what sample size you need, calculate it before using the Data Analysis tool (using the methods outlined at the top of this article). The Data Analysis tool can help you extract a sample, but it can't help you decide on the size. Why? There are many "human" factors that go into selecting a sample size including budget, prior research (you can use a sample size from previous research) and tables constructed from previous research. How to use Excel Sampling to find a Sample: Steps Step 1: Enter your data items into Excel. You can enter your data into rows or columns. Ensure the rows and columns are even; for example, enter data into column A to cell 12 and column B to cell 12. Step 2: Click "Data" and then click "Data Analysis." If you don't see Data Analysis on your toolbar, load the Data Analysis Toolpak. Step 3: Click "Sampling" and then click "OK." Data entered into a worksheet for Excel sampling: the rows and columns are even. Step 4: Click in the Input Range box and then select your entire data set. Step 5: Click either "Periodic Sampling" or "Random Sampling." If you choose periodic, enter the nth number (i.e. every 5) and if you choose random sampling, enter the sample size. Step 6: Choose an output range. For example, click the "New Worksheet" button and Excel will return the sample in a new worksheet. Step 7: Click "OK." That's it! Check out our YouTube channel for more Excel tips and help! Back to top Related Articles Ways to Reduce Sample Size. Effective Sample Size.. Unequal sample size. References Bartlett, J. et al. (2001). Organizational Research: Determining Appropriate Sample Size in Survey Research. Retrieved January 15, 2018 from: . Evans, M.; Hastings, N.; and Peacock, B. Statistical Distributions, 3rd ed. New York: Wiley, 2000. Israel, G. (n.d.) Determining Sample Size. University of Florida IFAS Extension. Article posted on Tarleton State University website. Retrieved January 13, 2018 from . ------------------------------------------------------------------------------Need help with a homework or test question? With Chegg Study, you can get step-by-step solutions to your questions from an expert in the field. Your first 30 minutes with a Chegg tutor is free! Comments? Need to post a correction? Please post a comment on our Facebook page.

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