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** OpenBook Sample Size Calculators**

In order to generalize from a random sample and avoid sampling errors or biases, a random sample needs to be of adequate size. What is adequate depends on several issues which often confuse people doing surveys for the first time. This is because what is important here is not the proportion of the research population that gets sampled, but the absolute size of the sample selected relative to the complexity of the population, the aims of the researcher and the kinds of statistical manipulation that will be used in data analysis (Taherdoost, 2016). While the larger the sample the lesser the likelihood that findings will be biased does hold, diminishing returns can quickly set in when samples get over a specific size which need to be balanced against the researcher’s resources (Gill et al., 2010).

To put it bluntly, larger sample sizes reduce sampling error but at a decreasing rate. Several statistical formulas or calculators are available for determining sample size.

Gill, J. Johnson, P. & Clark, M. (2010). *Research Methods for Managers*. Sage Publications

Taherdoost, H. (2016). *Sampling Methods in Research Methodology; How to Choose a Sampling Technique for Research*. International Journal of Advance Research in Management, 5(2), 18-27

**OpenBook Yamane Calculator (OY Calculator, 2022)**

**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 (1967) Formula is written as

**n = N / (1 + Ne ^{2}) **simplified and adjusted, to be more accurate, from Cochran’s (1963, 1975) Sample Size Formula:

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 |

By substituting for z = 1.96 and p = 0.5 in the simplified formula above, Taro Yamane Formula can be proved as follows:

**e** could be 0.10, **0.05** or 0.01. They are margin of errors, that can be tolerated in determining sample size, at confidence level of 90%, **95%** and 99% respectively. They are used in educational and social science research studies. The most commonly and widely used is **0.05.** The sample proportion ** p, **though varies, but by default is

**0.5**. If you are not familiar with confidence level, confidence interval or margin of error and sample proportion – the common terms in sample size and calculation, you can click here.

One of the advantages of using **OpenBook Yamane Calculator,** to accurately determine sample size, is when the total population is relatively large. Secondly, you don’t need to crack your brain of any complex formula and all its variables’ values especially if you don’t have relevant knowledge in statistics. Other advantage is that as large sample size reduces sampling error to validate research findings, there are always excess samples of 16 or 15 at total population of 300,000 and above, when compared with other Sample Size Calculators, which is enough to gather much more information or data, from the respondents, about a study. To prove this, the highest sample you would ever get using **OpenBook Yamane Calculator** from total population of 300,000 and above is 400 and the highest sample you would ever get using other Sample Size Calculators from also 300,000 total population and above is 384 or 385. So, there are always excess samples of 16 or 15 to get much more information about a study using **OpenBook Yamane Calculator.**

However, the population N is to be determined first from the study area. When the population is relatively large and the exact number is unknown, then 300,000 or more can be used because any sample size gotten cannot be greater than 400 or 385 (either Taro Yamane Formula at confidence interval of 0.05 or other Sample Size Formula at confidence level of 95%, confidence interval of 5% and sample proportion of 50%).

**To calculate the Sample Size n, using OY Calculator below: enter the Total Population N, then calculate by clicking on Calculate Button. To enter different Total Population N, click Reset Button.**

*Using OY Calculator, you may also change the default 0.05 in the margin of error e placeholder to your desired confidence interval by selecting either 0.10 or 0.01 as alternate scenario. The 300,000 in the right field of population N placeholder is to be used when the exact number is unknown at confidence interval of 0.05. The common usage of 0.05 confidence interval for a specific sample size result is to bring balance against the researcher’s resources relative to the complexity of the population. Statistically, large samples must be equal to or greater than 30 (Murray, 2009). As sample size is used to validate research findings, it must not be too small. If too small, it will not yield valid results. At the same time, if it’s too large, may be a waste of money and time.*

Murray, R. Spiegel et al (2009). *Probability and Statistics.* The McGraw-Hill Companies Inc.

The use of OpenBook Yamane Calculator can be referenced in your thesis or dissertation as:

OpenBook Yamane Calculator, 2022. *OpenBook Sample Size Calculators.* OpenBook Communications and Technologies, Nigeria. https://www.openbookpage.com/

**OpenBook Cochran Calculator (OC Calculator, 2022) and OpenBook Cochran Correction Calculator (OCC Calculator, 2024)**

(From Cochran’s Sample Size Formula without Adjustment or Modification of Z-score z, 1.96 at confidence interval or margin error e, 0.05 and sample proportion p, 0.5)

**To calculate the Sample Size n, using OC/OCC Calculator below: enter the Total Population N, then calculate by clicking on Calculate Button. To enter different Total Population N, click Reset Button.**

*Using OC Calculator, you may also change the default 1.96 and 0.05 in their respective placeholder to your desired confidence level of either 90% or 99% by selecting 1.65 or 2.58 and 0.10 or 0.01 respectively as alternate scenarios. The 300,000 in the right field of population N placeholder is to be used when the exact number is unknown at confidence interval of 0.05 and z-score value of 1.96. The common usage of 0.05 confidence interval and 1.96 z-score value for a specific sample size result is to bring balance against the researcher’s resources relative to the complexity of the population. Statistically, large samples must be equal to or greater than 30 (Murray, 2009). *

*As sample size is used to validate research findings, it must not be too small. If too small, it will not yield valid results. At the same time, if it’s too large, may be a waste of money and time.*

Murray, R. Spiegel et al (2009). *Probability and Statistics.* The McGraw-Hill Companies Inc.

**OpenBook Cochran Correction Calculator, 2024 (OCC Calculator, 2024)**

**Sample Size can also be calculated using Cochran Sample Size Formula with the application of FINITE POPULATION CORRECTIONs (FPCs) **

Cochran (1963, 1975) developed the equation to yield a representative sample for proportion of large sample.

**n _{0} = z² pq/e²**

which is valid where n_{0} is the sample size, z² is the abscissa of the normal curve that cuts off an area α at the tails (1 – α equals the desired confidence level is 95%), e is the desired level of precision, p is the estimated proportion of an attribute that is present in the population, and q is 1-p. The value for z is found in statistical tables which contain the area under the normal curve.

* Finite Population Correction for Proportions (If small population). *If the population is small then the sample size can be reduced slightly. This is because a given sample size provides proportionately more information for a small population than for a large population. The sample size (n

_{0}) can be adjusted as

**n = n _{0} / [1 + {(n_{0} – 1) / N}]**

where n is the sample size and N is the population size

Cochran (1977) introduced Finite Population Corrections (FPCs) based on the sampling fraction **f = n/N** where n is the sample size and N is the finite population size. In practice, FPCs may be ignored if f does not exceed 5%. Larger samples relative to their populations require FPCs because ignoring large sampling fractions results in biased standard errors (Cochran, 1977). Applied researchers should identify their target populations, examine their sampling fraction, and consider using FPCs because applying FPCs yields more accurate inferences for finite populations.

The use of OpenBook Cochran Calculator can be referenced in your thesis or dissertation as:

OpenBook Cochran Calculator, 2022. *OpenBook Sample Size Calculators. *OpenBook Communications and Technologies, Nigeria. https://www.openbookpage.com/

**OpenBook Krejcie-Morgan Calculator (OK-M Calculator, 2024)**

(From Krejcie and Morgan Sample Size Formula without Adjustment or Modification of Chi-Square x², 3.841 at confidence interval or margin error e, 0.05 and sample proportion p, 0.5)

Krejcie and Morgan (1970) Formula was introduced as an alternative formula in computing sample size for categorical data. The formula is written as:

**n = x²Np(1-p)/e²(N-1)+x²p(1-p)**

Where:

- n = Number of Samples,
- N = Total Population,
- e = Error Tolerance (level) or Margin of Error, 0.05
- p = Sample Proportion, 0.5
- x² = Chi-Square value found in Chi-Square Table, 3.841

The table below is Chi-Square Table for most use confidence level or confidence interval.

x²_{0.}_{90} | x²_{0}_{.95} | x²_{0}_{.99} | |

Degree of Freedom | 10% (0.10) | 5% (0.05) | 1% (0.01) |

1 | 2.706 | 3.841 | 6.635 |

**OpenBook Krejcie-Morgan Calculator** can be used as an alternative tool to confirm the cases of population and sample size not listed in Krejcie and Morgan Sample Size Table (1970), a well known table for sample size determination among behavioural and social science researchers.

**To calculate the Sample Size n, using OK-M Calculator below: enter the Total Population N, then calculate by clicking on Calculate Button. To enter different Total Population N, click Reset Button.**

*Using OK-M Calculator, you may also change the default 3.841 and 0.05 in their respective placeholder to your desired confidence level of either 90% or 99% by selecting 2.706 or 6.635 and 0.10 or 0.01 respectively as alternate scenarios. The 300,000 in the right field of population N placeholder is to be used when the exact number is unknown at confidence interval of 0.05 and chi-square x² value of 3.841. The common usage of 0.05 confidence interval and 3.841 chi-square x² value for a specific sample size result is to bring balance against the researcher’s resources relative to the complexity of the population. Statistically, large samples must be equal to or greater than 30 (Murray, 2009). *

*As sample size is used to validate research findings, it must not be too small. If too small, it will not yield valid results. At the same time, if it’s too large, may be a waste of money and time.*

Murray, R. Spiegel et al (2009). *Probability and Statistics.* The McGraw-Hill Companies Inc

The use of OpenBook Krejcie-Morgan Calculator can be referenced in your thesis or dissertation as:

OpenBook Krejcie-Morgan Calculator, 2024. *OpenBook Sample Size Calculators. *OpenBook Communications and Technologies, Nigeria. https://www.openbookpage.com/

**Gotten Your Sample Size from OY/OC/OK-M Calculator, What’s Next?**

Now that you have gotten your **sample size, **from any of the calculators above, for the number of copies of your questionnaires, to be administered to your respondents, let **OpenBook** have the Questionnaire Reliability Checking and SPSS Data Analysis done for you. SPSS Data Analysis using Descriptive and Inferential Statistics.

**Questionnaire Reliability Checking (Cronbach Alpha) is EXCLUSIVELY FREE! **Distance is not a barrier; you can upload your already ticked FORM/QUESTIONNAIRE to **whatsapp number: 2348028999115.**

Service Price of SSPS Data Analysis ranges from 15,000 to 70,000 in Nigeria Naira (NGN) and 100 to 470 in US Dollar (USD).

BSc/BA Research Data Analysis: NGN15,000/USD100.

MSc/MA Research Data Analysis: NGN30,000/USD200.

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You can make payment by clicking here.

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