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The right sample size - How we address this dilemma? Sample size calculation is the act of choosing the number of observations to include in a statistical sample. The sample size is an important feature of any empirical study in which the goal is to make inferences about a population from a sample. In practice, the sample size used in a study is determined based on the expense of data collection, and the need to have sufficient statistical power. In complicated studies there may be several different sample sizes involved in the study: for example, in as survey sampling involving stratified sampling there would be different sample sizes for each population. In a census, data are collected on the entire population; hence the sample size is equal to the population size. In experimental design, where a study may be divided into different treatment groups, there may be different sample sizes for each group. Sample size is statistically determined by using the following: 1. Degree of variation - s (sigma) or Standard Deviation for Continuous data & p or Proportion Defective in case of Discrete data. Higher the degree of variation, higher is the sample size required. 2. a (Alpha). Za = 1 - a = Confidence Level is the Probability of correctly accepting the null hypothesis (detecting no change when there is none). Typically a is set at .05 or 5%. Confidence Level of 95% means that if your audit on the sample shows a result of 70%, there is 95% chance that the audit results of the entire population will be 70%. Higher the a, lesser the sample size as one is willing to take more risk. 3. D (Delta). Margin of Error or Confidence interval – it is set as per how close you want to be to the Target. For example, you have rolled out a survey and have used a confidence interval of 4. If 47% percent of your sample says “OK” , you can be "sure" that if you had asked the question of the entire relevant population, between 43% (47-4) and 51% (47+4) would have also said “OK”. Higher the margin of error, lesser the sample size as one has settled to be less precise. 4. N or Population size (if known) – is used to “correct” the sample size for the known finite population. The following are used for sampling for hypothesis testing only. 5. b (Beta). 1- b = Power is the Probability of correctly accepting the alternative hypothesis (detecting a change when there is one). Typically b is set at .1 or 10%. Higher the b, lesser the sample size as one is willing to take more risk. 6. Whether One sided or 2-sided distribution; in other words, whether Alternate hypothesis is > or < instead of ≠ (does not equal). Sample Size is higher for one sided or “≠” alternate hypothesis testing. The formulae for calculating sample size are as follows: Continuous data formula: n = ((Za/2 * s)/D) ^2 Discrete data formula: n = ((Za/2)/D) ^2 * p (1-p) In case N (Population size) is known, multiply n (sample size) by (N-n) / (N-1) to determine corrected sample size. Now, all of the above is great & nice to know, but here the question is how to determine the right sample size. The Basic question: How can I make sure that the data I get really represents the population? • Can samples fool me? How will I know that I can trust my samples? • How many samples do I need? • Why so many? (How can we get away with so few?) As a quality professional, I have faced the above questions several times and frankly, even asked myself the same sometimes. A honest manager of a service center once asked me “Why even determine sample size – can't I just plan it as per my team’s availability?” The answer is “Absolutely not”. The right sampling strategy goes a long way in improving processes by reducing the cost of poor quality. Cost of poor quality is nothing but rework, callbacks, customer escalations, impact to downstream processing time & quality. Firstly, the Four Basic Sampling Strategies • Random Sample – Population Studies. Each unit has an equal probability of being selected in a sample. • Stratified Random Sample – Population Studies. Randomly sample within a stratified category or group. Sample sizes for each group are generally proportional to the relative size of the group. • Systematic Sampling – Population or Process Studies. Sample every nth unit. For example, collecting every 4th unit. • Rational Subgroup – Process Studies. Each unit is collected at point “A” in a process every nth hour. Usually multiple sequential units are collected. In a service center, we typically end up using the random sampling strategy, but it isn’t always the best one. For example, there are 10 processors / call centre execs in a process & out of them, there are 3 who make the maximum errors, let’s say their contribution to overall errors is between 80-90%. Shouldn’t I be using stratified random sampling in this case rather than auditing 10% for everyone. Similarly, the team works on 15 activities/ worktypes/ call types of varying complexities, so shouldn’t the sampling strategy be adjusted to allow more auditing for the more complex activities/ worktypes/ call types which have higher chances of error. Smaller sample sizes vs. Larger sample sizes: • Less Cost / Higher costs • Quicker data collection / Longer time to get data • Wider confidence intervals and/or more risk of missing the population parameter / Tighter confidence intervals and/or less risk of missing the population parameter. To sum it up, Sample size for determining population mean or proportion depends on: • What level of risk you’re willing to take? If your defects in your output have an impact on loss of human life or property, severely impacts the customer or has a compliance implication, the confidence level needs to be very high. • What size difference you want to detect? If your accuracy target is 99%, you need to take a margin of error of less than 1%. • How much variation is in the population? Consider that your team is performing at 99.5% accuracy vs. a target of 90%, in such a case you need lesser samples because you are almost sure that the team (population data) will meet the target. • Before collecting data, you should think about the sampling strategy and sample size requirements to ensure that you have an appropriate amount of data for drawing conclusions.