![]() The z-score of the sample mean is calculated as follows: Suppose, a random sample of 100 observations was taken from a population having mean μ = 70 and standard error (SE) of the mean is 15. Let’s take an example to understand z-score calculation better with sampling distribution (which will be used in the hypothesis testing technique known as Z-test). Where standard deviation of the sampling distribution is denoted as σ and the sample size is n. The standard error of the sample mean can be calculated as the following: SE is standard error (SE) of the sample mean. The population mean is denoted by μ and the sample mean is denoted by x̄. Μ is the mean of the observations in the population The formula for z-statistics in Z-test is the following: If the value of Z-statistics falls in the rejection region, it indicates that there is sufficient evidence to reject the null hypothesis. The value of Z-statistics is compared with the critical value of Z-statistics which is determined from the Z-table. The value of Z-statistics is used to determine whether to reject the null hypothesis or otherwise. Z-test for sampling distribution is used to determine whether the sample mean is statistically different from the population mean. ![]() Recall that the sampling distribution is defined as the distribution of all the possible samples that could be drawn from a population. Note that the sampling distribution is used in the hypothesis testing technique known as Z-test. When considering the sampling distribution, Z-score or Z-statistics is defined as the number of standard deviations between the sample mean and the population mean (mean of the sampling distribution). Z-score or Z-statistics for Sampling distribution to perform Z-test In the same way, Z=-1 represents that the observation is 1 standard deviation away from mean in the negative direction. Note that Z = +1 represents that the observation is 1 standard deviation away from mean in the positive direction. The figure below represents different values of Z-scores. When the mean and standard deviation of a data set are known, it is easy to convert them into Z-score for that particular sample or population. The process of converting raw observations into Z-score is also called as standardization or normalization. Observation 45 is -1.5 standard deviation away from the mean 90. The observation X = 45 will have Z-scores as follows: Suppose, the mean of data points in a sample is 90 and the standard deviation is 30. Let’s take an example to understand z-score calculation better. Σ is the standard deviation of the observations in the sample X̄ is the mean of the observations in the sample Alternatively, when defined for population, Z-score can be used to measure the number of standard deviations by which the data points differ from the population mean. When considering a sample of data, Z-score is used to measure the number of standard deviations by which the data points in the sample differ from the mean. The hypothesis tests where Z-statistics get used is one-sample Z-test and two-samples Z-test.Īdditionally, z-score at different confidence intervals can be used to estimate the population mean based on a given sample or difference in the population means based on two different samples. Z-score for sampling distributions: In case of sampling distribution, the goal is to perform Z-test and find the value of Z-score or Z-statistics for hypothesis testing and rejecting the null hypothesis or otherwise.Z-score for sample data: In case of sample data, the objective is to find number of standard deviations an observation is away from the sample mean.The Z-score formula differs when considering the sample data, or, when considering sampling distributions with an end goal of whether finding the deviation from mean, or, performing hypothesis testing respectively. Z-score / Z-statistics Concepts & Formula Z-score for Estimating Population Mean/Proportion.Z-score or Z-statistics for Sampling distribution to perform Z-test.Z-score / Z-statistics Concepts & Formula.
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