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Both Z-scores and T-scores are measures of standardization used in statistical analysis, particularly when comparing data points to a population or sample mean. They help determine how far or different a data point is from the mean.
A Z-score, also known as a standard score, quantifies the number of standard deviations a data point is from the mean of a distribution. It is used when the population standard deviation is known and the data is assumed to follow a normal distribution.
Z=σ(X−μ)
If an IQ test has a mean (μ) of 100 and a standard deviation (σ) of 15, and an individual scores 120, their Z-score is calculated as: Z=15(120−100)=1.33 This indicates the individual scored 1.33 standard deviations above the mean.
A T-score is used when the sample size is small (typically less than 30) and the population standard deviation is unknown. It measures how many standard errors the sample mean is from the population mean.
T=ns(Xˉ−μ)
If a sample of 10 students has a mean score (Xˉ) of 82, the population mean (μ) is 78, and the sample standard deviation (s) is 5, the T-score is: T=105(82−78)=2.53 This means the sample mean is 2.53 standard errors above the population mean.
Understanding when and how to use Z-scores and T-scores is essential for accurate statistical analysis and hypothesis testing. They provide a means to assess the relative standing of data points and sample means in a distribution.