t-scores versus z-scores (College Board AP® Statistics): Study Guide

Naomi C

Written by: Naomi C

Reviewed by: Dan Finlay

Updated on

t-scores versus z-scores

How do I know when to use a t-score or a z-score?

  • It is important to be able to recognise which critical value to use for the inference test that you are conducting

    • You will need to consider the population distribution, the population parameters given and the sample size

What happens if the population is normally distributed?

  • If a population is normally distributed and the population variance is known

    • then you should use a z-score

  • If a population is normally distributed but the population variance is unknown

    • then you should use a t-score when the sample size is small, n less than 30

  • In practice, if the population variance is unknown but the sample size is large, n greater than 30

    • t-scores can be used

    • but z-scores can also be used

      • especially as the t-scores in the table run out for high values of n

    • If a z-score is used, then the sample standard deviation, s, is used in place of the population standard deviation, sigma

What happens if the population is not normally distributed?

  • If a population is not normally distributed but n greater or equal than 30

    • then, due to the Central Limit theorem, you can use z-scores

    • Use the population standard deviation, sigma, if it is given

      • or the sample standard deviation, s, if sigma is not given

  • If a population is not normally distributed and n less than 30

    • then there is no statistical procedure you can use!

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Naomi C

Author: Naomi C

Expertise: Maths Content Creator

Naomi graduated from Durham University in 2007 with a Masters degree in Civil Engineering. She has taught Mathematics in the UK, Malaysia and Switzerland covering GCSE, IGCSE, A-Level and IB. She particularly enjoys applying Mathematics to real life and endeavours to bring creativity to the content she creates.

Dan Finlay

Reviewer: Dan Finlay

Expertise: Maths Subject Lead

Dan graduated from the University of Oxford with a First class degree in mathematics. As well as teaching maths for over 8 years, Dan has marked a range of exams for Edexcel, tutored students and taught A Level Accounting. Dan has a keen interest in statistics and probability and their real-life applications.

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