When to use a z score versus a t score ?

Question

How do I determine whether to use a z score versus a t score ?

Answer 1

So, when deciding whether to use a z or t score, one needs to understand the fundamental similarities and differences between the two types of tests. Both z scores and t scores are used in hypothesis testing to determine how far a sample mean deviates from a population mean. They are also used to make decisions about the significance of results in inferential statistics. However, choosing between the two depends on certain conditions related to the data.

Z Score (Z Statistic):

A z score is used when you know the population standard deviation (σ) and when the sample size is relatively large (typically n≥ 30). The z distribution assumes a normal distribution and is based on the population parameters.

When to Use a Z Score:

1. Known Population Standard Deviation (σ): Use a z score when the population standard deviation is known.

2. Large Sample Size (n ≥ 30): For large samples, the sampling distribution of the sample mean tends to be approximately normal, even if the population distribution is not perfectly normal (Central Limit Theorem).

3. Normally Distributed Population: If the population from which the sample is drawn is normally distributed, and the standard deviation is known, a z score is appropriate regardless of sample size.

T Score (T Statistic):

A t score is used when the population standard deviation is unknown and the sample size is small (typically n <30). The t distribution is broader and has heavier tails than the normal distribution, which accounts for the additional uncertainty in estimating the population standard deviation from the sample.

When to Use a T Score:

1. Unknown Population Standard Deviation (σ): Use a t score when you do not know the population standard deviation and must estimate it using the sample standard deviation (s).

2. Small Sample Size (n < 30): For smaller samples, the t distribution adjusts for the additional variability by having thicker tails.

3. Normally Distributed Population: The t score is most accurate when the population from which the sample is drawn is normally distributed, especially with smaller sample sizes.

Key Differences:

1. Known vs. Unknown Population Standard Deviation:

- Z Score: Use when σ is known.

- T Score: Use when σ is unknown and needs to be estimated with the sample standard deviation (s).

2. Sample Size:

- Z Score: Preferable for large samples (n ≥ 30).

- T Score: Necessary for small samples (n < 30).

3. Distribution:

- Z Score: Follows the standard normal distribution (mean = 0, standard deviation = 1).

- T Score: Follows the t distribution, which becomes closer to the normal distribution as the sample size increases. For very large samples, the t distribution approximates the normal distribution.

Summary:

- Use a z score when you know the population standard deviation and have a large sample size.

- Use a t score when the population standard deviation is unknown and the sample size is small.

I hope you found this answer to be helpful. Should you have any additional questions, please let me know. Thank you! Take care!

Dr. Christal-Joy Turner

Answer 2

How do you know when to use a z score (z test) or a t score (t test) in statistical data analysis?

Let’s first define what a z score and t score (calculated from the z test and t test respectively) represent:

1. Given an observation selected from a sample data distribution, the z score or t score tells you where that observation falls in relation to the mean.
2. An example would be an SAT score observed in a sample taken from the national population of scores. Where does the score fall in relation to the mean of the sample distribution.

So how do you know when to use a z test to get the z score versus a t test to get the t score?

1. if the population size is ≥ 30
2. if the standard deviation is known, use the z test.
3. If the standard deviation is unknown, use the t test.
4. If the sample < 30 use the t test.
5. The caveat is if you know that the population the sample is taken from is normally distributed, and the population standard deviation is known, you can also use the z test. However the t test will be more accurate when the sample size is < 30. As the sample size gets larger (≥ 30) the two tests converge to the same value.