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My organization recently did a morale survey. Results were given as t-scores. I had never seen that before, so I embarked on a web journey. I found a very similar survey. My question isn't about the validity of the approach; rather, it is about how to meaningfully interpret a t-score (as opposed to a z-score).

The t-score is similar to the z-score except that the displacement from the mean is normalized by the sample standard deviation. I get uncomfortable about this because a standard deviation under a t-distributino is not as meaningful to me a standard distribution under a normal distribution. Table 1 at this stats info page shows that the area under a standard deviation is not constant (except when the t-distribution approaches a normal distribution). So what exactly does a t-score convey?

Glen_b
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StatSmartWannaB
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1 Answers1

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It's still "the number of standard deviations from the mean" but in terms of sample quantities rather than population mean and standard deviation. So the general sense of what a "+1" or a "-0.5" are pretty similar to the interpretation for a z-score, but it's "noisier" in a sense.

Many people still call it a z-score (even though it's no longer distributed as a Z even with a normally-distributed population).

Glen_b
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  • That's exactly my concern/confusion. We focus on t-distributions when samples are small, but that's exactly when the area under a standard deviation becomes "noisy", i.e., dependent on more than just the t-score, i.e., additionally dependent on degrees of freedom. In order for the t-score to be meaningful, we need to carry around tables such as Table 1 in my original post. – StatSmartWannaB Dec 30 '15 at 21:42
  • That really depends on what you mean by "meaningful"; my answer describes the sense in which they can be interpreted. Why would you try to compute quantiles from either individual z-scores or individual t-scores? [If you need quantiles, one can find the sample quantiles immediately without any need for assuming normality of the original population.] – Glen_b Dec 30 '15 at 22:54
  • Agreed...I didn't have an exact definition of meaningful. The question was about what a t-score conveys, i.e., what meaning do other people derive from it. In my experience (admittedly, not broadly encompassing), the reason why you would want to know the number of standard deviations from the mean is to get a sense of how unlikely it is, i.e., the usual null hypothesis test. People tend to think about the area under the PDF, even though they may not have the exact number. They have mental anchor points at 1 and 2 sigmas out. – StatSmartWannaB Jan 01 '16 at 14:52
  • On the other hand, if you have reason to use a t-score instead of z-score, you're dealing with a family of PDFs, thus making the anchor points dependent on the degrees of freedom. So they are less meaningful in the sense described above without the DF figure. Hence, I wondered whether there was another reason for reporting t-scores. – StatSmartWannaB Jan 01 '16 at 14:52