For example, I have a normal distribution of grades with mean = 30, SD = 2 and my observation = 60. Then the z-score would be 15. I would interpret it as 60 is 15 SD away from the mean. However, I can also interpret the z-score as 15 grades away from the mean per SD. This line does not make any sense to me. I am not able to bridge the gap between 60 is 15 SD away from the mean and 15 grades away from the mean per SD. It is like a mental jump that I am not able to make.
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$\begingroup$ Not sure I understand z-score is '15 grades away from the mean'. Values on a standardized scale no longer 'grades' in the original sense. (Standard scores measure in terms of standard deviations.) $\endgroup$– BruceETCommented Jul 1, 2020 at 7:16
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$\begingroup$ I am also confused by the claim about grades. Could you provide more context? Was this a quote from a larger passage? That might help clarify the meaning. $\endgroup$– Sean RaleighCommented Jul 1, 2020 at 7:20
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$\begingroup$ @BruceET What I meant was that for example if we have 5 dollars to be divided among 2 people, we would divide 5/2 and the result would be 2.5, i.e. 2.5 dollars per person. Dividing it gave us how many dollars would go to one person. Similarly, 15 grades away from the mean per SD came from (distance of observation from mean)/SD, the way I thought, it would mean the distance of the observation from mean per SD. $\endgroup$– user290096Commented Jul 1, 2020 at 7:20
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$\begingroup$ @SeanRaleigh Its not from any text, I was trying to understand z-scores and could not grasp my head around the interpretation. $\endgroup$– user290096Commented Jul 1, 2020 at 7:27
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1 Answer
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I might know what's going on. The Z-score is defined as $$ z = \frac{x-\mu}{\sigma} $$ where $\mu$ is the mean and $\sigma$ is the standard deviation(SD).
In your example, $\mu = 30$, $x = 60$ and $\sigma = 2$.
Using the formula directly, the Z-score of your observation is
$$z = \frac{x-\mu}{\sigma} = \frac{60-30}{2} = 15$$
When you solve the equation for $x$, you get the formula, $$x = \mu + 15\sigma$$
Directly interpreting this formula, you can say that the score, $x$ is 15 SD away from the mean.
On the other hand, you can expand the Z-score formula in this way: $$ z = \frac{x}{\sigma} - \frac{\mu}{\sigma} $$
By rewriting the formula with the calculated Z-score, you get
$$ \begin{align} \frac{x}{\sigma} & = z + \frac{\mu}{\sigma} = 15 +\frac{\mu}{\sigma} \end{align} $$
With this formula, you can describe the score as the score per SD is 15 away from the mean per SD. In this version, all the quantities are unitless.
They are different ways of looking at the same Z-score.
answered Jul 1, 2020 at 7:36
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$\begingroup$ Thank you so much for this! $\endgroup$– user290096Commented Jul 2, 2020 at 21:17