I read about the 68-95-99.7 rule that shows
I also read that a Z value of 1.96 gives a confidence level of 95%
Is it correct to think of Z as the "number of standard deviations" ? Only I have never seen it mentioned that way.
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No. The z score is not 'the number of standard deviations'. Instead the z-score of a value is the number of standard deviations that value is above the mean. A z-score of 1.7 is 1.7 standard deviations above the mean. A z score of -1 is one standard deviation below the mean, and so on.
This is not mere nitpicking, it's essential to correctly conveying your meaning. I have seen exactly this imprecision in relation to z-scores lead to error on numerous occasion. Stats is not the place for woolly thinking and muddled words $-$ it is tricky enough when you say exactly what you mean.
Yes. A Z value of a particular data point tells you how many standard deviations it is from its mean. Z=0 means it has the same value as the population mean, Z=-1 means it is 1std lower than its mean etc. The probability that an observation will lie within the interval of its population mean plus/minus two times the standard deviation is 95%. This is the connection between z scores and confidence intervals.
Sometimes z-score refers to a quantile randomized z-score, where the quantiles of distribution are mapped to z of a standard normal, so by construction, z-score of one is bigger than 68.27% percent of values in the distribution, regardless of how many standard deviations from a mean a value is.