Chapter 5: z-Scores: Location of Scores and Standardized Distributions

  • we can use the mean and standard deviation to describe a single, original score (or a raw score), transforming it into a z-score or standard score; this helps us ID and describe the exact location of each score in a distribution.
    z-scores also help standardize an entire distribution—we use z-scores to standardize IQ tests, eg, so that the average is always 100 regardless of which IQ test you take, allowing us to easily compare scores from different tests. generally, we can consider this second purpose, standardizing, as taking different distributions and making them equivalent.

  • z-scores transform every raw score into a signed number so that (A) the sign tells us whether the score is located above (+) or below (-) the mean, and (2) the number tells the distance between the score and the mean in terms of the number of standard deviations. so, eg, on a standard IQ test, a score of 130 would be transformed into z = +2.00 because it’s exactly 2 standard deviations above the mean.

  • z-score: specifies the precise location of each X value within a distribution; sign signifies whether it’s above or below the mean, and numerical value specifies distance from mean by counting # of standard deviations between X and μ.

  • for more complicated z-scores, we can use the formula (recalling that X-μ is the deviation score, as described last chapter).

  • we can double-check z-score calculations just by making note of whereabouts the number ought to be—eg, if your score is only slightly above 1 standard deviation, the z-score should be just slightly over 1.

  • we can also rearrange the z-score formula to find the raw score, using the equation , with zσ representing the deviation of X. that said, it’s usually easier to just use the definition of a z-score and figure it out in your head.

  • the equation to find the z-score of a sample rather than a population is exactly the same but uses different abbreviations, as explained in previous chapters: .

  • we can also use a z-score to describe many questions about the scores and the distributions in which they are located! (1) we can also find the standard deviation if we have X, z, and μ. (2) we can find the mean if we have X, z, and σ. (3) we can find σ and μ if we know 2 X values and their corresponding z-values

    • they recommend drawing a picture to help organize what you know and what you need to find.

  • sometimes we transform an entire set of X scores into z-scores. that distribution then has the following properties:

    • shape: the distribution of the z-scores has the exact same distribution as the X scores, so the shape is retained.

    • the mean: z-score distributions always have a mean of 0, making comparison easier.

    • the standard deviation: the standard deviation will always be a value of 1, also making comparison easier.

  • also, we don’t have to actually transform each score—we can just relabel the x-axis!

  • transforming all X scores into z-scores results in a sample distribution of z-scores, denoted with Mz and sz. (note: we still treat it as a sample since it is, in fact, still a sample!)

  • standardized distribution: composed of scores that have been transformed to create predetermined values for µ and σ; used to make dissimilar distributions comparable. one example is the z-score distribution, where µ=0 and σ=1.00. another eg is IQ test distributions.

    • standardized score: the score assigned to an individual when their raw data has been transformed to fit onto a standardized distribution; an example is the z-score. another eg is IQ scores.

  • some people hate negatives and decimals, so we often also transform values to avoid that! the goal is to create a standardized distribution with “simple” values for the mean and standard deviations without changing any individual’s location on the graph. these kinds of things are often used for standardized tests like the SAT—the average is coordinated to be 500 with a standard distribution of 100. they also do this with IQ scores, where the average is 100 and the standard deviation is 15.

    • we do this process by (A) transforming the original scores into z-scores, and (2) transforming the z-scores into new X values so that the specific mean and standard deviation are attained. for B, you can just assign a number for the mean and sigma, then do the math to figure out what number corresponds to each z-score on that scale.

  • it’s not possible to test treatments on entire populations, so we take a sample, give them the experimental treatment, and then compare them with the rest of the population to see if there’s a noticeable difference between them. we can assess whether there’s a [noticeable] difference by z-scores, eg—those with extreme z-scores, eg ≥±2.00, are noticeably different. we learn how to set there parameters in the next chapter, probability.