z-score

(noun)

The standardized value of observation $x$ from a distribution that has mean $\mu$ and standard deviation $\sigma$.

Related Terms

  • Student's t-statistic
  • standard normal distribution
  • raw score
  • binomial distribution

Examples of z-score in the following topics:

  • Z-Scores and Location in a Distribution

    • Thus, a positive $z$-score represents an observation above the mean, while a negative $z$-score represents an observation below the mean.
    • $z$-scores are also called standard scores, $z$-values, normal scores or standardized variables.
    • The conversion of a raw score, $x$, to a $z$-score can be performed using the following equation:
    • $z$-scores for this standard normal distribution can be seen in between percentiles and $t$-scores.
    • Define $z$-scores and demonstrate how they are converted from raw scores
  • Standardizing with Z scores

    • If it is 1.5 standard deviations below the mean, then its Z score is -1.5.
    • Using µSAT = 1500, σSAT = 300, and xAnn = 1800, we find Ann's Z score:
    • Use Tom's ACT score, 24, along with the ACT mean and standard deviation to compute his Z score.
    • Observations above the mean always have positive Z scores while those below the mean have negative Z scores.
    • SAT score of 1500), then the Z score is 0.
  • Summary of Formulas

    • To find the kth percentile when the z-score is known: k = µ + ( z ) σ
  • The Standard Normal Distribution

    • The standard normal distribution is a normal distribution of standardized values called z-scores.
    • A z-score is measured in units of the standard deviation.
    • x = µ + ( z ) σ = 5 + ( 3 )( 2 ) = 11 (6.1)
    • The transformation z = (x − µ)/σ produces the distribution Z ∼ N ( 0,1 ) .
  • Z-scores

    • Values of x that are larger than the mean have positive z-scores and values of x that are smaller than the mean have negative z-scores.
    • If x equals the mean, then x has a z-score of 0.
    • The z-score when x = 10 pounds is z = 2.5 (verify).
    • The z-score for y = 4 is z = 2.
    • The z-scores are -1 and +1 for -6 and 6, respectively.
  • Constructing a normal probability plot (special topic)

    • Create a scatterplot of the observations (vertical) against the Z scores (horizontal).
    • If the observations are normally distributed, then their Z scores will approximately correspond to their percentiles and thus to the zi in Table 3.16.
    • The zi in Table 3.16 are not the Z scores of the observations but only correspond to the percentiles of the observations.
  • Normal probability table

    • Example 3.7 Ann from Example 3.2 earned a score of 1800 on her SAT with a corresponding Z = 1.
    • A normal probability table, which lists Z scores and corresponding percentiles, can be used to identify a percentile based on the Z score (and vice versa).
    • We use this table to identify the percentile corresponding to any particular Z score.
    • We can also find the Z score associated with a percentile.
    • We determine the Z score for the 80th percentile by combining the row and column Z values: 0.84.
  • Normal probability examples

    • However, the percentile describes those who had a Z score lower than 0.43.
    • TIP: always draw a picture first, and find the Z score second
    • Finally, the height x is found using the Z score formula with the known mean µ, standard deviation σ, and Z score Z = 0.92.
    • Finally, the height x is found using the Z score formula with the known mean µ, standard deviation σ, and Z score Z = 0.92.
    • 3.17: Remember: draw a picture first, then find the Z score.
  • Change of Scale

    • Standard scores are also called $z$-values, $z$-scores, normal scores, and standardized variables.
    • The $z$-score is only defined if one knows the population parameters.
    • The absolute value of $z$ represents the distance between the raw score and the population mean in units of the standard deviation.
    • $z$ is negative when the raw score is below the mean, positive when above.
    • Includes: standard deviations, cumulative percentages, percentile equivalents, $Z$-scores, $T$-scores, and standard nine.
  • 68-95-99.7 rule

    • SAT scores closely follow the normal model with mean µ = 1500 and standard deviation σ = 300. ( a) About what percent of test takers score 900 to 2100?
    • (b) What percent score between 1500 and 2100?
    • To find the area between Z = −1 and Z = 1, use the normal probability table to determine the areas below Z = −1 and above Z = 1.
    • Repeat this for Z = −2 to Z = 2 and also for Z = −3 to Z = 3.
    • (b) Since the normal model is symmetric, then half of the test takers from part (a) (95% / 2 = 47.5% of all test takers) will score 900 to 1500 while 47.5% score between 1500 and 2100.
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