standard score

(noun)

The number of standard deviations an observation or datum is above the mean.

Related Terms

Examples of standard score in the following topics:

  • Change of Scale

    • The standard score is the number of standard deviations an observation or datum is above the mean.
    • Thus, a positive standard score represents a datum above the mean, while a negative standard score represents a datum below the mean.
    • Standard scores are also called $z$-values, $z$-scores, normal scores, and standardized variables.
    • The absolute value of $z$ represents the distance between the raw score and the population mean in units of the standard deviation.
    • Includes: standard deviations, cumulative percentages, percentile equivalents, $Z$-scores, $T$-scores, and standard nine.

  • Z-Scores and Location in a Distribution

    • A $z$-score is the signed number of standard deviations an observation is above the mean of a distribution.
    • We obtain a $z$-score through a conversion process known as standardizing or normalizing.
    • $z$-scores are also called standard scores, $z$-values, normal scores or standardized variables.
    • $z$-scores are most frequently used to compare a sample to a standard normal deviate (standard normal distribution, with $\mu = 0$ and $\sigma =1$).
    • $z$-scores for this standard normal distribution can be seen in between percentiles and $t$-scores.

  • 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.
    • For example, if the mean of a normal distribution is 5 and the standard deviation is 2, the value 11 is 3 standard deviations above (or to the right of) the mean.
    • The mean for the standard normal distribution is 0 and the standard deviation is 1.
    • The value x comes from a normal distribution with mean µ and standard deviation σ.

  • Standardizing with Z scores

    • Table 3.4 shows the mean and standard deviation for total scores on the SAT and ACT.
    • If the observation is one standard deviation above the mean, its Z score is 1.
    • If it is 1.5 standard deviations below the mean, then its Z score is -1.5.
    • Use Tom's ACT score, 24, along with the ACT mean and standard deviation to compute his Z score.
    • (b) Use the Z score to determine how many standard deviations above or below the mean x falls.

  • Exercises

    • If scores are normally distributed with a mean of 35 and a standard deviation of 10, what percent of the scores is:
    • What are the mean and standard deviation of the standard normal distribution?
    • (a) What score would be needed to be in the 85th percentile?
    • One score is randomly sampled.
    • True/false: A Z-score represents the number of standard deviations above or below the mean.

  • Z-scores

    • The z-score tells you how many standard deviations that the value x is above (to the right of) or below (to the left of) the mean, µ.
    • If x equals the mean, then x has a z-score of 0.
    • This z-score tells you that x = 10 is ________ standard deviations to the ________ (right or left) of the mean _____ (What is the mean?
    • This z-score tells you that x = − 3 is ________ standard deviations to the __________ (right or left) of the mean.
    • The z-score for y = 4 is z = 2.

  • Confidence Interval for a Population Mean, Standard Deviation Known

    • Step By Step Example of a Confidence Interval for a Mean—Standard Deviation Known
    • Suppose scores on exams in statistics are normally distributed with an unknown population mean, and a population standard deviation of 3 points.
    • A random sample of 36 scores is taken and gives a sample mean (sample mean score) of 68.
    • We know the population standard deviation is 3.
    • Calculate the confidence interval for a mean given that standard deviation is known

  • 68-95-99.7 rule

    • The probability of being further than 4 standard deviations from the mean is about 1-in-30,000.
    • 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?
    • 3.23: (a) 900 and 2100 represent two standard deviations above and below the mean, which means about 95% of test takers will score between 900 and 2100.
    • (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.

  • Statistical Literacy

    • The standard for an athlete's admission, as reflected in SAT scores alone, is lower than the standard for non-athletes by as much as 20 percent, with the weight of this difference being carried by the so-called "revenue sports" of football and basketball.
    • Based on what you have learned in this chapter about measurement scales, does it make sense to compare SAT scores using percentages?
    • As you may know, the SAT has an arbitrarily-determined lower limit on test scores of 200.
    • Therefore, it is not meaningful to report SAT score differences in terms of percentages.
    • For example, consider the effect of subtracting 200 from every student's score so that the lowest possible score is 0.