standard score

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.

• IQ Tests

  • It is a statistical law that under a normal curve, 68% of scores will lie between -1 and +1 standard deviation, 95% of scores will lie between -2 and +2 standard deviations, and >99% percent of scores will fall between -3 and +3 standard deviations.
  • The scores of an IQ test are normally distributed so that one standard deviation is equal to 15 points; that is to say, when you go one standard deviation above the mean of 100, you get a score of 115.
  • When you go one standard deviation below the mean, you get a score of 85.
  • While all of these tests measure intelligence, not all of them label their standard scores as IQ scores.
  • IQ test scores tend to form a bell curve, with approximately 95% of the population scoring between two standard deviations of the mean score of 100.

• 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.