Transform raw scores into standard scores (z scores) and vice versa.
137
A standard score, called a z score, transforms a raw score so that it is expressed in terms of how many standard deviations it falls away from the mean. A positive z score means the score falls above the mean, a negative z score means it falls below the mean, and a z score of zero means the score falls right at the mean. z scores standardize scores, allowing different variables to be expressed in a common unit of measurement.
Describe the normal curve and calculate the likelihood of an outcome falling in specified areas of it.
The normal distribution is important because it is believed that many psychological variables are normally distributed. The normal curve is a specific bell-shaped curve, defined by the percentage of cases that fall in specified areas. About 34% of cases fall from the mean to 1 standard deviation above the mean, ≈14% fall from 1 to 2 standard deviations above the mean, and ≈2% fall from 2 standard deviations above to 3 above the mean. Because the normal distribution is symmetric, the same percentages fall below the mean. It is rare that a case has a score more than 3 standard deviations from the mean.
Appendix Table 1 lists the percentage of cases that fall in specified segments of the normal distribution. A flowchart, Figure 4.12, can be used as a guide to find the area under a normal curve that is above a z score, below a z score, or from the mean to a z score.
Transform raw scores and standard scores into percentile ranks and vice versa.
Percentile ranks tell the percentage of cases in a distribution that have scores at or below a given level. They provide easy-to-interpret information about how well a person performed on a test.
Calculate the probability of an outcome falling in a specified area under the normal curve.
Conclusions that statisticians draw are probabilistic.
Probability is defined as the number of ways a specific outcome or event can occur, divided by the total number of possible outcomes.
The normal curve can be divided into a middle section and extreme sections. A middle percentage of scores is symmetric around the midpoint, while an extreme percentage is evenly divided between the two tails of the distribution. The same z score, for example, defines the middle 60% and the extreme 40%. The middle 60% consists of the first 30% above the midpoint and the first 30% below the midpoint, while the extreme 40% consists of the 20% of scores above the middle 60% and the 20% below it.
z scores can be used to find the probability that a score, selected at random, falls in a certain section of the normal curve. For example, p = .50 that a score selected at random has a score at or below the mean.
The rare zone of a distribution is the part in the tails where scores rarely fall. Commonly, statisticians consider something rare if it happens less than 5% of the time. If rare is written as p < .05, then a common outcome is written as p > .05, meaning it happens more than 5% of the time. The common zone of a distribution is the central part where scores commonly fall.