Chapter 5- Standard Scores Flashcards
Probability
(symbolized as p) is the frequency of times an outcome occurs divided by the total number of possible outcomes
random event
any event in which the outcome observed can vary.
fixed event
any event in which the outcome observed is always the same.
sample space
the total number of possible outcomes that can occur in a given random event.
CHARACTERISTICS OF THE NORMAL DISTRIBUTION
The normal distribution is mathematically defined. The shape of a normal distribution is specified by an equation relating each score (distributed along the x-axis) with each frequency (distributed along the y-axis):
It is not necessary to memorize this formula. It is important to understand that rarely do behavioral data fall exactly within the limits of this formula. When we say that data are normally distributed, we mean that the data approximate a normal distribution. The normal distribution is so exact that it is simply impractical to think that behavior can fit exactly within the limits defined by this formula.
The normal distribution is theoretical. This characteristic
follows from the first in that it emphasizes that data can be normally distributed in theory—although rarely do we observe behaviors that are exactly normally distributed. Instead, behavioral data typically approximate a normal distribution. As you will see in this chapter, we can still use the normal distribution to describe behavior so long as the behaviors being described are approximately normally distributed.
The mean, median, and mode are all located at the 50th percentile. In a normal distribution, the mean, the median, and the mode are the same value at the center of the distribution. So half the data (50%) in a normal distribution fall above the mean, the median, and the mode, and half the data (50%) fall below these measures.
The normal distribution is symmetrical. The normal distribution is symmetrical in that the distribution of data above the mean is the same as the distribution of data below the mean. If you were to fold a normal curve in half, both sides of the curve would exactly overlap.
The mean can equal any value. The normal distribution can be defined by its mean and standard deviation. The mean of a normal distribution can equal any number from positive infinity (−∞) to negative infinity (−∞):
−∞ ≤ M ≤ +∞.
The standard deviation can equal any positive value. The standard deviation (SD) is a measure of variability. Data can vary (SD > 0) or not vary (SD = 0). A negative standard deviation is meaningless. In the normal distribution, then, the standard deviation can be any positive value greater than 0.
The total area under the curve of a normal distribution is equal to 1.0. The area under the normal curve has the same characteristics as probability: Portions of it vary between 0 and 1 and can never be negative. In this way, the area under the normal curve can be used to determine the probabilities at different points along the distribution. In Characteristic 3, we stated that 50% of all data fall above and 50% fall below the mean. This is the same as saying that half (.50) of the area under the normal curve falls above and half of the area (.50) falls below the mean. The total area, then, is equal to 1.0. Figure 5.2 shows the proportions of area under the normal curve 3 SD above and below the mean (±3 SD).
The tails of a normal distribution are asymptotic. In a normal distribution, the tails are asymptotic, meaning that as you travel away from the mean the tails of the distribution are always approaching the x-axis but never touch it. Because the tails of the normal distribution go out to infinity, this characteristic allows for the possibility of outliers (or scores far from the mean) in a data set.
FYI
standard normal distribution, or z distribution
a normal distribution with a mean equal to 0 and a standard deviation equal to 1. The standard normal distribution is distributed in z score units along the x-axis.
z score
a value on the x-axis of a standard normal distribution. The numerical value of a z score specifies the distance or the number of standard deviations that a value is above or below the mean.
standard normal transformation or z transformation
a formula used to convert any normal distribution with any mean and any variance to a standard normal distribution with a mean equal to 0 and a standard deviation equal to 1.
unit normal table or z table
a type of probability distribution table displaying a list of z scores and the corresponding probabilities (or proportions of area) associated with each z score listed
asymptotic
approach closely but never reaching ( like the tails on a normal distribution)
