PSYCH 104 - CH6 Probability, Normal Distributions, and z Scores TERMS
8 characteristics of normal distribution
1. It is mathematically defined (by an equation) 2. It is theoretical (data can be normally distributed in theory) 3. The mean, the median, and the mode are all located at the 50th percentile (midpoint) 4. It is symmetrical (the distribution of data above the mean is the same as the distribution of data below the mean) 5. The mean can equal any value (-∞ ≤ M ≤ + ∞) 6. The standard deviation can equal any positive value. 7. The total area under the curve of a normal distribution is equal to 1.0 8. The tails of a normal distribution are asymptotic (always approaching the x-axis but never touch it)
Normal distribution is defined by the percentage of scores that fall within specific region
Empirical rule 68% = 1 SD 95% = 2 SD 99.7% = 3 SD
Normal distribution is classified by 2 parameters
Mean (μ) = LOCATION. Changing μ shifts the distribution left/right Standard deviation (σ) = SPREAD Changing σ increases/decreases the spread
To find the cutoff score for a given proportion, we follow two steps:
Step 1: Locate a z score associated with a given proportion in the unit normal table. Step 2: Transform the z score into a raw score (x).
To locate the proportion, and therefore the probability, of scores in any normal distribution, we follow two steps:
Step 1: Transform a raw score (x) into a z score. (z = x−μ/σ) Step 2: Locate the corresponding proportion for the z score in the unit normal table.
Column C lists the area from a z score toward the tail.
The first value for the area listed in Column C is .5000 (total area above the mean)
Cut scores are selected points on the score scale of a test.
The points are used to determine whether a particular test score is sufficient for some purpose.
Column A lists the z scores.
The table lists only positive z scores (at or above the mean) from z = 0 at the mean to z = 4.00 above the mean.
The area at each z score is given as
a proportion in the unit normal table
The normal distribution, also called the symmetrical, Gaussian, or bell-shaped distribution, is
a theoretical distribution in which scores are symmetrically distributed above and below the mean, the median, and the mode at the center of the distribution.
The unit normal table or z table is
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.
The z transformation formula converts
any normal distribution with any mean and variance to the standard normal distribution with a mean equal to 0 and variance equal to 1.
In the standard normal distribution, z scores below the mean
are negative.
In the standard normal distribution, z scores above the mean
are positive
Because the normal distribution is symmetrical, probabilities associated with positive z scores
are the same for corresponding negative z scores.
The normal distribution is a specific
bell-shaped curve defined by the percentage of cases that fall in specific areas under the curve.
In a normal distribution the score occurrence
decrease symmetrically
Normal distribution is used to
determine the probability of a certain outcome in relation to all other outcomes.
Column B lists the area between a z score and the mean.
first value .0000 z score =1.00 is .3413
In a normal distribution scores
further away from the midpoint appear less frequently than those around the midpoint.
Standard normal distribution or z distribution
is a normal distribution with a mean μ = 0 and a standard deviation σ = 1.
The behavioral data that researchers measure often tend to approximate a
normal distribution.
The tails of a normal distribution never touch the x-axis, so it is possible to observe
outliers in a data set that is normally distributed.
To find the probabilities of scores in a normal distribution, you must know
the mean and standard deviation in that distribution.
Proportions of the area under a normal curve are used to determine
the probabilities for normally distributed data.
A z score is a value on
the x-axis of a standard normal distribution.
95% of all data fall within 2 SD of the mean in a normal distribution,
these data are normal or likely.
Only 5% of data fall in the tails beyond about 2SD from the mean in a normal distribution,
these data are not normal or likely.
The mean in any normal distribution corresponds
to a z score equal to 0. (when M = x, the solution for the z transformation is 0)
We use the z transformation
to locate where a score in any normal distribution would be in the standard normal distribution.
Not all bell-shaped curves are normal distribution
true
The unit normal table can be used to locate a cutoff score for a given proportion for data that are normally distributed.
true
The unit normal table can be used to locate scores that fall within a given proportion or percentile.
true
Statistically speaking, normal behavior is defined by the statistical norm, which is data that fall
within about 2 SD of the mean in a normal distribution.
Formula for calculating x when z is known
x = μ+(z*σ) for a population of scores, or x = M+(z*SD) for a sample of scores .
z transformation formula
z = x−μ/σ for a population of scores, or z = x−M/SD for a sample of scores .
The standard normal distribution is distributed in
z score units along the x-axis.
