Ch 6: z-scores & the Standard Normal Distribution

Question 1

What is a Z-Score?

Answer 1

A standardized score that indicates how many standard deviations away from the mean a data point lies; combine information about where the distribution is located (the mean/center) with how wide the distribution is (the standard deviation/spread) to interpret a raw score (x); will tell us how far the score is away from the mean in units of standard deviations and in what direction; has two parts: the sign (positive or negative) and the magnitude (the actual number); the sign tells you in which half of the distribution it falls: a positive sign (or no sign) indicates that the score is above the mean and on the right hand-side or upper end of the distribution, and a negative sign tells you the score is below the mean and on the left-hand side or lower end of the distribution; The magnitude of the number tells you, in units of standard deviations, how far away the score is from the center or mean; the magnitude generally falls between -3 and 3.

Question 2

What is Normal Distribution?

Answer 2

A symmetric, bell-shaped distribution defined by its mean and standard deviation; is sometimes called the “bell curve;” It is also called the “Gaussian curve;” is described in terms of two parameters: the mean (which you can think of as the location of the peak), and the standard distribution (which specifies the width of the distribution); The bell-like shape of the distribution never
changes, only its location and width; can differ in their means and in their standard deviations; what is consistent about all is the shape and proportion of scores within a given distance along the x-axis

Question 3

What is a Standard Normal Distribution?

Answer 3

A normal distribution with a mean of 0 and standard deviation of 1; (also known as the Unit Normal Distribution)

Question 4

What is a Percentile Rank?

Answer 4

The percentage of scores in a distribution that fall below a particular score.

Question 5

What is Standardization?

Answer 5

The process of converting raw scores to z-scores; A z-score will tell you exactly where in the standard normal distribution a value is located, and any normal distribution can be converted into a standard normal distribution by converting all of the scores in the distribution into z-scores,

Question 6

What is the Empirical Rule?

Answer 6

States that 68%, 95%, and 99.7% of data fall within 1, 2, and 3 standard deviations of the mean
* 68% of all scores will fall between a Z score of -1.00 and
+1.00
* 95% of all scores will fall between a Z score of -2.00 and
+2.00
* 99.7% of all scores will fall between a Z score of -3.00 and
+3.00
* 50% of all scores lie above/below a Z score of 0.00

Question 7

What is the Gaussian Distribution?

Answer 7

Another name for the normal distribution, named after Carl Friedrich Gauss.

Question 8

What is the Area Under the Curve?

Answer 8

The proportion of scores falling within specified z-score boundaries; bounded by (defined by, delineated by, etc.) by a single z-score or pair of z-scores; the total area under the curve of a distribution is always equal to 1.0; these areas under the curve can be added together or subtracted from 1 to find the proportion in other areas

Question 9

What is a Raw Score?

Answer 9

An original, unstandardized score in its original units.

Question 10

What is an Outlier?

Answer 10

A score typically defined as having a z-score beyond ±3 (or sometimes ±2); sometimes defined as scores that
have z scores less than −3.00 or greater than +3.00; they are defined as scores that are more than three standard deviations from the mean

Question 11

What are the Seven Features of a Normal Distribution?

Answer 11

• Normal distributions are symmetric around their mean.
• The mean, median, and mode of a normal distribution are
  equal.
• The area under the normal curve is equal to 1.0.
• Normal distributions are denser in the center and less
  dense in the tails.
• Normal distributions are defined by two parameters, the
  mean (μ) and the standard deviation (σ).
• 68% of the area of a normal distribution is within one
  standard deviation of the mean.
• Approximately 95% of the area of a normal distribution is
  within two standard deviations of the mean.

Question 12

Formulas to Calculate Z-Scores:

Answer 12

z = x - μ / σ (Population)
z = x - x̄ / s (Sample)
note: same formula just with appropriate symbols for mean and standard deviation depending on either population or sample data
z = value - mean / standard deviation

Question 13

Formulas for Transforming z to x:

Answer 13

Z Score to Raw Score - Sample
X = (Z) (SD) + M
Z Score to Raw Score - Population
X = (Z) (σ) + μ