unit 1 - chapter 6 - the normal distribution Flashcards
standardized testing is…
Standardized testing is a normal distribution
properties of the normal distribution
Continuous distribution
Symmetrical
Mean = median = mode
Area under the curve = 1
Defined by two variables:
Mean = 0
Standard deviation = 1
+/-1 o = 68.27
+/-2 o = 95.45
+/- 3 o = 99.73
1, 2 and 3 standard deviations
+/-1 o = 68.27
+/-2 o = 95.45
+/- 3 o = 99.73
normal probability density function
The normal probability density function, a continuous distribution, is the most important of all the distributions. It is widely used and even more widely abused. Its graph is bell-shaped.
You see the bell curve in almost all disciplines. Some of these include psychology, business, economics, the sciences, nursing, and, of course, mathematics.
Some of your instructors may use the normal distribution to help determine your grade.
Most IQ scores are normally distributed. Often real-estate prices fit a normal distribution.
normal distribution
The normal distribution is extremely important, but it cannot be applied to everything in the real world. Remember here that we are still talking about the distribution of population data.
This is a discussion of probability and thus it is the population data that may be normally distributed, and if it is, then this is how we can find probabilities of specific events just as we did for population data that may be binomially distributed or Poisson distributed.
normal distribution and its two parameters
The normal distribution has two parameters (two numerical descriptive measures): the mean (μ) and the standard deviation (σ).
If X is a quantity to be measured that has a normal distribution with mean (μ) and standard deviation (σ), we designate this by writing the following formula of the normal probability density function:
standard normal distribution
since the area under the curve must equal one, a change in the standard deviation, σ, causes a change in the shape of the normal curve; the curve becomes fatter and wider or skinnier and taller depending on σ.
A change in μ causes the graph to shift to the left or right. This means there are an infinite number of normal probability distributions. One of special interest is called the standard normal distribution.
z-scores
The z-score tells you how many standard deviations the value x is above (to the right of) or below (to the left of) the mean, μ.
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.
The mean for the standard normal distribution is zero, and the standard deviation is one. What this does is dramatically simplify the mathematical calculation of probabilities.
z = x - mew / sd
empirical rule
If X is a random variable and has a normal distribution with mean µ and standard deviation σ, then the Empirical Rule states the following:
About 68% of the x values lie between –1σ and +1σ of the mean µ (within one standard deviation of the mean).
About 95% of the x values lie between –2σ and +2σ of the mean µ (within two standard deviations of the mean).
About 99.7% of the x values lie between –3σ and +3σ of the mean µ (within three standard deviations of the mean). Notice that almost all the x values lie within three standard deviations of the mean.
standardizing formula
x - |x - mew | / sd
This is still dealing with probability so we always are dealing with the population, with known parameter values and a known distribution. It is also important to note that because the normal distribution is symmetrical it does not matter if the z-score is positive or negative when calculating a probability. One standard deviation to the left (negative Z-score) covers the same area as one standard deviation to the right (positive Z-score). This fact is why the Standard Normal tables do not provide areas for the left side of the distribution.
Where the vertical lines in the equation means the absolute value of the number.
What the standardizing formula is really doing is computing the number of standard deviations X is from the mean of its own distribution.
