AP STATS. DENSITY CURVES and NORMAL DISTRIBUTIONS. The histogram displays the Grade equivalent vocabulary scores for 7 th graders on the Iowa Test of Basic Skills. The scores of students on this national test have a regular distribution .
The histogram displays the Grade equivalent vocabulary scores for 7th graders on the Iowa Test of Basic Skills. The scores of students on this national test have a regular distribution.
Describe the overall shape of distributions
Idealized mathematical models for distributions
Show patterns that are accurate enough for practical purposes
Always on or above the horizontal axis
The total area under the curve is exactly 1
Areas under the curve represent relative frequencies of observations
Because a Density Curve is an Idealized Description of the distribution of data, we must distinguish between:
The Mean , and standard deviation (s ) ; computed from the actual observations
The mean (μ ) and standard deviation (σ ) of the idealized distribution.
Consider the unusual density curve:
Find the % of the data in the following intervals
0 < X < 0.6 ?
0.2 < X < 0.4 ?
0 < X < 0.8 ?
The Mean ( μ ) - A measure of center or location. The mean can be any + value. The mean is in the same location as the median.
The Standard Deviation ( σ ) – A measure of spread. The standard deviation must be a positive number.
Together, the Mean and the Standard Deviation define a specific normal distribution.
The standard deviation can be located visually by finding the INFLECTION POINTS on either side of the mean
The INFLECTION POINTS of the curve are the places where the CONCAVITY changes.
In the normal distribution with a mean (mu) and standard deviation (sigma):
68% of all observations lie within one standard deviation of the mean
95% of all observations lie within two standard deviations of the mean
99.7% of all observations fall within three standard deviations of the mean.
Because normal distributions are used so frequently, a short notation is often used to describe the parameters of mean and standard deviation.
N ( μ , σ )
For example: N ( 64.5, 2.5 ) indicates a normal distribution with a mean = 64.5 and a standard deviation of 2.5.
Percentile is a familiar term because it is so frequently used in the reporting of standardized test scores. Percentiles are used when we are interested in seeing where an individual observation stands in relation to other observations in the distribution.
An observations PERCENTILE is the percent of the distribution that lies to the LEFT of the observation
2.6 – 2.9
2.11 – 2.18