AP StatisticsChapter 2 Notes

Measures of Relative Standing Percentiles

The percent of data that lies at or below a particular value.

e.g. standardized test score reports baby weight/height/head size.

Standardized value (z-score)

z = x - meanstd dev

= x - x

s

Z-score Z-score tells you how many standard

deviations away from the mean a given observation is.

Z-scores are more useful with symmetric distributions.

Density Curve An idealized mathematical model used to

represent a distribution. Always on or above the horizontal axis. Has an area of exactly 1 underneath it.

The area under the curve for any given interval is equal to the proportion of all observations that fall in that interval. median: equal areas point Mean: balance point

Notation used for density curves Observed data Idealized Data

(Sample) (population) Statistics Parameters

MEAN μ

standard deviation σ

x

s

Normal Distribution Always symmetric, but

the exact shape depends upon μ and σ.

Change in curvature (point of inflection) shows where 1 standard deviation from the mean is located.

Empirical Rule (68-95-99.7 Rule)

Example IQ scores are Normally

distributed with a mean of 100 and a std dev of 15.

What % of people have IQ scores… Between 70 and 130? Less than 85? Greater than 145? Less than 115? Between 55 and 70?

Probability Calculations We define a Normal distribution by its mean and

standard deviation. N(μ, σ)

If we standardize the distribution by calculating z-scores, we create the distribution: N(0,1).

The z-table provides the percentiles associated with various z-scores.

When performing a calculation, be sure to draw a sketch of the region under the Normal curve that you are working with, and answer the question in context.

Examples IQ scores are Normally distributed with a mean of

100 and a std dev of 15. What percent of people have IQ scores less than

82? Less than 121? What percent of people have IQ scores greater

than 107? What percent of people have IQ scores between

88 and 104? A person is considered a genius if they are in the

top 2% in terms of IQ. What IQ score does a person need to be considered a genius?

Using the calculator To calculate the % of observations within a

certain interval, use the z-table or the graphing calculator. 2nd Vars (Dist), choose option 2. Normalcdf (min, max, μ, σ)

To calculate raw data scores from percentiles: 2nd Vars (Dist), choose option 3. invNorm(%, μ, σ)