Download AP STATISTIC EXAM REVIEW I: EXPLORING DATA (20–30

AP STATISTIC
EXAM REVIEW I: EXPLORING DATA (20–30%)
CHAPTER 2 REVIEW:
DESCRIBING LOCATION IN A DISTRIBUTION
DENSITY CURVES AND NORMAL DISTRIBUTIONS
• Two ways of describing an individual’s location within a distribution are percentiles and 𝒛-scores. An
observation’s percentile is the percent of the distribution that is below the value of that
observation. To standardize any observation 𝑥, subtract the mean of the distribution and then
divide the difference by the standard deviation. The resulting 𝑧-score
𝑥 − 𝑚𝑒𝑎𝑛
𝑧=
𝑠𝑡𝑎𝑛𝑑𝑎𝑟𝑑 𝑑𝑒𝑣𝑖𝑎𝑡𝑖𝑜𝑛
says how many standard deviations 𝑥 lies above or below the distribution mean. We can also use
percentiles and 𝑧-scores to compare the location of individuals in different distributions.
• A cumulative relative frequency graph allows us to examine location within a distribution.
Cumulative relative frequency graphs begin by grouping the observations into equal-width classes
(much like the process of making a histogram). The completed graph shows the accumulating
percent of observations as you move through the classes in increasing order.
• It is common to transform data, especially when changing units of measurement. When you add a
constant 𝑎 to all the values in a data set, measures of center (median and mean) and location
(quartiles and percentiles) increase by 𝑎. Measures of spread do not change. When you multiply all
the values in a data set by a positive constant 𝑏, measures of center, location, and spread are
multiplied by 𝑏. Neither of these transformations changes the shape of the distribution.
• We can describe the overall pattern of a distribution by a density curve. A density curve always
remains on or above the horizontal axis and has total area 1 underneath it. An area under a density
curve gives the proportion of observations that fall in an interval of values.
• A density curve is an idealized description of the overall pattern of a distribution that smooths out the
irregularities in the actual data. We write the mean of a density curve as 𝜇 and the standard
deviation of a density curve as 𝜎 to distinguish them from the mean 𝑥̅ and the standard deviation
𝑠𝑥 of the actual data.
• The mean and the median of a density curve can be located by eye. The mean 𝜇 is the balance point
of the curve. The median divides the area under the curve in half. The standard deviation 𝜎 cannot
be located by eye on most density curves.
• The mean and median are equal for symmetric density curves. The mean of a skewed curve is located
farther toward the long tail than the median is.
• The normal distributions are described by a special family of bell-shaped, symmetric density curves,
called normal curves. The mean 𝜇 and standard deviation 𝜎 completely specify a Normal
distribution 𝑁(𝜇, 𝜎). The mean is the center of the curve, and 𝜎 is the distance from 𝜇 to the
change-of-curvature points on either side.
• All Normal distributions obey the 68–95–99.7 rule, which describes what percent of observations lie
within one, two, and three standard deviations of the mean.
AP STATISTIC
EXAM REVIEW I: EXPLORING DATA (20–30%)
• All Normal distributions are the same when measurements are standardized. If x follows a Normal
distribution with mean 𝜇 and standard deviation 𝜎, we can standardize using
𝑥−𝜇
𝑧=
𝜎
The variable z has the standard normal distribution with mean 0 and standard deviation 1.
• You can use your calculator to perform Normal calculations quickly.
• To perform certain inference procedures in later chapters, we will need to know that the data come
from populations that are approximately Normally distributed. To assess Normality for a given set of
data, we first observe the shape of a dotplot, stemplot, or histogram. Then we can check how well
the data fit the 68–95–99.7 rule for Normal distributions. Another good method for assessing
Normality is to construct a Normal probability plot.
**CHAPTER 2 AP EXAM TIPS**
• Normal probability plots are not included on the AP Statistics topic outline. However, these graphs are
very useful for assessing Normality. You may use them on the AP exam if you wish—just be sure that
you know what you’re looking for (a linear pattern).