Download 2.3 Notes

AP Statistics
2.3 Measures of Central Tendency
Measure of Central Tendency: a value that represents a typical, or central, entry of a data set.
: sum of data entries divided by the number of entries
Population Mean:
Sample Mean:
NOTATIONS:
∑ The uppercase Greek letter sigma; indicates a summation of values
x A variable that represents quantitative data entries
N Number of entries in a population
n Number of entries in a sample
𝜇 The lowercase Greek letter mu; the population mean
𝑥̅
Read as “x bar”; the sample mean
: the middle data entry when the data is in order.
~Mean of two middle data entries if set has an even number of data entries
~Equal number of data values above and below
: the data entry that occurs with the greatest frequency.



If none, the data has
.
If two entries have the greatest frequency, the data set is
.
Only measure of central tendency that can be used to describe data at the nominal level of
measurement.
: data entry that is far removed from the other entries in the data set.
~ typically effects the mean
~ typically does not effect the median, mode
Example: Find the mean, the median, the mode, and the outlier (if they exist) of the ages of our class. Which
measure of central tendency best describes a typical entry?
Now find the mean, the median, and the mode of the set of data above, excluding any outliers. What was
effected?
: the mean of a data set whose entries have varying weights.
Formula:
𝑥̅ =
∑(𝑥𝑤)
∑𝑤
where w is the weight of each entry x
Example: You are taking a class in
which your grade is determined
from five sources: 50% from your
test mean, 15% from your
midterm, 20% from your final
exam, 10% from your computer
lab work, and 5% from your
homework. Your scores are 86
(test mean), 96(midterm), 82 (final
exam), 98 (computer lab), and 100
(homework). What is the
weighted mean of your scores?
Example: Johnny goes to Butler
Senior High School. Each nine
weeks is worth 20% while the two
exams are worth 10% each.
Johnny’s nine week grades are 94,
91, 88, 83 and his exam grades are
88 and 56. What is Johnny’s
weighted mean?
Let’s say that Johnny had an
extended illness that kept him in
the hospital through the end of
May and June. As a result, Johnny
was exempt from the final exam.
Now what is Johnny’s weighted
mean?
AP Statistics
2.3 Continued
: the approximation of a mean of a sample by using the information from a
frequency distribution.
Formula:
𝑥̅ =
∑(𝑥𝑓)
𝑛
where x is the midpoint and f is the frequency of a class
Example: Find the mean of the frequency distribution below.
Shapes of Distributions:
_____________________: Vertical line can be drawn through the middle of a graph of the distribution and the
resulting halves are approximately mirror images.
~mean, median, and mode are equal
______________________: All entries (or all classes) in the distribution have equal frequencies.
~Uniform is also symmetric, so the mean, median, and mode are equal
Frequency Distribution is SKEWED if the
“tail” of the graph elongates more to one side
than the other.
_____________________ (Negatively Skewed): tail extends to the left

mean is less than the median and the median is usually less than
the mode
_____________________ (Positively Skewed): tail extends to the right

mean is greater than the median and the median is usually
greater than the mode
The mean will always
fall in the direction the
distribution is skewed.
Skewed left  mean is
to the left of the median
Skewed right  mean is
to the right of the
median