Download Section 4

Section 4.1 ~ What is Average?
Objective: In this section you will understand the difference between the three most
common measures of central tendency; mean, median, and mode. You will learn how
each is affected by outliers and also when it is appropriate to use a weighted mean.
Essential Questions:
1. In your own words, explain what mean, median, and mode are representing.
2. How is mean calculated?
3. How do you find the median of an odd number set? Of an even number set?
4. How do you find the mode of a data set? Will there always be a mode?
5. What measure of central tendency is affected by outliers?
6. What is the major advantage of using median as opposed to mean in some
circumstances?
7. What is the rounding rule?
8. When is weighted mean used? How is it calculated?
Mean:
The mean is most commonly referred to as the _________________ and is found by the
following formula:
Ex. ~ The following values represent the weights of five wrestlers:
188
162
190
150
176
Find the average weight of these five wrestlers.
Median:
The median is the ___________________ number in a data set.
To find the median, you must first arrange the values in ascending (or descending) order:
A data set that has an odd number of values will have exactly 1 value in the
middle.
Ex ~ Find the median of the data set: 4 3 6 3 6
A data set that has an even number of values will have 2 values in the middle
which means that the median will be in the middle of those two values (or the
average of those two values).
Ex ~ Find the median of the data set: 4 3 6 3 6 5
Mode:
The mode is the __________________ value (or group of values) in a data set.
A data set may have one mode, more than one mode, or no mode
Ex ~ Find the mode in the data set: 4 3 6 10 6 3 6
Ex ~ Find the mode in the data set: 4 3 6 3 10 6
Ex ~ Find the mode in the data set 4 3 6 2 10
The mode is most commonly used for qualitative data at the nominal level since
neither the mean nor the median can be found for qualitative data
Ex. ~ the brand of shoes each student is wearing in this class
Rounding Rule:
Example 1:
Eight grocery stores sell the PR energy bar for the following prices:
$1.09 $1.29 $1.29 $1.35 $1.39 $1.49 $1.59 $1.79
Find the mean, median, and mode for these prices.
Outlier:
An outlier does not affect the median or mode because:
An outlier does affect the mean because:
Ex. ~ The following values represent the contract offer received for five
graduating college seniors in the NBA (zero means that they didn’t receive an
offer)
$0
$0
$0
$0
$3,500,000
Comparison of mean, median, and mode:
Measure
MEAN
MEDIAN
MODE
Definition
sum of all values
total number of values
middle value
most frequent value
Takes every value
into account?
Yes
No (except for the
ordering of every
number)
No
Affected by
Outliers?
Yes
No (since the an
outlier won’t fall in
the middle of a data
set)
No (since an outlier
won’t occur most
frequently)
Advantages
Commonly
understood as the
average; works well
with many statistical
methods
When there are
outliers finding the
median may be more
representative of an
“average” than the
mean
Most appropriate for
qualitative data at the
nominal level
Example 2:
A track coach wants to determine an appropriate heart rate for her athletes during their
workouts. She chooses five of her best runners and asks them to wear heart monitors
during a workout. In the middle of the workout, she reads the following heart rates for
the five athletes: 130, 135, 140, 145, and 325. Which is a better measure of the average
in this case – the mean or the median? Why?
Weighted mean:
Formula for weighted mean:
Ex. ~ Suppose your course grade is based on four tests and one final exam. Each test
counts as 15% of your final grade and the final exam counts as 40% of your final
grade. Your test scores are 75, 80, 84, and 88 and your final exam score is a 96.
What is your final grade?
Example 3:
Each quarter grade is calculated with a 80% weight on tests and quizzes and a 20%
weight on class work, homework, and class participation. Furthermore, your course
grade is calculated with a 40% weight on quarter 3, a 40% weight on quarter 4, and a
20% weight on the final exam.
Suppose that Joe’s quiz/test average in quarter 3 was an 86% and his homework/class
work/ class participation grade was 95%. In quarter 4 his quiz/test average was a 92%
and his homework/class work/class participation grade was a 92%, and he scored an 80%
on his final exam. What was Joe’s final grade?
Example 4:
Each quarter grade is calculated with a 80% weight on tests and quizzes and a 20%
weight on class work, homework, and class participation. Furthermore, your course
grade is calculated with a 40% weight on quarter 3, a 40% weight on quarter 4, and a
20% weight on the final exam. If Amy got an 88% for quarter 3 and a 93% for quarter 4,
what would she need to get on the final exam in order to receive at least a 90% for her
final grade?