1. Section2.3 – Measures of Central Tendency
SWBAT:
Identify and analyze patterns of distributions using shape, center and spread.

2. Measures of Central Tendency (MCT)
A value that represents a typical, or central, entry of a data set. The most commonly used are:
Mean: average
Median: middle value of an ordered data set
Mode: the data item occurring most frequently

3. MEAN
In statistics, mean is designated differently depending on if the dataset is a sample or population:
Sample Mean Population Mean
“x bar” μ (pronounced mu)
n = sample size N = population size

4. Mean Sensitive to the influence of outliers
When extreme outliers are present, median is a better measure of central tendency
EX) Calculate mean and median for:
45, 83, 90, 79, 81, 83, 90, 88
and for:
83, 90, 79, 81, 83, 90, 88

5. Let’s try some less obvious MEAN calculations:
The mean scores for a statistics course (by major) are given: What is the mean score for the class?
9 engineering majors: 85
5 math majors: 90
13 business majors: 81
9(85) + 5(90) + 13(81) = 27

6. MEAN calculations, cont’d
Find the mean of the frequency distribution:
Height
(in inches)
Frequency 4
63 – 65 5
66 – 48 8
69 – 71 1

7. Finding the MEAN of a Frequency Distribution
MEAN calculations, cont’d
Finding the MEAN of a Frequency Distribution
In Words
In symbols
Find the midpoint of each class.
2. Find the sum of the products of the midpoints and the frequencies.
∑(x • f)
3. Find the sum of the frequencies
n = ∑ f
4. Find the mean of the frequency distribution

8. MEAN calculations, cont’d
Height
(in inches)
Frequency
Midpt (x)
(x • f) 4 61
244
63 – 65 5 64
320
66 – 48 8 67
536
69 – 71 1 70
______ 18
___________
1170

9. More bars on the left of the peak More bars on the right of the peak
MEAN calculations, cont’d:
3) Find the mean of the histograms:
More bars on the left of the peak
More bars on the right of the peak
Skewed Right
Skewed Left
Symmetric
Since a histogram is just a graphical representation of a distribution, you use the same process as finding the mean of the distribution…. but…..

10. MEAN calculations cont’d:
…there are some generalities based on shape
Symmetric
Uniform
Mean (as well as median) will be at the center.

11. MEAN calculations, cont’d
Skewed histogram means….
Skewed Left
Skewed Right
Mean
Mode
Mode
Mean
Median
Median
**Mean always fall in the direction the distribution is skewed**

12. YOU TRY…. Find the mean of the histogram below:

13. Sample Weighted Mean Pop. Weighted Mean
The mean of a data set whose entries have varying weights.
Sample Weighted Mean Pop. Weighted Mean
w = weight of each entry
--weights may not sum to 100%
--if weights sum to 100%, then ∑w = 1

14. Weighted Mean, cont’d
Ex 1) A class is graded based on weighted mean as follows:
Homework: 20%
Quizzes: 35%
Tests: 45%
Let’s say scores are 95 on homework, 82 on quizzes and 79 on tests. What is the weighted mean?
What if there were no test scores yet?

15. Weighted Mean, cont’d 70.9 + .2x = 90 .2x = 19.1 X = 95.5
Ex 2) Rachel is taking a class in which her grade is determined as follows:
50% from her test mean
15% from her midterm
20% from her final
15% from her homework
Her scores are 86 (tests), 96 (midterm), and 100 (homework). What does Rachel need to get on her final exam to receive a 90% in class?
x = 90
.2x = 19.1
X = 95.5
Rachel needs to get a 95.5% on the final to earn a 90% in the class.

16. Weighted Mean, cont’d
Ex 3) For the month of April, a checking account has a balance of $523 for 24 days, $2415 for 2 days and $250 for 4 days. What is the account’s mean daily balance for April?

17. For large datasets: MEDIANS…. There’s an easier way to do it!
Arrange in order
The location of the median is found by counting
data items up from the bottom of the set.
Ex) Ages at concert:
24, 27, 19, 21, 18, 23, 21, 20, 19, 33, 30, 29, 21
18, 24, 26, 38, 19, 35, 34, 33, 30, 21, 27, 30
18, 18, 19, 19, 19, 20, 21, 21, 21, 21, 23, 24, 24, 26, 27, 27, 29, 30, 30, 30, 33, 33, 34, 35, 38

18. MEDIAN calculations, cont’d:
Works for stem-and-leaf plots too:

19. MEDIAN calculations, cont’d:
Works for histograms too: